id sid tid token lemma pos american_scientific_journal-707 1 1 american american PROPN american_scientific_journal-707 1 2 scientific scientific ADJ american_scientific_journal-707 1 3 research research NOUN american_scientific_journal-707 1 4 journal journal NOUN american_scientific_journal-707 1 5 for for ADP american_scientific_journal-707 1 6 engineering engineering NOUN american_scientific_journal-707 1 7 , , PUNCT american_scientific_journal-707 1 8 technology technology NOUN american_scientific_journal-707 1 9 , , PUNCT american_scientific_journal-707 1 10 and and CCONJ american_scientific_journal-707 1 11 sciences science NOUN american_scientific_journal-707 1 12 ( ( PUNCT american_scientific_journal-707 1 13 asrjets asrjet NOUN american_scientific_journal-707 1 14 ) ) PUNCT american_scientific_journal-707 1 15 issn issn PROPN american_scientific_journal-707 1 16 ( ( PUNCT american_scientific_journal-707 1 17 print print NOUN american_scientific_journal-707 1 18 ) ) PUNCT american_scientific_journal-707 1 19 2313 2313 NUM american_scientific_journal-707 1 20 - - SYM american_scientific_journal-707 1 21 4410 4410 NUM american_scientific_journal-707 1 22 , , PUNCT american_scientific_journal-707 1 23 issn issn PROPN american_scientific_journal-707 1 24 ( ( PUNCT american_scientific_journal-707 1 25 online online ADJ american_scientific_journal-707 1 26 ) ) PUNCT american_scientific_journal-707 1 27 2313 2313 NUM american_scientific_journal-707 1 28 - - SYM american_scientific_journal-707 1 29 4402 4402 NUM american_scientific_journal-707 1 30 Β© Β© PROPN american_scientific_journal-707 1 31 global global ADJ american_scientific_journal-707 1 32 society society NOUN american_scientific_journal-707 1 33 of of ADP american_scientific_journal-707 1 34 scientific scientific ADJ american_scientific_journal-707 1 35 research research NOUN american_scientific_journal-707 1 36 and and CCONJ american_scientific_journal-707 1 37 researchers researcher NOUN american_scientific_journal-707 1 38 http://asrjetsjournal.org/ http://asrjetsjournal.org/ PRON american_scientific_journal-707 1 39 updating update VERB american_scientific_journal-707 1 40 the the DET american_scientific_journal-707 1 41 fundamental fundamental ADJ american_scientific_journal-707 1 42 theorem theorem NOUN american_scientific_journal-707 1 43 of of ADP american_scientific_journal-707 1 44 homomorphism homomorphism PROPN american_scientific_journal-707 1 45 of of ADP american_scientific_journal-707 1 46 general general ADJ american_scientific_journal-707 1 47 universal universal ADJ american_scientific_journal-707 1 48 algebras algebras PROPN american_scientific_journal-707 1 49 gezahagne gezahagne PROPN american_scientific_journal-707 1 50 mulat mulat PROPN american_scientific_journal-707 1 51 addis addis PROPN american_scientific_journal-707 1 52 * * PROPN american_scientific_journal-707 1 53 , , PUNCT american_scientific_journal-707 1 54 department department NOUN american_scientific_journal-707 1 55 of of ADP american_scientific_journal-707 1 56 mathematics mathematic NOUN american_scientific_journal-707 1 57 , , PUNCT american_scientific_journal-707 1 58 dilla dilla PROPN american_scientific_journal-707 1 59 university university PROPN american_scientific_journal-707 1 60 , , PUNCT american_scientific_journal-707 1 61 dilla dilla PROPN american_scientific_journal-707 1 62 , , PUNCT american_scientific_journal-707 1 63 ethiopia ethiopia PROPN american_scientific_journal-707 1 64 email email NOUN american_scientific_journal-707 1 65 : : PUNCT american_scientific_journal-707 1 66 buttu412@yahoo.com buttu412@yahoo.com X american_scientific_journal-707 1 67 abstract abstract PROPN american_scientific_journal-707 1 68 from from ADP american_scientific_journal-707 1 69 the the DET american_scientific_journal-707 1 70 fundamental fundamental ADJ american_scientific_journal-707 1 71 theorem theorem NOUN american_scientific_journal-707 1 72 of of ADP american_scientific_journal-707 1 73 homomorphisms homomorphism NOUN american_scientific_journal-707 1 74 , , PUNCT american_scientific_journal-707 1 75 it it PRON american_scientific_journal-707 1 76 is be AUX american_scientific_journal-707 1 77 well well ADV american_scientific_journal-707 1 78 known know VERB american_scientific_journal-707 1 79 that that SCONJ american_scientific_journal-707 1 80 any any DET american_scientific_journal-707 1 81 homomorphism homomorphism NOUN american_scientific_journal-707 1 82 of of ADP american_scientific_journal-707 1 83 groups group NOUN american_scientific_journal-707 1 84 ( ( PUNCT american_scientific_journal-707 1 85 or or CCONJ american_scientific_journal-707 1 86 rings ring NOUN american_scientific_journal-707 1 87 , , PUNCT american_scientific_journal-707 1 88 or or CCONJ american_scientific_journal-707 1 89 modules module NOUN american_scientific_journal-707 1 90 , , PUNCT american_scientific_journal-707 1 91 or or CCONJ american_scientific_journal-707 1 92 vector vector NOUN american_scientific_journal-707 1 93 spaces space NOUN american_scientific_journal-707 1 94 and and CCONJ american_scientific_journal-707 1 95 of of ADP american_scientific_journal-707 1 96 general general ADJ american_scientific_journal-707 1 97 universal universal ADJ american_scientific_journal-707 1 98 algebras algebra NOUN american_scientific_journal-707 1 99 ) ) PUNCT american_scientific_journal-707 1 100 can can AUX american_scientific_journal-707 1 101 be be AUX american_scientific_journal-707 1 102 decomposed decompose VERB american_scientific_journal-707 1 103 as as ADP american_scientific_journal-707 1 104 a a DET american_scientific_journal-707 1 105 composition composition NOUN american_scientific_journal-707 1 106 of of ADP american_scientific_journal-707 1 107 a a DET american_scientific_journal-707 1 108 monomorphism monomorphism NOUN american_scientific_journal-707 1 109 and and CCONJ american_scientific_journal-707 1 110 an an DET american_scientific_journal-707 1 111 epimorphism epimorphism NOUN american_scientific_journal-707 1 112 . . PUNCT american_scientific_journal-707 2 1 this this DET american_scientific_journal-707 2 2 paper paper NOUN american_scientific_journal-707 2 3 provides provide VERB american_scientific_journal-707 2 4 the the DET american_scientific_journal-707 2 5 uniqueness uniqueness NOUN american_scientific_journal-707 2 6 of of ADP american_scientific_journal-707 2 7 such such ADJ american_scientific_journal-707 2 8 decomposition decomposition NOUN american_scientific_journal-707 2 9 up up ADP american_scientific_journal-707 2 10 to to ADP american_scientific_journal-707 2 11 the the DET american_scientific_journal-707 2 12 level level NOUN american_scientific_journal-707 2 13 of of ADP american_scientific_journal-707 2 14 associates associate NOUN american_scientific_journal-707 2 15 in in ADP american_scientific_journal-707 2 16 the the DET american_scientific_journal-707 2 17 case case NOUN american_scientific_journal-707 2 18 of of ADP american_scientific_journal-707 2 19 general general ADJ american_scientific_journal-707 2 20 universal universal ADJ american_scientific_journal-707 2 21 algebras algebras PROPN american_scientific_journal-707 2 22 keywords keyword NOUN american_scientific_journal-707 2 23 : : PUNCT american_scientific_journal-707 2 24 algebra algebra NOUN american_scientific_journal-707 2 25 of of ADP american_scientific_journal-707 2 26 a a DET american_scientific_journal-707 2 27 given give VERB american_scientific_journal-707 2 28 type type NOUN american_scientific_journal-707 2 29 ; ; PUNCT american_scientific_journal-707 2 30 homomorphisms homomorphism NOUN american_scientific_journal-707 2 31 ; ; PUNCT american_scientific_journal-707 2 32 kernel kernel NOUN american_scientific_journal-707 2 33 of of ADP american_scientific_journal-707 2 34 a a DET american_scientific_journal-707 2 35 homomorphism homomorphism NOUN american_scientific_journal-707 2 36 ; ; PUNCT american_scientific_journal-707 2 37 congruence congruence NOUN american_scientific_journal-707 2 38 relations relation NOUN american_scientific_journal-707 2 39 and and CCONJ american_scientific_journal-707 2 40 an an DET american_scientific_journal-707 2 41 associate associate NOUN american_scientific_journal-707 2 42 of of ADP american_scientific_journal-707 2 43 a a DET american_scientific_journal-707 2 44 homomorphism homomorphism NOUN american_scientific_journal-707 2 45 . . PUNCT american_scientific_journal-707 3 1 1 1 X american_scientific_journal-707 3 2 . . X american_scientific_journal-707 3 3 introduction introduction NOUN american_scientific_journal-707 3 4 it it PRON american_scientific_journal-707 3 5 is be AUX american_scientific_journal-707 3 6 well well ADV american_scientific_journal-707 3 7 known know VERB american_scientific_journal-707 3 8 that that SCONJ american_scientific_journal-707 3 9 , , PUNCT american_scientific_journal-707 3 10 if if SCONJ american_scientific_journal-707 3 11 𝑓:𝐺 𝑓:𝐺 PROPN american_scientific_journal-707 3 12 β†’ β†’ SYM american_scientific_journal-707 3 13 𝐺 𝐺 PROPN american_scientific_journal-707 3 14 β€² β€² NOUN american_scientific_journal-707 3 15 is be AUX american_scientific_journal-707 3 16 a a DET american_scientific_journal-707 3 17 homomorphism homomorphism NOUN american_scientific_journal-707 3 18 of of ADP american_scientific_journal-707 3 19 groups group NOUN american_scientific_journal-707 3 20 ( ( PUNCT american_scientific_journal-707 3 21 or or CCONJ american_scientific_journal-707 3 22 rings ring NOUN american_scientific_journal-707 3 23 or or CCONJ american_scientific_journal-707 3 24 modules module NOUN american_scientific_journal-707 3 25 ) ) PUNCT american_scientific_journal-707 3 26 the the DET american_scientific_journal-707 3 27 quotient quotient NOUN american_scientific_journal-707 3 28 group group NOUN american_scientific_journal-707 3 29 𝐺 𝐺 PROPN american_scientific_journal-707 3 30 ker𝑓 ker𝑓 PROPN american_scientific_journal-707 3 31 οΏ½ οΏ½ PROPN american_scientific_journal-707 3 32 is be AUX american_scientific_journal-707 3 33 isomorphic isomorphic ADJ american_scientific_journal-707 3 34 to to ADP american_scientific_journal-707 3 35 the the DET american_scientific_journal-707 3 36 image image NOUN american_scientific_journal-707 3 37 of of ADP american_scientific_journal-707 3 38 𝑓 𝑓 PRON american_scientific_journal-707 3 39 which which PRON american_scientific_journal-707 3 40 is be AUX american_scientific_journal-707 3 41 a a DET american_scientific_journal-707 3 42 subgroup subgroup NOUN american_scientific_journal-707 3 43 of of ADP american_scientific_journal-707 3 44 the the DET american_scientific_journal-707 3 45 codomain codomain ADJ american_scientific_journal-707 3 46 group group NOUN american_scientific_journal-707 3 47 𝐺 𝐺 PROPN american_scientific_journal-707 3 48 β€². β€². NOUN american_scientific_journal-707 3 49 this this DET american_scientific_journal-707 3 50 isomorphism isomorphism NOUN american_scientific_journal-707 3 51 𝑔 𝑔 PROPN american_scientific_journal-707 3 52 is be AUX american_scientific_journal-707 3 53 simply simply ADV american_scientific_journal-707 3 54 induced induce VERB american_scientific_journal-707 3 55 by by ADP american_scientific_journal-707 3 56 𝑓 𝑓 PRON american_scientific_journal-707 3 57 , , PUNCT american_scientific_journal-707 3 58 in in ADP american_scientific_journal-707 3 59 the the DET american_scientific_journal-707 3 60 sense sense NOUN american_scientific_journal-707 3 61 that that SCONJ american_scientific_journal-707 3 62 , , PUNCT american_scientific_journal-707 3 63 𝑔 𝑔 PROPN american_scientific_journal-707 3 64 can can AUX american_scientific_journal-707 3 65 be be AUX american_scientific_journal-707 3 66 defined define VERB american_scientific_journal-707 3 67 by by ADP american_scientific_journal-707 3 68 𝑔(π‘Ž 𝑔(π‘Ž NOUN american_scientific_journal-707 3 69 + + CCONJ american_scientific_journal-707 3 70 ker ker PROPN american_scientific_journal-707 3 71 𝑓 𝑓 X american_scientific_journal-707 3 72 ) ) PUNCT american_scientific_journal-707 3 73 = = SYM american_scientific_journal-707 3 74 𝑓(π‘Ž 𝑓(π‘Ž NOUN american_scientific_journal-707 3 75 ) ) PUNCT american_scientific_journal-707 3 76 for for ADP american_scientific_journal-707 3 77 any any DET american_scientific_journal-707 3 78 coset coset NOUN american_scientific_journal-707 4 1 π‘Ž π‘Ž X american_scientific_journal-707 4 2 + + NUM american_scientific_journal-707 4 3 ker ker NOUN american_scientific_journal-707 4 4 𝑓 𝑓 PRON american_scientific_journal-707 4 5 , , PUNCT american_scientific_journal-707 4 6 π‘Ž π‘Ž PROPN american_scientific_journal-707 4 7 ∈ ∈ PROPN american_scientific_journal-707 4 8 𝐺 𝐺 NOUN american_scientific_journal-707 4 9 also also ADV american_scientific_journal-707 4 10 , , PUNCT american_scientific_journal-707 4 11 we we PRON american_scientific_journal-707 4 12 have have VERB american_scientific_journal-707 4 13 the the DET american_scientific_journal-707 4 14 natural natural ADJ american_scientific_journal-707 4 15 epimorphism epimorphism NOUN american_scientific_journal-707 4 16 β„Ž:𝐺 β„Ž:𝐺 PROPN american_scientific_journal-707 4 17 β†’ β†’ SYM american_scientific_journal-707 4 18 𝐺 𝐺 PROPN american_scientific_journal-707 4 19 ker ker NOUN american_scientific_journal-707 4 20 𝑓 𝑓 DET american_scientific_journal-707 4 21 οΏ½ οΏ½ PROPN american_scientific_journal-707 4 22 , , PUNCT american_scientific_journal-707 4 23 defined define VERB american_scientific_journal-707 4 24 by by ADP american_scientific_journal-707 4 25 β„Ž(π‘Ž β„Ž(π‘Ž PROPN american_scientific_journal-707 4 26 ) ) PUNCT american_scientific_journal-707 4 27 = = PUNCT american_scientific_journal-707 5 1 π‘Ž π‘Ž X american_scientific_journal-707 6 1 + + X american_scientific_journal-707 6 2 ker𝑓 ker𝑓 NOUN american_scientific_journal-707 6 3 for for ADP american_scientific_journal-707 6 4 all all DET american_scientific_journal-707 6 5 π‘Ž π‘Ž DET american_scientific_journal-707 6 6 ∈ ∈ NOUN american_scientific_journal-707 6 7 𝐺. 𝐺. NOUN american_scientific_journal-707 6 8 in in ADP american_scientific_journal-707 6 9 other other ADJ american_scientific_journal-707 6 10 words word NOUN american_scientific_journal-707 6 11 , , PUNCT american_scientific_journal-707 6 12 we we PRON american_scientific_journal-707 6 13 have have VERB american_scientific_journal-707 6 14 a a DET american_scientific_journal-707 6 15 decomposition𝑓 decomposition𝑓 NOUN american_scientific_journal-707 6 16 = = SYM american_scientific_journal-707 6 17 𝑔 𝑔 PROPN american_scientific_journal-707 6 18 ∘ ∘ PROPN american_scientific_journal-707 6 19 β„Ž;where β„Ž;where PROPN american_scientific_journal-707 6 20 g g PROPN american_scientific_journal-707 6 21 is be AUX american_scientific_journal-707 6 22 a a DET american_scientific_journal-707 6 23 monomorphism monomorphism NOUN american_scientific_journal-707 6 24 and and CCONJ american_scientific_journal-707 6 25 h h NOUN american_scientific_journal-707 6 26 is be AUX american_scientific_journal-707 6 27 an an DET american_scientific_journal-707 6 28 epimorphism epimorphism NOUN american_scientific_journal-707 6 29 . . PUNCT american_scientific_journal-707 7 1 this this PRON american_scientific_journal-707 7 2 is be AUX american_scientific_journal-707 7 3 known know VERB american_scientific_journal-707 7 4 as as ADP american_scientific_journal-707 7 5 the the DET american_scientific_journal-707 7 6 fundamental fundamental ADJ american_scientific_journal-707 7 7 theorem theorem NOUN american_scientific_journal-707 7 8 of of ADP american_scientific_journal-707 7 9 homomorphisms homomorphism NOUN american_scientific_journal-707 7 10 [ [ X american_scientific_journal-707 7 11 3 3 NUM american_scientific_journal-707 7 12 , , PUNCT american_scientific_journal-707 7 13 11].this 11].this PROPN american_scientific_journal-707 7 14 result result NOUN american_scientific_journal-707 7 15 can can AUX american_scientific_journal-707 7 16 be be AUX american_scientific_journal-707 7 17 extended extend VERB american_scientific_journal-707 7 18 to to ADP american_scientific_journal-707 7 19 any any DET american_scientific_journal-707 7 20 homomorphism homomorphism PROPN american_scientific_journal-707 7 21 𝑓:𝐴 𝑓:𝐴 NUM american_scientific_journal-707 7 22 β†’ β†’ SYM american_scientific_journal-707 7 23 𝐴′ 𝐴′ PROPN american_scientific_journal-707 7 24 of of ADP american_scientific_journal-707 7 25 universal universal ADJ american_scientific_journal-707 7 26 algebras algebra NOUN american_scientific_journal-707 7 27 of of ADP american_scientific_journal-707 7 28 the the DET american_scientific_journal-707 7 29 same same ADJ american_scientific_journal-707 7 30 type type NOUN american_scientific_journal-707 7 31 by by ADP american_scientific_journal-707 7 32 considering consider VERB american_scientific_journal-707 7 33 the the DET american_scientific_journal-707 7 34 binary binary PROPN american_scientific_journal-707 7 35 relation relation PROPN american_scientific_journal-707 7 36 πœƒ πœƒ NOUN american_scientific_journal-707 7 37 on on ADP american_scientific_journal-707 7 38 the the DET american_scientific_journal-707 7 39 domain domain NOUN american_scientific_journal-707 7 40 algebra algebra NOUN american_scientific_journal-707 7 41 a a PRON american_scientific_journal-707 7 42 defined define VERB american_scientific_journal-707 7 43 by by ADP american_scientific_journal-707 7 44 , , PUNCT american_scientific_journal-707 7 45 πœƒ πœƒ X american_scientific_journal-707 7 46 = = SYM american_scientific_journal-707 7 47 { { PUNCT american_scientific_journal-707 7 48 ( ( PUNCT american_scientific_journal-707 7 49 π‘Ž π‘Ž NOUN american_scientific_journal-707 7 50 , , PUNCT american_scientific_journal-707 7 51 𝑏 𝑏 NOUN american_scientific_journal-707 7 52 ) ) PUNCT american_scientific_journal-707 7 53 ∈ ∈ PROPN american_scientific_journal-707 7 54 𝐴 𝐴 PROPN american_scientific_journal-707 7 55 Γ— Γ— PROPN american_scientific_journal-707 7 56 𝐴 𝐴 PROPN american_scientific_journal-707 7 57 ∢ ∢ NOUN american_scientific_journal-707 7 58 𝑓(π‘Ž 𝑓(π‘Ž ADJ american_scientific_journal-707 7 59 ) ) PUNCT american_scientific_journal-707 7 60 = = SYM american_scientific_journal-707 7 61 𝑓(𝑏 𝑓(𝑏 NOUN american_scientific_journal-707 7 62 ) ) PUNCT american_scientific_journal-707 7 63 } } PUNCT american_scientific_journal-707 8 1 ----------------------------------------------------------------------- ----------------------------------------------------------------------- PUNCT american_scientific_journal-707 8 2 * * PUNCT american_scientific_journal-707 8 3 corresponding correspond VERB american_scientific_journal-707 8 4 author author NOUN american_scientific_journal-707 8 5 . . PUNCT american_scientific_journal-707 9 1 e e X american_scientific_journal-707 9 2 - - NOUN american_scientific_journal-707 9 3 mail mail NOUN american_scientific_journal-707 9 4 address address NOUN american_scientific_journal-707 9 5 : : PUNCT american_scientific_journal-707 9 6 buttu412@yahoo.com buttu412@yahoo.com X american_scientific_journal-707 9 7 .. .. PUNCT american_scientific_journal-707 9 8 61 61 NUM american_scientific_journal-707 9 9 http://asrjetsjournal.org/ http://asrjetsjournal.org/ NOUN american_scientific_journal-707 9 10 mailto:buttu412@yahoo.com mailto:buttu412@yahoo.com PROPN american_scientific_journal-707 9 11 american american PROPN american_scientific_journal-707 9 12 scientific scientific PROPN american_scientific_journal-707 9 13 research research PROPN american_scientific_journal-707 9 14 journal journal NOUN american_scientific_journal-707 9 15 for for ADP american_scientific_journal-707 9 16 engineering engineering NOUN american_scientific_journal-707 9 17 , , PUNCT american_scientific_journal-707 9 18 technology technology NOUN american_scientific_journal-707 9 19 , , PUNCT american_scientific_journal-707 9 20 and and CCONJ american_scientific_journal-707 9 21 sciences science NOUN american_scientific_journal-707 9 22 ( ( PUNCT american_scientific_journal-707 9 23 asrjets asrjet NOUN american_scientific_journal-707 9 24 ) ) PUNCT american_scientific_journal-707 9 25 ( ( PUNCT american_scientific_journal-707 9 26 2014 2014 NUM american_scientific_journal-707 9 27 ) ) PUNCT american_scientific_journal-707 9 28 volume volume NOUN american_scientific_journal-707 9 29 10 10 NUM american_scientific_journal-707 9 30 , , PUNCT american_scientific_journal-707 9 31 no no DET american_scientific_journal-707 9 32 1 1 NUM american_scientific_journal-707 9 33 , , PUNCT american_scientific_journal-707 9 34 pp pp ADV american_scientific_journal-707 9 35 61 61 NUM american_scientific_journal-707 9 36 - - SYM american_scientific_journal-707 9 37 73 73 NUM american_scientific_journal-707 9 38 figure1 figure1 NUM american_scientific_journal-707 9 39 : : PUNCT american_scientific_journal-707 9 40 decomposition decomposition NOUN american_scientific_journal-707 9 41 of of ADP american_scientific_journal-707 9 42 homomorphism homomorphism NOUN american_scientific_journal-707 9 43 of of ADP american_scientific_journal-707 9 44 groups group NOUN american_scientific_journal-707 9 45 and and CCONJ american_scientific_journal-707 9 46 the the DET american_scientific_journal-707 9 47 quotient quotient NOUN american_scientific_journal-707 9 48 algebra algebra PROPN american_scientific_journal-707 9 49 𝐴 𝐴 PROPN american_scientific_journal-707 9 50 πœƒ πœƒ PROPN american_scientific_journal-707 9 51 οΏ½ οΏ½ PROPN american_scientific_journal-707 9 52 . . PUNCT american_scientific_journal-707 10 1 in in ADP american_scientific_journal-707 10 2 the the DET american_scientific_journal-707 10 3 general general ADJ american_scientific_journal-707 10 4 case case NOUN american_scientific_journal-707 10 5 of of ADP american_scientific_journal-707 10 6 universal universal ADJ american_scientific_journal-707 10 7 algebras algebra NOUN american_scientific_journal-707 10 8 , , PUNCT american_scientific_journal-707 10 9 this this DET american_scientific_journal-707 10 10 πœƒ πœƒ NOUN american_scientific_journal-707 10 11 is be AUX american_scientific_journal-707 10 12 defined define VERB american_scientific_journal-707 10 13 as as ADP american_scientific_journal-707 10 14 the the DET american_scientific_journal-707 10 15 kernel kernel NOUN american_scientific_journal-707 10 16 of of ADP american_scientific_journal-707 10 17 𝑓 𝑓 PRON american_scientific_journal-707 10 18 and and CCONJ american_scientific_journal-707 10 19 is be AUX american_scientific_journal-707 10 20 actually actually ADV american_scientific_journal-707 10 21 a a DET american_scientific_journal-707 10 22 congruence congruence NOUN american_scientific_journal-707 10 23 relation relation NOUN american_scientific_journal-707 10 24 on on ADP american_scientific_journal-707 10 25 the the DET american_scientific_journal-707 10 26 domain domain NOUN american_scientific_journal-707 10 27 algebra algebra NOUN american_scientific_journal-707 11 1 a a PRON american_scientific_journal-707 11 2 and and CCONJ american_scientific_journal-707 11 3 𝐴 𝐴 PROPN american_scientific_journal-707 11 4 πœƒ πœƒ PRON american_scientific_journal-707 11 5 οΏ½ οΏ½ PROPN american_scientific_journal-707 11 6 is be AUX american_scientific_journal-707 11 7 the the DET american_scientific_journal-707 11 8 set set NOUN american_scientific_journal-707 11 9 of of ADP american_scientific_journal-707 11 10 all all DET american_scientific_journal-707 11 11 congruence congruence NOUN american_scientific_journal-707 11 12 classes class NOUN american_scientific_journal-707 11 13 of of ADP american_scientific_journal-707 11 14 elements element NOUN american_scientific_journal-707 11 15 of of ADP american_scientific_journal-707 11 16 𝐴 𝐴 PROPN american_scientific_journal-707 11 17 corresponding correspond VERB american_scientific_journal-707 11 18 to to ADP american_scientific_journal-707 11 19 πœƒ πœƒ PROPN american_scientific_journal-707 11 20 [ [ X american_scientific_journal-707 11 21 8 8 NUM american_scientific_journal-707 11 22 ] ] PUNCT american_scientific_journal-707 11 23 . . PUNCT american_scientific_journal-707 12 1 in in ADP american_scientific_journal-707 12 2 the the DET american_scientific_journal-707 12 3 familiar familiar ADJ american_scientific_journal-707 12 4 cases case NOUN american_scientific_journal-707 12 5 of of ADP american_scientific_journal-707 12 6 homomorphisms homomorphism NOUN american_scientific_journal-707 12 7 of of ADP american_scientific_journal-707 12 8 groups group NOUN american_scientific_journal-707 12 9 ( ( PUNCT american_scientific_journal-707 12 10 or or CCONJ american_scientific_journal-707 12 11 rings ring NOUN american_scientific_journal-707 12 12 or or CCONJ american_scientific_journal-707 12 13 modules module NOUN american_scientific_journal-707 12 14 ) ) PUNCT american_scientific_journal-707 12 15 the the DET american_scientific_journal-707 12 16 kernel kernel NOUN american_scientific_journal-707 12 17 of of ADP american_scientific_journal-707 12 18 f f PROPN american_scientific_journal-707 12 19 is be AUX american_scientific_journal-707 12 20 a a DET american_scientific_journal-707 12 21 normal normal ADJ american_scientific_journal-707 12 22 subgroup subgroup NOUN american_scientific_journal-707 12 23 ( ( PUNCT american_scientific_journal-707 12 24 or or CCONJ american_scientific_journal-707 12 25 an an DET american_scientific_journal-707 12 26 ideal ideal NOUN american_scientific_journal-707 12 27 or or CCONJ american_scientific_journal-707 12 28 a a DET american_scientific_journal-707 12 29 submodule submodule NOUN american_scientific_journal-707 12 30 respectively respectively ADV american_scientific_journal-707 12 31 ) ) PUNCT american_scientific_journal-707 12 32 of of ADP american_scientific_journal-707 12 33 the the DET american_scientific_journal-707 12 34 domain domain NOUN american_scientific_journal-707 12 35 and and CCONJ american_scientific_journal-707 12 36 the the DET american_scientific_journal-707 12 37 congruence congruence PROPN american_scientific_journal-707 12 38 classes class NOUN american_scientific_journal-707 12 39 are be AUX american_scientific_journal-707 12 40 precisely precisely ADV american_scientific_journal-707 12 41 the the DET american_scientific_journal-707 12 42 cosets coset NOUN american_scientific_journal-707 12 43 of of ADP american_scientific_journal-707 12 44 the the DET american_scientific_journal-707 12 45 conventional conventional ADJ american_scientific_journal-707 12 46 kernels kernel NOUN american_scientific_journal-707 12 47 [ [ X american_scientific_journal-707 12 48 5 5 NUM american_scientific_journal-707 12 49 , , PUNCT american_scientific_journal-707 12 50 9 9 NUM american_scientific_journal-707 12 51 , , PUNCT american_scientific_journal-707 12 52 10 10 NUM american_scientific_journal-707 12 53 ] ] PUNCT american_scientific_journal-707 12 54 . . PUNCT american_scientific_journal-707 13 1 in in ADP american_scientific_journal-707 13 2 all all DET american_scientific_journal-707 13 3 these these DET american_scientific_journal-707 13 4 cases case NOUN american_scientific_journal-707 13 5 there there PRON american_scientific_journal-707 13 6 is be VERB american_scientific_journal-707 13 7 an an DET american_scientific_journal-707 13 8 order order NOUN american_scientific_journal-707 13 9 isomorphism isomorphism NOUN american_scientific_journal-707 13 10 of of ADP american_scientific_journal-707 13 11 the the DET american_scientific_journal-707 13 12 set set NOUN american_scientific_journal-707 13 13 of of ADP american_scientific_journal-707 13 14 normal normal ADJ american_scientific_journal-707 13 15 subgroups(or subgroups(or ADJ american_scientific_journal-707 13 16 ideals ideal NOUN american_scientific_journal-707 13 17 or or CCONJ american_scientific_journal-707 13 18 submodules submodule NOUN american_scientific_journal-707 13 19 ) ) PUNCT american_scientific_journal-707 13 20 onto onto ADP american_scientific_journal-707 13 21 the the DET american_scientific_journal-707 13 22 set set NOUN american_scientific_journal-707 13 23 of of ADP american_scientific_journal-707 13 24 congruence congruence PROPN american_scientific_journal-707 13 25 relations relation NOUN american_scientific_journal-707 13 26 on on ADP american_scientific_journal-707 13 27 the the DET american_scientific_journal-707 13 28 domain domain NOUN american_scientific_journal-707 13 29 algebra algebra NOUN american_scientific_journal-707 13 30 . . PUNCT american_scientific_journal-707 14 1 in in ADP american_scientific_journal-707 14 2 this this DET american_scientific_journal-707 14 3 scenario scenario NOUN american_scientific_journal-707 14 4 , , PUNCT american_scientific_journal-707 14 5 we we PRON american_scientific_journal-707 14 6 can can AUX american_scientific_journal-707 14 7 say say VERB american_scientific_journal-707 14 8 that that SCONJ american_scientific_journal-707 14 9 any any DET american_scientific_journal-707 14 10 homomorphism homomorphism PROPN american_scientific_journal-707 14 11 f f PROPN american_scientific_journal-707 14 12 of of ADP american_scientific_journal-707 14 13 algebras algebra NOUN american_scientific_journal-707 14 14 of of ADP american_scientific_journal-707 14 15 any any DET american_scientific_journal-707 14 16 type type NOUN american_scientific_journal-707 14 17 can can AUX american_scientific_journal-707 14 18 be be AUX american_scientific_journal-707 14 19 decomposed decompose VERB american_scientific_journal-707 14 20 as as ADP american_scientific_journal-707 14 21 a a DET american_scientific_journal-707 14 22 composition composition NOUN american_scientific_journal-707 14 23 of of ADP american_scientific_journal-707 14 24 a a DET american_scientific_journal-707 14 25 monomorphism monomorphism NOUN american_scientific_journal-707 14 26 and and CCONJ american_scientific_journal-707 14 27 an an DET american_scientific_journal-707 14 28 epimorphism epimorphism NOUN american_scientific_journal-707 14 29 . . PUNCT american_scientific_journal-707 15 1 it it PRON american_scientific_journal-707 15 2 is be AUX american_scientific_journal-707 15 3 natural natural ADJ american_scientific_journal-707 15 4 to to PART american_scientific_journal-707 15 5 question question VERB american_scientific_journal-707 15 6 ourselves ourselves PRON american_scientific_journal-707 15 7 that that SCONJ american_scientific_journal-707 15 8 , , PUNCT american_scientific_journal-707 15 9 is be AUX american_scientific_journal-707 15 10 this this DET american_scientific_journal-707 15 11 decomposition decomposition NOUN american_scientific_journal-707 15 12 done do VERB american_scientific_journal-707 15 13 uniquely uniquely ADV american_scientific_journal-707 15 14 in in ADP american_scientific_journal-707 15 15 one one NUM american_scientific_journal-707 15 16 way way NOUN american_scientific_journal-707 15 17 ? ? PUNCT american_scientific_journal-707 16 1 of of ADV american_scientific_journal-707 16 2 course course ADV american_scientific_journal-707 16 3 not not PART american_scientific_journal-707 16 4 in in ADP american_scientific_journal-707 16 5 one one NUM american_scientific_journal-707 16 6 and and CCONJ american_scientific_journal-707 16 7 only only ADV american_scientific_journal-707 16 8 one one NUM american_scientific_journal-707 16 9 way way NOUN american_scientific_journal-707 16 10 and and CCONJ american_scientific_journal-707 16 11 this this DET american_scientific_journal-707 16 12 paper paper NOUN american_scientific_journal-707 16 13 provides provide VERB american_scientific_journal-707 16 14 the the DET american_scientific_journal-707 16 15 uniqueness uniqueness NOUN american_scientific_journal-707 16 16 of of ADP american_scientific_journal-707 16 17 such such DET american_scientific_journal-707 16 18 a a DET american_scientific_journal-707 16 19 decomposition decomposition NOUN american_scientific_journal-707 16 20 of of ADP american_scientific_journal-707 16 21 homomorphismsupto homomorphismsupto NOUN american_scientific_journal-707 16 22 the the DET american_scientific_journal-707 16 23 level level NOUN american_scientific_journal-707 16 24 of of ADP american_scientific_journal-707 16 25 associate associate NOUN american_scientific_journal-707 16 26 in in ADP american_scientific_journal-707 16 27 the the DET american_scientific_journal-707 16 28 case case NOUN american_scientific_journal-707 16 29 of of ADP american_scientific_journal-707 16 30 homomorphisms homomorphism NOUN american_scientific_journal-707 16 31 of of ADP american_scientific_journal-707 16 32 general general ADJ american_scientific_journal-707 16 33 universal universal ADJ american_scientific_journal-707 16 34 algebras algebra NOUN american_scientific_journal-707 16 35 . . PUNCT american_scientific_journal-707 17 1 2 2 X american_scientific_journal-707 17 2 . . NOUN american_scientific_journal-707 17 3 method method NOUN american_scientific_journal-707 17 4 / / SYM american_scientific_journal-707 17 5 approach approach NOUN american_scientific_journal-707 17 6 let let VERB american_scientific_journal-707 17 7 a a PRON american_scientific_journal-707 17 8 and and CCONJ american_scientific_journal-707 17 9 b b NOUN american_scientific_journal-707 17 10 be be AUX american_scientific_journal-707 17 11 algebras algebra NOUN american_scientific_journal-707 17 12 of of ADP american_scientific_journal-707 17 13 a a DET american_scientific_journal-707 17 14 given give VERB american_scientific_journal-707 17 15 type type NOUN american_scientific_journal-707 17 16 β„± β„± PROPN american_scientific_journal-707 17 17 and and CCONJ american_scientific_journal-707 17 18 let let VERB american_scientific_journal-707 17 19 𝑓:𝐴 𝑓:𝐴 NOUN american_scientific_journal-707 17 20 β†’ β†’ SYM american_scientific_journal-707 17 21 𝐡 𝐡 PROPN american_scientific_journal-707 17 22 , , PUNCT american_scientific_journal-707 17 23 be be AUX american_scientific_journal-707 17 24 any any DET american_scientific_journal-707 17 25 homomorphism homomorphism NOUN american_scientific_journal-707 17 26 . . PUNCT american_scientific_journal-707 18 1 let let VERB american_scientific_journal-707 18 2 us we PRON american_scientific_journal-707 18 3 consider consider VERB american_scientific_journal-707 18 4 the the DET american_scientific_journal-707 18 5 binary binary PROPN american_scientific_journal-707 18 6 relation relation NOUN american_scientific_journal-707 18 7 𝐾𝑓 𝐾𝑓 PROPN american_scientific_journal-707 18 8 defined define VERB american_scientific_journal-707 18 9 by by ADP american_scientific_journal-707 18 10 : : PUNCT american_scientific_journal-707 18 11 𝐾𝑓 𝐾𝑓 PROPN american_scientific_journal-707 18 12 = = PUNCT american_scientific_journal-707 18 13 { { PUNCT american_scientific_journal-707 18 14 ( ( PUNCT american_scientific_journal-707 18 15 π‘₯,𝑦 π‘₯,𝑦 NOUN american_scientific_journal-707 18 16 ) ) PUNCT american_scientific_journal-707 18 17 ∈ ∈ PROPN american_scientific_journal-707 18 18 𝐴 𝐴 PROPN american_scientific_journal-707 18 19 Γ— Γ— PROPN american_scientific_journal-707 18 20 𝐴 𝐴 PROPN american_scientific_journal-707 18 21 ∢ ∢ NOUN american_scientific_journal-707 18 22 𝑓(π‘₯ 𝑓(π‘₯ NOUN american_scientific_journal-707 18 23 ) ) PUNCT american_scientific_journal-707 18 24 = = SYM american_scientific_journal-707 18 25 𝑓(𝑦 𝑓(𝑦 PROPN american_scientific_journal-707 18 26 ) ) PUNCT american_scientific_journal-707 18 27 } } PUNCT american_scientific_journal-707 18 28 then then ADV american_scientific_journal-707 18 29 𝐾𝑓 𝐾𝑓 PROPN american_scientific_journal-707 18 30 becomes become VERB american_scientific_journal-707 18 31 a a DET american_scientific_journal-707 18 32 congruence congruence NOUN american_scientific_journal-707 18 33 relation relation NOUN american_scientific_journal-707 18 34 on on ADP american_scientific_journal-707 18 35 𝐴 𝐴 PROPN american_scientific_journal-707 18 36 [ [ X american_scientific_journal-707 18 37 6 6 NUM american_scientific_journal-707 18 38 , , PUNCT american_scientific_journal-707 18 39 11 11 NUM american_scientific_journal-707 18 40 ] ] PUNCT american_scientific_journal-707 18 41 . . PUNCT american_scientific_journal-707 19 1 also also ADV american_scientific_journal-707 19 2 , , PUNCT american_scientific_journal-707 19 3 the the DET american_scientific_journal-707 19 4 natural natural ADJ american_scientific_journal-707 19 5 map map NOUN american_scientific_journal-707 19 6 β„Ž:𝐴 β„Ž:𝐴 PROPN american_scientific_journal-707 19 7 β†’ β†’ SYM american_scientific_journal-707 19 8 𝐴 𝐴 PROPN american_scientific_journal-707 19 9 𝐾𝑓 𝐾𝑓 PROPN american_scientific_journal-707 19 10 οΏ½ οΏ½ PROPN american_scientific_journal-707 19 11 , , PUNCT american_scientific_journal-707 19 12 defined define VERB american_scientific_journal-707 19 13 by by ADP american_scientific_journal-707 19 14 β„Ž(π‘Ž β„Ž(π‘Ž PROPN american_scientific_journal-707 19 15 ) ) PUNCT american_scientific_journal-707 19 16 = = PUNCT american_scientific_journal-707 20 1 𝐾𝑓(π‘Ž 𝐾𝑓(π‘Ž X american_scientific_journal-707 20 2 ) ) PUNCT american_scientific_journal-707 20 3 , , PUNCT american_scientific_journal-707 20 4 the the DET american_scientific_journal-707 20 5 congruence congruence ADJ american_scientific_journal-707 20 6 class class NOUN american_scientific_journal-707 20 7 of of ADP american_scientific_journal-707 20 8 π‘Ž π‘Ž NOUN american_scientific_journal-707 20 9 corresponding corresponding NOUN american_scientific_journal-707 20 10 to to ADP american_scientific_journal-707 20 11 𝐾𝑓 𝐾𝑓 PROPN american_scientific_journal-707 20 12 , , PUNCT american_scientific_journal-707 20 13 is be AUX american_scientific_journal-707 20 14 an an DET american_scientific_journal-707 20 15 epimorphism epimorphism NOUN american_scientific_journal-707 20 16 and and CCONJ american_scientific_journal-707 20 17 the the DET american_scientific_journal-707 20 18 function function NOUN american_scientific_journal-707 20 19 𝑔 𝑔 NOUN american_scientific_journal-707 20 20 : : PUNCT american_scientific_journal-707 20 21 𝐴 𝐴 PROPN american_scientific_journal-707 20 22 𝐾𝑓 𝐾𝑓 PROPN american_scientific_journal-707 20 23 οΏ½ οΏ½ PROPN american_scientific_journal-707 20 24 β†’ β†’ SYM american_scientific_journal-707 20 25 𝐡 𝐡 PROPN american_scientific_journal-707 20 26 , , PUNCT american_scientific_journal-707 20 27 defined define VERB american_scientific_journal-707 20 28 by by ADP american_scientific_journal-707 20 29 𝑔 𝑔 PROPN american_scientific_journal-707 20 30 οΏ½ οΏ½ PROPN american_scientific_journal-707 20 31 𝐾𝑓(π‘Ž 𝐾𝑓(π‘Ž PART american_scientific_journal-707 20 32 ) ) PUNCT american_scientific_journal-707 20 33 οΏ½ οΏ½ PROPN american_scientific_journal-707 20 34 = = SYM american_scientific_journal-707 20 35 𝑓(π‘Ž 𝑓(π‘Ž PROPN american_scientific_journal-707 20 36 ) ) PUNCT american_scientific_journal-707 20 37 for for ADP american_scientific_journal-707 20 38 all all PRON american_scientific_journal-707 20 39 π‘Ž π‘Ž PRON american_scientific_journal-707 20 40 ∈ ∈ PROPN american_scientific_journal-707 20 41 𝐴 𝐴 PROPN american_scientific_journal-707 20 42 , , PUNCT american_scientific_journal-707 20 43 is be AUX american_scientific_journal-707 20 44 a a DET american_scientific_journal-707 20 45 monomorphism monomorphism NOUN american_scientific_journal-707 20 46 . . PUNCT american_scientific_journal-707 21 1 now now ADV american_scientific_journal-707 21 2 , , PUNCT american_scientific_journal-707 21 3 we we PRON american_scientific_journal-707 21 4 have have VERB american_scientific_journal-707 21 5 the the DET american_scientific_journal-707 21 6 decomposition decomposition NOUN american_scientific_journal-707 21 7 𝑓 𝑓 DET american_scientific_journal-707 21 8 = = SYM american_scientific_journal-707 21 9 𝑔 𝑔 PROPN american_scientific_journal-707 21 10 ∘ ∘ ADV american_scientific_journal-707 21 11 β„Ž. β„Ž. NOUN american_scientific_journal-707 21 12 to to PART american_scientific_journal-707 21 13 discuss discuss VERB american_scientific_journal-707 21 14 about about ADP american_scientific_journal-707 21 15 the the DET american_scientific_journal-707 21 16 uniqueness uniqueness NOUN american_scientific_journal-707 21 17 of of ADP american_scientific_journal-707 21 18 the the DET american_scientific_journal-707 21 19 monomorphism𝑔 monomorphism𝑔 PROPN american_scientific_journal-707 21 20 and and CCONJ american_scientific_journal-707 21 21 the the DET american_scientific_journal-707 21 22 epimorphismβ„Ž epimorphismβ„Ž ADV american_scientific_journal-707 21 23 in in ADP american_scientific_journal-707 21 24 this this DET american_scientific_journal-707 21 25 decomposition decomposition NOUN american_scientific_journal-707 21 26 , , PUNCT american_scientific_journal-707 21 27 we we PRON american_scientific_journal-707 21 28 first first ADV american_scientific_journal-707 21 29 define define VERB american_scientific_journal-707 21 30 the the DET american_scientific_journal-707 21 31 notion notion NOUN american_scientific_journal-707 21 32 of of ADP american_scientific_journal-707 21 33 an an DET american_scientific_journal-707 21 34 associate associate NOUN american_scientific_journal-707 21 35 of of ADP american_scientific_journal-707 21 36 a a DET american_scientific_journal-707 21 37 homomorphism homomorphism NOUN american_scientific_journal-707 21 38 of of ADP american_scientific_journal-707 21 39 algebras algebras PROPN american_scientific_journal-707 21 40 and and CCONJ american_scientific_journal-707 21 41 then then ADV american_scientific_journal-707 21 42 we we PRON american_scientific_journal-707 21 43 prove prove VERB american_scientific_journal-707 21 44 the the DET american_scientific_journal-707 21 45 uniqueness uniqueness NOUN american_scientific_journal-707 21 46 of of ADP american_scientific_journal-707 21 47 such such DET american_scientific_journal-707 21 48 a a DET american_scientific_journal-707 21 49 decomposition decomposition NOUN american_scientific_journal-707 21 50 upto upto ADP american_scientific_journal-707 21 51 the the DET american_scientific_journal-707 21 52 level level NOUN american_scientific_journal-707 21 53 of of ADP american_scientific_journal-707 21 54 associates associate NOUN american_scientific_journal-707 21 55 ; ; PUNCT american_scientific_journal-707 21 56 that that PRON american_scientific_journal-707 21 57 is is ADV american_scientific_journal-707 21 58 , , PUNCT american_scientific_journal-707 21 59 , , PUNCT american_scientific_journal-707 21 60 if𝑓 if𝑓 ADV american_scientific_journal-707 21 61 can can AUX american_scientific_journal-707 21 62 be be AUX american_scientific_journal-707 21 63 decomposed decompose VERB american_scientific_journal-707 21 64 in in ADP american_scientific_journal-707 21 65 two two NUM american_scientific_journal-707 21 66 ways way NOUN american_scientific_journal-707 21 67 as as ADP american_scientific_journal-707 21 68 𝑓 𝑓 PRON american_scientific_journal-707 21 69 = = X american_scientific_journal-707 21 70 π‘”π‘œβ„Ž π‘”π‘œβ„Ž PROPN american_scientific_journal-707 21 71 and and CCONJ american_scientific_journal-707 21 72 𝑓 𝑓 PRON american_scientific_journal-707 21 73 = = PUNCT american_scientific_journal-707 21 74 π‘”β€²π‘œβ„Žβ€² π‘”β€²π‘œβ„Žβ€² PROPN american_scientific_journal-707 21 75 , , PUNCT american_scientific_journal-707 21 76 where where SCONJ american_scientific_journal-707 21 77 𝑔 𝑔 PROPN american_scientific_journal-707 21 78 and and CCONJ american_scientific_journal-707 21 79 𝑔′ 𝑔′ NOUN american_scientific_journal-707 21 80 are be AUX american_scientific_journal-707 21 81 monomorphisms monomorphism NOUN american_scientific_journal-707 21 82 and and CCONJ american_scientific_journal-707 21 83 β„Ž β„Ž NOUN american_scientific_journal-707 21 84 and and CCONJ american_scientific_journal-707 21 85 β„Žβ€² β„Žβ€² NOUN american_scientific_journal-707 21 86 are be AUX american_scientific_journal-707 21 87 epimorphisms epimorphism NOUN american_scientific_journal-707 21 88 then then ADV american_scientific_journal-707 21 89 we we PRON american_scientific_journal-707 21 90 will will AUX american_scientific_journal-707 21 91 prove prove VERB american_scientific_journal-707 21 92 that that SCONJ american_scientific_journal-707 21 93 62 62 NUM american_scientific_journal-707 21 94 american american ADJ american_scientific_journal-707 21 95 scientific scientific ADJ american_scientific_journal-707 21 96 research research NOUN american_scientific_journal-707 21 97 journal journal NOUN american_scientific_journal-707 21 98 for for ADP american_scientific_journal-707 21 99 engineering engineering NOUN american_scientific_journal-707 21 100 , , PUNCT american_scientific_journal-707 21 101 technology technology NOUN american_scientific_journal-707 21 102 , , PUNCT american_scientific_journal-707 21 103 and and CCONJ american_scientific_journal-707 21 104 sciences science NOUN american_scientific_journal-707 21 105 ( ( PUNCT american_scientific_journal-707 21 106 asrjets asrjet NOUN american_scientific_journal-707 21 107 ) ) PUNCT american_scientific_journal-707 21 108 ( ( PUNCT american_scientific_journal-707 21 109 2014 2014 NUM american_scientific_journal-707 21 110 ) ) PUNCT american_scientific_journal-707 21 111 volume volume NOUN american_scientific_journal-707 21 112 10 10 NUM american_scientific_journal-707 21 113 , , PUNCT american_scientific_journal-707 21 114 no no DET american_scientific_journal-707 21 115 1 1 NUM american_scientific_journal-707 21 116 , , PUNCT american_scientific_journal-707 21 117 pp pp ADV american_scientific_journal-707 21 118 61 61 NUM american_scientific_journal-707 21 119 - - SYM american_scientific_journal-707 21 120 73 73 NUM american_scientific_journal-707 21 121 𝑔~𝑔′and 𝑔~𝑔′and PROPN american_scientific_journal-707 21 122 β„Ž~β„Žβ€². β„Ž~β„Žβ€². PROPN american_scientific_journal-707 21 123 this this PRON american_scientific_journal-707 21 124 says say VERB american_scientific_journal-707 21 125 that that SCONJ american_scientific_journal-707 21 126 , , PUNCT american_scientific_journal-707 21 127 the the DET american_scientific_journal-707 21 128 decomposition decomposition NOUN american_scientific_journal-707 21 129 of of ADP american_scientific_journal-707 21 130 a a DET american_scientific_journal-707 21 131 homomorphism𝑓 homomorphism𝑓 NOUN american_scientific_journal-707 21 132 as as ADP american_scientific_journal-707 21 133 a a DET american_scientific_journal-707 21 134 composition composition NOUN american_scientific_journal-707 21 135 of of ADP american_scientific_journal-707 21 136 amonomorphism amonomorphism NOUN american_scientific_journal-707 21 137 and and CCONJ american_scientific_journal-707 21 138 anepimorphism anepimorphism NOUN american_scientific_journal-707 21 139 is be AUX american_scientific_journal-707 21 140 unique unique ADJ american_scientific_journal-707 21 141 upto upto ADJ american_scientific_journal-707 21 142 associate associate NOUN american_scientific_journal-707 21 143 . . PUNCT american_scientific_journal-707 22 1 in in ADP american_scientific_journal-707 22 2 this this DET american_scientific_journal-707 22 3 vein vein NOUN american_scientific_journal-707 22 4 we we PRON american_scientific_journal-707 22 5 generalize generalize VERB american_scientific_journal-707 22 6 and and CCONJ american_scientific_journal-707 22 7 unify unify VERB american_scientific_journal-707 22 8 all all DET american_scientific_journal-707 22 9 the the DET american_scientific_journal-707 22 10 fundamental fundamental ADJ american_scientific_journal-707 22 11 theorems theorem NOUN american_scientific_journal-707 22 12 of of ADP american_scientific_journal-707 22 13 homomorphisms homomorphism NOUN american_scientific_journal-707 22 14 and and CCONJ american_scientific_journal-707 22 15 isomorphisms isomorphism NOUN american_scientific_journal-707 22 16 . . PUNCT american_scientific_journal-707 23 1 3 3 X american_scientific_journal-707 23 2 . . NUM american_scientific_journal-707 23 3 preliminaries preliminary NOUN american_scientific_journal-707 23 4 3.1 3.1 NUM american_scientific_journal-707 23 5 . . PUNCT american_scientific_journal-707 24 1 definition definition NOUN american_scientific_journal-707 24 2 of of ADP american_scientific_journal-707 24 3 algebras algebras PROPN american_scientific_journal-707 24 4 definition definition NOUN american_scientific_journal-707 24 5 3.1.1 3.1.1 NUM american_scientific_journal-707 24 6 . . PUNCT american_scientific_journal-707 25 1 fora fora PROPN american_scientific_journal-707 25 2 nonempty nonempty ADV american_scientific_journal-707 25 3 set set VERB american_scientific_journal-707 25 4 a a PRON american_scientific_journal-707 25 5 and and CCONJ american_scientific_journal-707 25 6 a a DET american_scientific_journal-707 25 7 nonnegative nonnegative ADJ american_scientific_journal-707 25 8 integer integer NOUN american_scientific_journal-707 25 9 n n CCONJ american_scientific_journal-707 25 10 , , PUNCT american_scientific_journal-707 25 11 we we PRON american_scientific_journal-707 25 12 define define VERB american_scientific_journal-707 25 13 𝐴0 𝐴0 PROPN american_scientific_journal-707 25 14 = = PUNCT american_scientific_journal-707 25 15 { { PUNCT american_scientific_journal-707 25 16 βˆ… βˆ… NOUN american_scientific_journal-707 25 17 } } PUNCT american_scientific_journal-707 25 18 , , PUNCT american_scientific_journal-707 25 19 and and CCONJ american_scientific_journal-707 25 20 for for ADP american_scientific_journal-707 25 21 𝑛 𝑛 PROPN american_scientific_journal-707 25 22 > > SYM american_scientific_journal-707 25 23 0 0 PROPN american_scientific_journal-707 25 24 , , PUNCT american_scientific_journal-707 25 25 𝐴𝑛is 𝐴𝑛is VERB american_scientific_journal-707 25 26 the the DET american_scientific_journal-707 25 27 set set NOUN american_scientific_journal-707 25 28 of of ADP american_scientific_journal-707 25 29 𝑛-tuples 𝑛-tuple NOUN american_scientific_journal-707 25 30 of of ADP american_scientific_journal-707 25 31 elements element NOUN american_scientific_journal-707 25 32 from from ADP american_scientific_journal-707 25 33 a. a. NOUN american_scientific_journal-707 25 34 an an PRON american_scientific_journal-707 25 35 𝑛 𝑛 PROPN american_scientific_journal-707 25 36 βˆ’ βˆ’ PROPN american_scientific_journal-707 25 37 π‘Žπ‘Ÿπ‘¦ π‘Žπ‘Ÿπ‘¦ NOUN american_scientific_journal-707 25 38 operation operation NOUN american_scientific_journal-707 25 39 ( ( PUNCT american_scientific_journal-707 25 40 or or CCONJ american_scientific_journal-707 25 41 function function NOUN american_scientific_journal-707 25 42 ) ) PUNCT american_scientific_journal-707 25 43 on on ADP american_scientific_journal-707 25 44 a a PRON american_scientific_journal-707 25 45 is be AUX american_scientific_journal-707 25 46 any any DET american_scientific_journal-707 25 47 function function NOUN american_scientific_journal-707 25 48 f f NOUN american_scientific_journal-707 25 49 from from ADP american_scientific_journal-707 25 50 an an PRON american_scientific_journal-707 25 51 to to ADP american_scientific_journal-707 25 52 a a PRON american_scientific_journal-707 25 53 ; ; PUNCT american_scientific_journal-707 25 54 n n X american_scientific_journal-707 25 55 is be AUX american_scientific_journal-707 25 56 the the DET american_scientific_journal-707 25 57 arity(or arity(or ADJ american_scientific_journal-707 25 58 rank rank NOUN american_scientific_journal-707 25 59 ) ) PUNCT american_scientific_journal-707 25 60 of of ADP american_scientific_journal-707 25 61 f. f. PROPN american_scientific_journal-707 25 62 a a DET american_scientific_journal-707 25 63 finitaryoperation finitaryoperation NOUN american_scientific_journal-707 25 64 is be AUX american_scientific_journal-707 25 65 an an DET american_scientific_journal-707 25 66 𝑛ary 𝑛ary ADJ american_scientific_journal-707 25 67 operation operation NOUN american_scientific_journal-707 25 68 , , PUNCT american_scientific_journal-707 25 69 for for ADP american_scientific_journal-707 25 70 some some DET american_scientific_journal-707 25 71 n. n. NOUN american_scientific_journal-707 25 72 the the DET american_scientific_journal-707 25 73 image image NOUN american_scientific_journal-707 25 74 of of ADP american_scientific_journal-707 25 75 ( ( PUNCT american_scientific_journal-707 25 76 π‘Ž1 π‘Ž1 PROPN american_scientific_journal-707 25 77 , , PUNCT american_scientific_journal-707 25 78 π‘Ž2 π‘Ž2 PROPN american_scientific_journal-707 25 79 , , PUNCT american_scientific_journal-707 25 80 . . PUNCT american_scientific_journal-707 25 81 . . PUNCT american_scientific_journal-707 26 1 . . PUNCT american_scientific_journal-707 27 1 , , PUNCT american_scientific_journal-707 27 2 π‘Žπ‘› π‘Žπ‘› PROPN american_scientific_journal-707 27 3 ) ) PUNCT american_scientific_journal-707 27 4 under under ADP american_scientific_journal-707 27 5 an an DET american_scientific_journal-707 27 6 𝑛 𝑛 PROPN american_scientific_journal-707 27 7 βˆ’ary βˆ’ary ADJ american_scientific_journal-707 27 8 operation operation NOUN american_scientific_journal-707 27 9 𝑓is 𝑓is PROPN american_scientific_journal-707 27 10 denoted denote VERB american_scientific_journal-707 27 11 by by ADP american_scientific_journal-707 27 12 𝑓(π‘Ž1 𝑓(π‘Ž1 PROPN american_scientific_journal-707 27 13 , , PUNCT american_scientific_journal-707 27 14 π‘Ž2 π‘Ž2 PROPN american_scientific_journal-707 27 15 , , PUNCT american_scientific_journal-707 27 16 . . PUNCT american_scientific_journal-707 27 17 . . PUNCT american_scientific_journal-707 28 1 . . PUNCT american_scientific_journal-707 29 1 , , PUNCT american_scientific_journal-707 29 2 π‘Žπ‘›).an π‘Žπ‘›).an PROPN american_scientific_journal-707 29 3 operation operation NOUN american_scientific_journal-707 29 4 𝑓on 𝑓on NOUN american_scientific_journal-707 30 1 𝐴is 𝐴is PROPN american_scientific_journal-707 30 2 called call VERB american_scientific_journal-707 30 3 a a DET american_scientific_journal-707 30 4 nullaryoperation nullaryoperation NOUN american_scientific_journal-707 30 5 ( ( PUNCT american_scientific_journal-707 30 6 or or CCONJ american_scientific_journal-707 30 7 constant constant ADJ american_scientific_journal-707 30 8 ) ) PUNCT american_scientific_journal-707 30 9 if if SCONJ american_scientific_journal-707 30 10 its its PRON american_scientific_journal-707 30 11 arity arity NOUN american_scientific_journal-707 30 12 is be AUX american_scientific_journal-707 30 13 zero zero NUM american_scientific_journal-707 30 14 ; ; PUNCT american_scientific_journal-707 30 15 it it PRON american_scientific_journal-707 30 16 is be AUX american_scientific_journal-707 30 17 completely completely ADV american_scientific_journal-707 30 18 determined determine VERB american_scientific_journal-707 30 19 by by ADP american_scientific_journal-707 30 20 the the DET american_scientific_journal-707 30 21 image image NOUN american_scientific_journal-707 30 22 𝑓(βˆ… 𝑓(βˆ… PROPN american_scientific_journal-707 30 23 ) ) PUNCT american_scientific_journal-707 30 24 in in ADP american_scientific_journal-707 30 25 a a PRON american_scientific_journal-707 30 26 of of ADP american_scientific_journal-707 30 27 the the DET american_scientific_journal-707 30 28 only only ADJ american_scientific_journal-707 30 29 element element NOUN american_scientific_journal-707 30 30 βˆ… βˆ… NOUN american_scientific_journal-707 30 31 in in ADP american_scientific_journal-707 30 32 𝐴0 𝐴0 PROPN american_scientific_journal-707 30 33 , , PUNCT american_scientific_journal-707 30 34 and and CCONJ american_scientific_journal-707 30 35 as as ADP american_scientific_journal-707 30 36 such such ADJ american_scientific_journal-707 30 37 it it PRON american_scientific_journal-707 30 38 is be AUX american_scientific_journal-707 30 39 convenient convenient ADJ american_scientific_journal-707 30 40 to to PART american_scientific_journal-707 30 41 identify identify VERB american_scientific_journal-707 30 42 it it PRON american_scientific_journal-707 30 43 with with ADP american_scientific_journal-707 30 44 the the DET american_scientific_journal-707 30 45 element element NOUN american_scientific_journal-707 30 46 𝑓(βˆ… 𝑓(βˆ… PROPN american_scientific_journal-707 30 47 ) ) PUNCT american_scientific_journal-707 30 48 . . PUNCT american_scientific_journal-707 31 1 thus thus ADV american_scientific_journal-707 31 2 a a DET american_scientific_journal-707 31 3 nullary nullary ADJ american_scientific_journal-707 31 4 operation operation NOUN american_scientific_journal-707 31 5 is be AUX american_scientific_journal-707 31 6 thought think VERB american_scientific_journal-707 31 7 of of ADP american_scientific_journal-707 31 8 as as ADP american_scientific_journal-707 31 9 an an DET american_scientific_journal-707 31 10 element element NOUN american_scientific_journal-707 31 11 of of ADP american_scientific_journal-707 31 12 a. a. NOUN american_scientific_journal-707 31 13 an an DET american_scientific_journal-707 31 14 operation operation NOUN american_scientific_journal-707 31 15 𝑓on 𝑓on VERB american_scientific_journal-707 31 16 a a PRON american_scientific_journal-707 31 17 is be AUX american_scientific_journal-707 31 18 unary unary ADJ american_scientific_journal-707 31 19 , , PUNCT american_scientific_journal-707 31 20 binary binary ADJ american_scientific_journal-707 31 21 , , PUNCT american_scientific_journal-707 31 22 or or CCONJ american_scientific_journal-707 31 23 ternary ternary ADJ american_scientific_journal-707 31 24 if if SCONJ american_scientific_journal-707 31 25 its its PRON american_scientific_journal-707 31 26 arity arity NOUN american_scientific_journal-707 31 27 is be AUX american_scientific_journal-707 31 28 1 1 NUM american_scientific_journal-707 31 29 , , PUNCT american_scientific_journal-707 31 30 2 2 NUM american_scientific_journal-707 31 31 , , PUNCT american_scientific_journal-707 31 32 or or CCONJ american_scientific_journal-707 31 33 3 3 NUM american_scientific_journal-707 31 34 , , PUNCT american_scientific_journal-707 31 35 respectively respectively ADV american_scientific_journal-707 31 36 . . PUNCT american_scientific_journal-707 32 1 [ [ X american_scientific_journal-707 32 2 1 1 X american_scientific_journal-707 32 3 ] ] PUNCT american_scientific_journal-707 32 4 definition definition NOUN american_scientific_journal-707 32 5 3.1.2 3.1.2 NUM american_scientific_journal-707 32 6 . . PUNCT american_scientific_journal-707 33 1 a a DET american_scientific_journal-707 33 2 type type NOUN american_scientific_journal-707 33 3 ( ( PUNCT american_scientific_journal-707 33 4 or or CCONJ american_scientific_journal-707 33 5 language language NOUN american_scientific_journal-707 33 6 ) ) PUNCT american_scientific_journal-707 33 7 of of ADP american_scientific_journal-707 33 8 algebras algebras PROPN american_scientific_journal-707 33 9 is be AUX american_scientific_journal-707 33 10 a a DET american_scientific_journal-707 33 11 set set ADJ american_scientific_journal-707 33 12 β„± β„± PROPN american_scientific_journal-707 33 13 of of ADP american_scientific_journal-707 33 14 function function NOUN american_scientific_journal-707 33 15 symbols symbol NOUN american_scientific_journal-707 33 16 such such ADJ american_scientific_journal-707 33 17 that that SCONJ american_scientific_journal-707 33 18 a a DET american_scientific_journal-707 33 19 nonnegative nonnegative ADJ american_scientific_journal-707 33 20 integer integer NOUN american_scientific_journal-707 33 21 𝑛is 𝑛is NOUN american_scientific_journal-707 33 22 assigned assign VERB american_scientific_journal-707 33 23 to to ADP american_scientific_journal-707 33 24 each each DET american_scientific_journal-707 33 25 member member NOUN american_scientific_journal-707 33 26 𝑓of 𝑓of PROPN american_scientific_journal-707 33 27 β„±. β„±. PROPN american_scientific_journal-707 33 28 this this DET american_scientific_journal-707 33 29 integer integer NOUN american_scientific_journal-707 33 30 is be AUX american_scientific_journal-707 33 31 called call VERB american_scientific_journal-707 33 32 the the DET american_scientific_journal-707 33 33 arity arity NOUN american_scientific_journal-707 33 34 ( ( PUNCT american_scientific_journal-707 33 35 or or CCONJ american_scientific_journal-707 33 36 rank rank NOUN american_scientific_journal-707 33 37 ) ) PUNCT american_scientific_journal-707 33 38 of of ADP american_scientific_journal-707 33 39 𝑓 𝑓 PROPN american_scientific_journal-707 33 40 , , PUNCT american_scientific_journal-707 33 41 and and CCONJ american_scientific_journal-707 33 42 𝑓is 𝑓is PROPN american_scientific_journal-707 33 43 said say VERB american_scientific_journal-707 33 44 to to PART american_scientific_journal-707 33 45 be be AUX american_scientific_journal-707 33 46 an an DET american_scientific_journal-707 33 47 𝑛 𝑛 PROPN american_scientific_journal-707 33 48 βˆ’ary βˆ’ary ADJ american_scientific_journal-707 33 49 operation operation NOUN american_scientific_journal-707 33 50 symbol symbol NOUN american_scientific_journal-707 33 51 . . PUNCT american_scientific_journal-707 34 1 the the DET american_scientific_journal-707 34 2 subset subset NOUN american_scientific_journal-707 34 3 of of ADP american_scientific_journal-707 34 4 𝑛 𝑛 PROPN american_scientific_journal-707 34 5 βˆ’ary βˆ’ary ADJ american_scientific_journal-707 34 6 function function NOUN american_scientific_journal-707 34 7 symbols symbol NOUN american_scientific_journal-707 34 8 in in ADP american_scientific_journal-707 34 9 f f PROPN american_scientific_journal-707 34 10 is be AUX american_scientific_journal-707 34 11 denoted denote VERB american_scientific_journal-707 34 12 by by ADP american_scientific_journal-707 34 13 ℱ𝑛.[1 ℱ𝑛.[1 PROPN american_scientific_journal-707 34 14 ] ] PUNCT american_scientific_journal-707 34 15 definition definition NOUN american_scientific_journal-707 35 1 3.1.3 3.1.3 NUM american_scientific_journal-707 35 2 . . PUNCT american_scientific_journal-707 36 1 if if SCONJ american_scientific_journal-707 36 2 β„± β„± PROPN american_scientific_journal-707 36 3 is be AUX american_scientific_journal-707 36 4 a a DET american_scientific_journal-707 36 5 type type NOUN american_scientific_journal-707 36 6 of of ADP american_scientific_journal-707 36 7 algebras algebra NOUN american_scientific_journal-707 36 8 then then ADV american_scientific_journal-707 36 9 an an DET american_scientific_journal-707 36 10 algebra algebra NOUN american_scientific_journal-707 36 11 a a PRON american_scientific_journal-707 36 12 of of ADP american_scientific_journal-707 36 13 type type NOUN american_scientific_journal-707 36 14 β„± β„± PROPN american_scientific_journal-707 36 15 is be AUX american_scientific_journal-707 36 16 an an DET american_scientific_journal-707 36 17 ordered order VERB american_scientific_journal-707 36 18 pair pair NOUN american_scientific_journal-707 36 19 〈 〈 NOUN american_scientific_journal-707 36 20 𝐴,𝐹 𝐴,𝐹 NOUN american_scientific_journal-707 36 21 βŒͺ βŒͺ NOUN american_scientific_journal-707 36 22 where where SCONJ american_scientific_journal-707 36 23 a a PRON american_scientific_journal-707 36 24 is be AUX american_scientific_journal-707 36 25 a a DET american_scientific_journal-707 36 26 nonempty nonempty ADV american_scientific_journal-707 36 27 set set VERB american_scientific_journal-707 36 28 and and CCONJ american_scientific_journal-707 36 29 𝐹is 𝐹is ADV american_scientific_journal-707 36 30 a a DET american_scientific_journal-707 36 31 family family NOUN american_scientific_journal-707 36 32 of of ADP american_scientific_journal-707 36 33 finitary finitary ADJ american_scientific_journal-707 36 34 operations operation NOUN american_scientific_journal-707 36 35 on on ADP american_scientific_journal-707 36 36 a a DET american_scientific_journal-707 36 37 indexed index VERB american_scientific_journal-707 36 38 by by ADP american_scientific_journal-707 36 39 the the DET american_scientific_journal-707 36 40 language language NOUN american_scientific_journal-707 36 41 β„± β„± PROPN american_scientific_journal-707 36 42 such such ADJ american_scientific_journal-707 36 43 that that SCONJ american_scientific_journal-707 36 44 corresponding correspond VERB american_scientific_journal-707 36 45 to to ADP american_scientific_journal-707 36 46 each each PRON american_scientific_journal-707 36 47 𝑛 𝑛 PROPN american_scientific_journal-707 36 48 βˆ’ary βˆ’ary ADJ american_scientific_journal-707 36 49 function function NOUN american_scientific_journal-707 36 50 symbol symbol NOUN american_scientific_journal-707 36 51 𝑓in 𝑓in PROPN american_scientific_journal-707 36 52 β„± β„± PROPN american_scientific_journal-707 36 53 there there PRON american_scientific_journal-707 36 54 is be VERB american_scientific_journal-707 36 55 an an DET american_scientific_journal-707 36 56 𝑛 𝑛 PROPN american_scientific_journal-707 36 57 βˆ’ary βˆ’ary ADJ american_scientific_journal-707 36 58 operation operation NOUN american_scientific_journal-707 36 59 𝑓𝐴 𝑓𝐴 PROPN american_scientific_journal-707 36 60 on on ADP american_scientific_journal-707 36 61 a. a. NOUN american_scientific_journal-707 36 62 the the DET american_scientific_journal-707 36 63 set set NOUN american_scientific_journal-707 36 64 a a PRON american_scientific_journal-707 36 65 is be AUX american_scientific_journal-707 36 66 called call VERB american_scientific_journal-707 36 67 the the DET american_scientific_journal-707 36 68 universe universe NOUN american_scientific_journal-707 36 69 ( ( PUNCT american_scientific_journal-707 36 70 or or CCONJ american_scientific_journal-707 36 71 underlying underlying ADJ american_scientific_journal-707 36 72 set set NOUN american_scientific_journal-707 36 73 ) ) PUNCT american_scientific_journal-707 36 74 of of ADP american_scientific_journal-707 36 75 〈 〈 PROPN american_scientific_journal-707 36 76 𝐴,𝐹 𝐴,𝐹 NOUN american_scientific_journal-707 36 77 βŒͺ βŒͺ NOUN american_scientific_journal-707 36 78 , , PUNCT american_scientific_journal-707 36 79 and and CCONJ american_scientific_journal-707 36 80 the the DET american_scientific_journal-707 36 81 𝑓𝐴 𝑓𝐴 NOUN american_scientific_journal-707 36 82 ’s ’s PART american_scientific_journal-707 36 83 are be AUX american_scientific_journal-707 36 84 called call VERB american_scientific_journal-707 36 85 the the DET american_scientific_journal-707 36 86 fundamental fundamental ADJ american_scientific_journal-707 36 87 operations operation NOUN american_scientific_journal-707 36 88 of of ADP american_scientific_journal-707 36 89 a.[1 a.[1 ADJ american_scientific_journal-707 36 90 ] ] PUNCT american_scientific_journal-707 36 91 if if SCONJ american_scientific_journal-707 36 92 𝐹 𝐹 PROPN american_scientific_journal-707 36 93 is be AUX american_scientific_journal-707 36 94 finite finite ADJ american_scientific_journal-707 36 95 , , PUNCT american_scientific_journal-707 36 96 say say VERB american_scientific_journal-707 36 97 β„± β„± PROPN american_scientific_journal-707 36 98 = = SYM american_scientific_journal-707 36 99 { { PUNCT american_scientific_journal-707 36 100 𝑓1 𝑓1 PROPN american_scientific_journal-707 36 101 , , PUNCT american_scientific_journal-707 36 102 𝑓2 𝑓2 NOUN american_scientific_journal-707 36 103 , , PUNCT american_scientific_journal-707 36 104 . . PUNCT american_scientific_journal-707 36 105 . . PUNCT american_scientific_journal-707 37 1 . . PUNCT american_scientific_journal-707 38 1 , , PUNCT american_scientific_journal-707 38 2 π‘“π‘˜ π‘“π‘˜ ADP american_scientific_journal-707 38 3 } } PUNCT american_scientific_journal-707 38 4 , , PUNCT american_scientific_journal-707 38 5 we we PRON american_scientific_journal-707 38 6 often often ADV american_scientific_journal-707 38 7 write write VERB american_scientific_journal-707 38 8 〈 〈 PROPN american_scientific_journal-707 38 9 𝐴 𝐴 PROPN american_scientific_journal-707 38 10 , , PUNCT american_scientific_journal-707 38 11 𝑓1 𝑓1 ADJ american_scientific_journal-707 38 12 , , PUNCT american_scientific_journal-707 38 13 𝑓2 𝑓2 NOUN american_scientific_journal-707 38 14 , , PUNCT american_scientific_journal-707 38 15 . . PUNCT american_scientific_journal-707 38 16 . . PUNCT american_scientific_journal-707 39 1 . . PUNCT american_scientific_journal-707 40 1 , , PUNCT american_scientific_journal-707 40 2 π‘“π‘˜βŒͺfor π‘“π‘˜βŒͺfor ADP american_scientific_journal-707 40 3 〈 〈 NOUN american_scientific_journal-707 40 4 𝐴,𝐹 𝐴,𝐹 NOUN american_scientific_journal-707 40 5 βŒͺ βŒͺ NOUN american_scientific_journal-707 40 6 , , PUNCT american_scientific_journal-707 40 7 usually usually ADV american_scientific_journal-707 40 8 adopting adopt VERB american_scientific_journal-707 40 9 the the DET american_scientific_journal-707 40 10 convention convention NOUN american_scientific_journal-707 40 11 : : PUNCT american_scientific_journal-707 40 12 π‘Žπ‘Ÿπ‘–π‘‘π‘¦ π‘Žπ‘Ÿπ‘–π‘‘π‘¦ PROPN american_scientific_journal-707 40 13 𝑓1 𝑓1 PROPN american_scientific_journal-707 40 14 β‰₯ β‰₯ NUM american_scientific_journal-707 40 15 π‘Žπ‘Ÿπ‘–π‘‘π‘¦ π‘Žπ‘Ÿπ‘–π‘‘π‘¦ PROPN american_scientific_journal-707 40 16 𝑓2 𝑓2 PROPN american_scientific_journal-707 40 17 β‰₯ β‰₯ NOUN american_scientific_journal-707 40 18 Β· Β· PUNCT american_scientific_journal-707 40 19 Β· Β· PUNCT american_scientific_journal-707 40 20 Β· Β· PUNCT american_scientific_journal-707 40 21 β‰₯ β‰₯ NUM american_scientific_journal-707 40 22 π‘Žπ‘Ÿπ‘–π‘‘π‘¦ π‘Žπ‘Ÿπ‘–π‘‘π‘¦ PROPN american_scientific_journal-707 40 23 π‘“π‘˜. π‘“π‘˜. PROPN american_scientific_journal-707 40 24 3.2 3.2 NUM american_scientific_journal-707 40 25 . . PUNCT american_scientific_journal-707 41 1 homomorphisms homomorphism NOUN american_scientific_journal-707 41 2 of of ADP american_scientific_journal-707 41 3 algebras algebras PROPN american_scientific_journal-707 41 4 definition definition NOUN american_scientific_journal-707 41 5 3.2.1 3.2.1 NUM american_scientific_journal-707 41 6 : : PUNCT american_scientific_journal-707 41 7 let let VERB american_scientific_journal-707 41 8 𝐴 𝐴 PROPN american_scientific_journal-707 41 9 and and CCONJ american_scientific_journal-707 41 10 𝐡 𝐡 NOUN american_scientific_journal-707 41 11 be be AUX american_scientific_journal-707 41 12 algebras algebra NOUN american_scientific_journal-707 41 13 of of ADP american_scientific_journal-707 41 14 a a DET american_scientific_journal-707 41 15 given give VERB american_scientific_journal-707 41 16 type type NOUN american_scientific_journal-707 41 17 β„±. β„±. PROPN american_scientific_journal-707 41 18 a a DET american_scientific_journal-707 41 19 mapping mapping NOUN american_scientific_journal-707 41 20 𝛼 𝛼 NOUN american_scientific_journal-707 41 21 : : PUNCT american_scientific_journal-707 41 22 𝐴 𝐴 PROPN american_scientific_journal-707 41 23 β†’ β†’ SYM american_scientific_journal-707 41 24 𝐡 𝐡 PROPN american_scientific_journal-707 41 25 is be AUX american_scientific_journal-707 41 26 called call VERB american_scientific_journal-707 41 27 a a DET american_scientific_journal-707 41 28 homomorphism homomorphism NOUN american_scientific_journal-707 41 29 , , PUNCT american_scientific_journal-707 41 30 if if SCONJ american_scientific_journal-707 41 31 the the DET american_scientific_journal-707 41 32 following following NOUN american_scientific_journal-707 41 33 are be AUX american_scientific_journal-707 41 34 satisfied satisfied ADJ american_scientific_journal-707 41 35 . . PUNCT american_scientific_journal-707 42 1 [ [ X american_scientific_journal-707 42 2 1 1 NUM american_scientific_journal-707 42 3 , , PUNCT american_scientific_journal-707 42 4 4 4 NUM american_scientific_journal-707 42 5 & & CCONJ american_scientific_journal-707 42 6 7 7 NUM american_scientific_journal-707 42 7 ] ] X american_scientific_journal-707 42 8 i. i. NOUN american_scientific_journal-707 42 9 if if SCONJ american_scientific_journal-707 42 10 𝑓 𝑓 DET american_scientific_journal-707 42 11 ∈ ∈ PROPN american_scientific_journal-707 42 12 β„± β„± PROPN american_scientific_journal-707 42 13 is be AUX american_scientific_journal-707 42 14 a a DET american_scientific_journal-707 42 15 nullary nullary ADJ american_scientific_journal-707 42 16 operation operation NOUN american_scientific_journal-707 42 17 symbol symbol NOUN american_scientific_journal-707 42 18 , , PUNCT american_scientific_journal-707 42 19 then then ADV american_scientific_journal-707 42 20 𝛼(𝑓𝐴 𝛼(𝑓𝐴 NUM american_scientific_journal-707 42 21 ) ) PUNCT american_scientific_journal-707 43 1 = = PROPN american_scientific_journal-707 43 2 𝑓𝐡 𝑓𝐡 PROPN american_scientific_journal-707 43 3 ii ii PROPN american_scientific_journal-707 43 4 . . PUNCT american_scientific_journal-707 44 1 if if SCONJ american_scientific_journal-707 44 2 𝑓 𝑓 DET american_scientific_journal-707 44 3 ∈ ∈ PROPN american_scientific_journal-707 44 4 β„± β„± PROPN american_scientific_journal-707 44 5 is be AUX american_scientific_journal-707 44 6 an an DET american_scientific_journal-707 44 7 n n CCONJ american_scientific_journal-707 44 8 - - PUNCT american_scientific_journal-707 44 9 ary ary NOUN american_scientific_journal-707 44 10 operation operation NOUN american_scientific_journal-707 44 11 symbol,𝑛 symbol,𝑛 PROPN american_scientific_journal-707 44 12 > > X american_scientific_journal-707 44 13 0 0 PUNCT american_scientific_journal-707 44 14 and and CCONJ american_scientific_journal-707 44 15 π‘Ž1 π‘Ž1 PROPN american_scientific_journal-707 44 16 , , PUNCT american_scientific_journal-707 44 17 π‘Ž2 π‘Ž2 NOUN american_scientific_journal-707 44 18 , , PUNCT american_scientific_journal-707 44 19 … … PUNCT american_scientific_journal-707 44 20 . . PUNCT american_scientific_journal-707 44 21 . . PUNCT american_scientific_journal-707 45 1 π‘Žπ‘› π‘Žπ‘› PROPN american_scientific_journal-707 45 2 ∈ ∈ PROPN american_scientific_journal-707 45 3 𝐴 𝐴 PROPN american_scientific_journal-707 45 4 , , PUNCT american_scientific_journal-707 45 5 then then ADV american_scientific_journal-707 45 6 𝛼(𝑓𝐴(π‘Ž1 𝛼(𝑓𝐴(π‘Ž1 ADJ american_scientific_journal-707 45 7 , , PUNCT american_scientific_journal-707 45 8 π‘Ž2 π‘Ž2 PROPN american_scientific_journal-707 45 9 , , PUNCT american_scientific_journal-707 45 10 … … PUNCT american_scientific_journal-707 45 11 . . PUNCT american_scientific_journal-707 45 12 . . PUNCT american_scientific_journal-707 46 1 π‘Žπ‘› π‘Žπ‘› X american_scientific_journal-707 46 2 ) ) PUNCT american_scientific_journal-707 46 3 = = SYM american_scientific_journal-707 47 1 𝑓𝐡(𝛼(π‘Ž1),𝛼(π‘Ž2 𝑓𝐡(𝛼(π‘Ž1),𝛼(π‘Ž2 PROPN american_scientific_journal-707 47 2 ) ) PUNCT american_scientific_journal-707 47 3 , , PUNCT american_scientific_journal-707 47 4 … … PUNCT american_scientific_journal-707 47 5 … … PUNCT american_scientific_journal-707 47 6 , , PUNCT american_scientific_journal-707 47 7 𝛼(π‘Žπ‘› 𝛼(π‘Žπ‘› NOUN american_scientific_journal-707 47 8 ) ) PUNCT american_scientific_journal-707 47 9 ) ) PUNCT american_scientific_journal-707 47 10 theorem theorem VERB american_scientific_journal-707 47 11 3.2.1 3.2.1 NUM american_scientific_journal-707 47 12 : : PUNCT american_scientific_journal-707 47 13 let let VERB american_scientific_journal-707 47 14 a a DET american_scientific_journal-707 47 15 , , PUNCT american_scientific_journal-707 47 16 b b NOUN american_scientific_journal-707 47 17 and and CCONJ american_scientific_journal-707 47 18 c c PROPN american_scientific_journal-707 47 19 be be AUX american_scientific_journal-707 47 20 algebras algebra NOUN american_scientific_journal-707 47 21 of of ADP american_scientific_journal-707 47 22 a a DET american_scientific_journal-707 47 23 given give VERB american_scientific_journal-707 47 24 type type NOUN american_scientific_journal-707 47 25 β„±. β„±. PROPN american_scientific_journal-707 47 26 let let VERB american_scientific_journal-707 47 27 𝛼:𝐴 𝛼:𝐴 NOUN american_scientific_journal-707 47 28 β†’ β†’ SYM american_scientific_journal-707 47 29 𝐡 𝐡 NOUN american_scientific_journal-707 47 30 and and CCONJ american_scientific_journal-707 47 31 𝛽:𝐡 𝛽:𝐡 PROPN american_scientific_journal-707 47 32 β†’ β†’ SYM american_scientific_journal-707 47 33 𝐢 𝐢 PROPN american_scientific_journal-707 47 34 be be AUX american_scientific_journal-707 47 35 homomorphisms homomorphism NOUN american_scientific_journal-707 47 36 . . PUNCT american_scientific_journal-707 48 1 then then ADV american_scientific_journal-707 48 2 π›½π‘œπ›Ό:𝐴 π›½π‘œπ›Ό:𝐴 PROPN american_scientific_journal-707 48 3 β†’ β†’ SYM american_scientific_journal-707 48 4 𝐢 𝐢 PROPN american_scientific_journal-707 48 5 is be AUX american_scientific_journal-707 48 6 also also ADV american_scientific_journal-707 48 7 a a DET american_scientific_journal-707 48 8 homomorphism homomorphism NOUN american_scientific_journal-707 48 9 [ [ X american_scientific_journal-707 48 10 7 7 NUM american_scientific_journal-707 48 11 ] ] PUNCT american_scientific_journal-707 48 12 . . PUNCT american_scientific_journal-707 49 1 63 63 NUM american_scientific_journal-707 49 2 american american ADJ american_scientific_journal-707 49 3 scientific scientific ADJ american_scientific_journal-707 49 4 research research NOUN american_scientific_journal-707 49 5 journal journal NOUN american_scientific_journal-707 49 6 for for ADP american_scientific_journal-707 49 7 engineering engineering NOUN american_scientific_journal-707 49 8 , , PUNCT american_scientific_journal-707 49 9 technology technology NOUN american_scientific_journal-707 49 10 , , PUNCT american_scientific_journal-707 49 11 and and CCONJ american_scientific_journal-707 49 12 sciences science NOUN american_scientific_journal-707 49 13 ( ( PUNCT american_scientific_journal-707 49 14 asrjets asrjet NOUN american_scientific_journal-707 49 15 ) ) PUNCT american_scientific_journal-707 49 16 ( ( PUNCT american_scientific_journal-707 49 17 2014 2014 NUM american_scientific_journal-707 49 18 ) ) PUNCT american_scientific_journal-707 49 19 volume volume NOUN american_scientific_journal-707 49 20 10 10 NUM american_scientific_journal-707 49 21 , , PUNCT american_scientific_journal-707 49 22 no no DET american_scientific_journal-707 49 23 1 1 NUM american_scientific_journal-707 49 24 , , PUNCT american_scientific_journal-707 49 25 pp pp ADV american_scientific_journal-707 49 26 61 61 NUM american_scientific_journal-707 49 27 - - SYM american_scientific_journal-707 49 28 73 73 NUM american_scientific_journal-707 49 29 proof proof NOUN american_scientific_journal-707 49 30 : : PUNCT american_scientific_journal-707 49 31 let let VERB american_scientific_journal-707 49 32 𝑓 𝑓 PRON american_scientific_journal-707 49 33 ∈ ∈ NOUN american_scientific_journal-707 49 34 β„± β„± PROPN american_scientific_journal-707 49 35 be be VERB american_scientific_journal-707 49 36 a a DET american_scientific_journal-707 49 37 nullary nullary ADJ american_scientific_journal-707 49 38 operation operation NOUN american_scientific_journal-707 49 39 symbol symbol NOUN american_scientific_journal-707 49 40 . . PUNCT american_scientific_journal-707 50 1 then then ADV american_scientific_journal-707 50 2 𝛽 𝛽 X american_scientific_journal-707 50 3 ∘ ∘ PROPN american_scientific_journal-707 50 4 𝛼(𝑓𝐴 𝛼(𝑓𝐴 NOUN american_scientific_journal-707 50 5 ) ) PUNCT american_scientific_journal-707 50 6 = = SYM american_scientific_journal-707 50 7 𝛽(𝛼(𝑓𝐴 𝛽(𝛼(𝑓𝐴 NOUN american_scientific_journal-707 50 8 ) ) PUNCT american_scientific_journal-707 50 9 ) ) PUNCT american_scientific_journal-707 50 10 = = PUNCT american_scientific_journal-707 50 11 𝛽(𝑓𝐡 𝛽(𝑓𝐡 X american_scientific_journal-707 50 12 ) ) PUNCT american_scientific_journal-707 50 13 ( ( PUNCT american_scientific_journal-707 50 14 ∡ ∡ X american_scientific_journal-707 50 15 𝛼is 𝛼is PROPN american_scientific_journal-707 50 16 a a DET american_scientific_journal-707 50 17 homomorphism homomorphism NOUN american_scientific_journal-707 50 18 ) ) PUNCT american_scientific_journal-707 50 19 = = SYM american_scientific_journal-707 50 20 𝑓𝐢 𝑓𝐢 NOUN american_scientific_journal-707 50 21 ( ( PUNCT american_scientific_journal-707 50 22 ∡ ∡ X american_scientific_journal-707 50 23 𝛽is 𝛽i VERB american_scientific_journal-707 50 24 a a DET american_scientific_journal-707 50 25 homomorphism homomorphism NOUN american_scientific_journal-707 50 26 ) ) PUNCT american_scientific_journal-707 50 27 let𝑓 let𝑓 NOUN american_scientific_journal-707 50 28 ∈ ∈ PROPN american_scientific_journal-707 50 29 β„± β„± PROPN american_scientific_journal-707 50 30 be be VERB american_scientific_journal-707 50 31 an an DET american_scientific_journal-707 50 32 n n CCONJ american_scientific_journal-707 50 33 - - PUNCT american_scientific_journal-707 50 34 ary ary NOUN american_scientific_journal-707 50 35 operation operation NOUN american_scientific_journal-707 50 36 symbol symbol NOUN american_scientific_journal-707 50 37 , , PUNCT american_scientific_journal-707 50 38 𝑛 𝑛 PROPN american_scientific_journal-707 50 39 > > X american_scientific_journal-707 50 40 0 0 PUNCT american_scientific_journal-707 50 41 and and CCONJ american_scientific_journal-707 50 42 π‘Ž1 π‘Ž1 PROPN american_scientific_journal-707 50 43 , , PUNCT american_scientific_journal-707 50 44 π‘Ž2 π‘Ž2 NOUN american_scientific_journal-707 50 45 , , PUNCT american_scientific_journal-707 50 46 … … PUNCT american_scientific_journal-707 50 47 . . PUNCT american_scientific_journal-707 50 48 . . PUNCT american_scientific_journal-707 51 1 π‘Žπ‘› π‘Žπ‘› PROPN american_scientific_journal-707 51 2 ∈ ∈ PROPN american_scientific_journal-707 51 3 𝐴. 𝐴. PROPN american_scientific_journal-707 52 1 then then ADV american_scientific_journal-707 52 2 , , PUNCT american_scientific_journal-707 52 3 𝛽 𝛽 NOUN american_scientific_journal-707 52 4 ∘ ∘ NOUN american_scientific_journal-707 52 5 𝛼(𝑓𝐴(π‘Ž1 𝛼(𝑓𝐴(π‘Ž1 ADJ american_scientific_journal-707 52 6 , , PUNCT american_scientific_journal-707 52 7 π‘Ž2 π‘Ž2 PROPN american_scientific_journal-707 52 8 , , PUNCT american_scientific_journal-707 52 9 … … PUNCT american_scientific_journal-707 52 10 . . PUNCT american_scientific_journal-707 52 11 . . PUNCT american_scientific_journal-707 53 1 π‘Žπ‘› π‘Žπ‘› X american_scientific_journal-707 53 2 ) ) PUNCT american_scientific_journal-707 53 3 = = SYM american_scientific_journal-707 53 4 𝛽(𝛼(𝑓𝐴(π‘Ž1 𝛽(𝛼(𝑓𝐴(π‘Ž1 PROPN american_scientific_journal-707 53 5 , , PUNCT american_scientific_journal-707 53 6 π‘Ž2 π‘Ž2 NOUN american_scientific_journal-707 53 7 , , PUNCT american_scientific_journal-707 53 8 … … PUNCT american_scientific_journal-707 53 9 . . PUNCT american_scientific_journal-707 53 10 . . PUNCT american_scientific_journal-707 54 1 π‘Žπ‘› π‘Žπ‘› X american_scientific_journal-707 54 2 ) ) PUNCT american_scientific_journal-707 54 3 ) ) PUNCT american_scientific_journal-707 55 1 = = PUNCT american_scientific_journal-707 55 2 𝛽(𝑓𝐡(𝛼(π‘Ž1),𝛼(π‘Ž2 𝛽(𝑓𝐡(𝛼(π‘Ž1),𝛼(π‘Ž2 PROPN american_scientific_journal-707 55 3 ) ) PUNCT american_scientific_journal-707 55 4 , , PUNCT american_scientific_journal-707 55 5 … … PUNCT american_scientific_journal-707 55 6 , , PUNCT american_scientific_journal-707 55 7 𝛼(π‘Žπ‘› 𝛼(π‘Žπ‘› NOUN american_scientific_journal-707 55 8 ) ) PUNCT american_scientific_journal-707 55 9 ) ) PUNCT american_scientific_journal-707 55 10 ) ) PUNCT american_scientific_journal-707 56 1 ( ( PUNCT american_scientific_journal-707 56 2 ∡ ∡ X american_scientific_journal-707 56 3 𝛼is 𝛼is PROPN american_scientific_journal-707 56 4 a a DET american_scientific_journal-707 56 5 homomorphism homomorphism NOUN american_scientific_journal-707 56 6 ) ) PUNCT american_scientific_journal-707 56 7 = = SYM american_scientific_journal-707 56 8 𝑓𝐢(𝛽(𝛼(π‘Ž1 𝑓𝐢(𝛽(𝛼(π‘Ž1 PROPN american_scientific_journal-707 56 9 ) ) PUNCT american_scientific_journal-707 56 10 ) ) PUNCT american_scientific_journal-707 56 11 , , PUNCT american_scientific_journal-707 56 12 𝛽 𝛽 NOUN american_scientific_journal-707 56 13 οΏ½ οΏ½ PROPN american_scientific_journal-707 56 14 𝛼(π‘Ž2 𝛼(π‘Ž2 NOUN american_scientific_journal-707 56 15 ) ) PUNCT american_scientific_journal-707 56 16 οΏ½ οΏ½ PROPN american_scientific_journal-707 56 17 , , PUNCT american_scientific_journal-707 56 18 … … PUNCT american_scientific_journal-707 56 19 … … PUNCT american_scientific_journal-707 56 20 𝛽 𝛽 NOUN american_scientific_journal-707 56 21 οΏ½ οΏ½ NOUN american_scientific_journal-707 56 22 𝛼(π‘Žπ‘› 𝛼(π‘Žπ‘› NUM american_scientific_journal-707 56 23 ) ) PUNCT american_scientific_journal-707 56 24 οΏ½ οΏ½ PROPN american_scientific_journal-707 56 25 ) ) PUNCT american_scientific_journal-707 56 26 ( ( PUNCT american_scientific_journal-707 56 27 ∡ ∡ X american_scientific_journal-707 56 28 𝛽is 𝛽i VERB american_scientific_journal-707 56 29 a a DET american_scientific_journal-707 56 30 homomorphism homomorphism NOUN american_scientific_journal-707 56 31 ) ) PUNCT american_scientific_journal-707 57 1 = = SYM american_scientific_journal-707 57 2 𝑓𝐢(𝛽 𝑓𝐢(𝛽 PROPN american_scientific_journal-707 57 3 ∘ ∘ NUM american_scientific_journal-707 57 4 𝛼(π‘Ž1 𝛼(π‘Ž1 ADJ american_scientific_journal-707 57 5 ) ) PUNCT american_scientific_journal-707 57 6 , , PUNCT american_scientific_journal-707 57 7 𝛽 𝛽 NOUN american_scientific_journal-707 57 8 ∘ ∘ NOUN american_scientific_journal-707 57 9 𝛼(π‘Ž2 𝛼(π‘Ž2 NOUN american_scientific_journal-707 57 10 ) ) PUNCT american_scientific_journal-707 57 11 , , PUNCT american_scientific_journal-707 57 12 … … PUNCT american_scientific_journal-707 57 13 … … PUNCT american_scientific_journal-707 57 14 … … PUNCT american_scientific_journal-707 57 15 … … PUNCT american_scientific_journal-707 57 16 𝛽 𝛽 PRON american_scientific_journal-707 57 17 ∘ ∘ NOUN american_scientific_journal-707 57 18 𝛼(π‘Žπ‘› 𝛼(π‘Žπ‘› NUM american_scientific_journal-707 57 19 ) ) PUNCT american_scientific_journal-707 57 20 ) ) PUNCT american_scientific_journal-707 58 1 therefore therefore ADV american_scientific_journal-707 58 2 , , PUNCT american_scientific_journal-707 58 3 π›½π‘œπ›Ό:𝐴 π›½π‘œπ›Ό:𝐴 PROPN american_scientific_journal-707 58 4 β†’ β†’ SYM american_scientific_journal-707 58 5 𝐢is 𝐢is PROPN american_scientific_journal-707 58 6 also also ADV american_scientific_journal-707 58 7 a a DET american_scientific_journal-707 58 8 homomorphism homomorphism NOUN american_scientific_journal-707 58 9 . . PUNCT american_scientific_journal-707 59 1 definition definition NOUN american_scientific_journal-707 59 2 3.2.2 3.2.2 NUM american_scientific_journal-707 59 3 : : PUNCT american_scientific_journal-707 59 4 let let VERB american_scientific_journal-707 59 5 a a PRON american_scientific_journal-707 59 6 and and CCONJ american_scientific_journal-707 59 7 b b NOUN american_scientific_journal-707 59 8 be be AUX american_scientific_journal-707 59 9 algebras algebra NOUN american_scientific_journal-707 59 10 of of ADP american_scientific_journal-707 59 11 a a DET american_scientific_journal-707 59 12 given give VERB american_scientific_journal-707 59 13 type type NOUN american_scientific_journal-707 59 14 β„± β„± PROPN american_scientific_journal-707 59 15 and and CCONJ american_scientific_journal-707 59 16 𝛼 𝛼 ADP american_scientific_journal-707 59 17 : : PUNCT american_scientific_journal-707 59 18 𝐴 𝐴 PROPN american_scientific_journal-707 59 19 β†’ β†’ SYM american_scientific_journal-707 59 20 𝐡 𝐡 NOUN american_scientific_journal-707 59 21 be be AUX american_scientific_journal-707 59 22 any any DET american_scientific_journal-707 59 23 function function NOUN american_scientific_journal-707 59 24 . . PUNCT american_scientific_journal-707 60 1 then then ADV american_scientific_journal-707 60 2 i. i. PROPN american_scientific_journal-707 60 3 𝛼 𝛼 PROPN american_scientific_journal-707 60 4 is be AUX american_scientific_journal-707 60 5 called call VERB american_scientific_journal-707 60 6 a a DET american_scientific_journal-707 60 7 monomorphism monomorphism NOUN american_scientific_journal-707 60 8 , , PUNCT american_scientific_journal-707 60 9 if if SCONJ american_scientific_journal-707 60 10 it it PRON american_scientific_journal-707 60 11 is be AUX american_scientific_journal-707 60 12 an an DET american_scientific_journal-707 60 13 injective injective ADJ american_scientific_journal-707 60 14 homomorphism homomorphism NOUN american_scientific_journal-707 60 15 . . PUNCT american_scientific_journal-707 61 1 ii ii PROPN american_scientific_journal-707 61 2 . . PUNCT american_scientific_journal-707 62 1 𝛼 𝛼 PROPN american_scientific_journal-707 62 2 is be AUX american_scientific_journal-707 62 3 called call VERB american_scientific_journal-707 62 4 an an DET american_scientific_journal-707 62 5 epimorphism epimorphism NOUN american_scientific_journal-707 62 6 , , PUNCT american_scientific_journal-707 62 7 if if SCONJ american_scientific_journal-707 62 8 it it PRON american_scientific_journal-707 62 9 is be AUX american_scientific_journal-707 62 10 a a DET american_scientific_journal-707 62 11 surjective surjective ADJ american_scientific_journal-707 62 12 homomorphism homomorphism NOUN american_scientific_journal-707 62 13 . . PUNCT american_scientific_journal-707 63 1 iii iii X american_scientific_journal-707 63 2 . . PUNCT american_scientific_journal-707 64 1 𝛼 𝛼 PROPN american_scientific_journal-707 64 2 is be AUX american_scientific_journal-707 64 3 called call VERB american_scientific_journal-707 64 4 an an DET american_scientific_journal-707 64 5 isomorphism isomorphism NOUN american_scientific_journal-707 64 6 , , PUNCT american_scientific_journal-707 64 7 if if SCONJ american_scientific_journal-707 64 8 it it PRON american_scientific_journal-707 64 9 is be AUX american_scientific_journal-707 64 10 a a DET american_scientific_journal-707 64 11 bijective bijective ADJ american_scientific_journal-707 64 12 homomorphism.[1 homomorphism.[1 NOUN american_scientific_journal-707 64 13 , , PUNCT american_scientific_journal-707 64 14 6 6 NUM american_scientific_journal-707 64 15 & & CCONJ american_scientific_journal-707 64 16 7 7 NUM american_scientific_journal-707 64 17 ] ] SYM american_scientific_journal-707 64 18 3.3 3.3 NUM american_scientific_journal-707 64 19 . . PUNCT american_scientific_journal-707 65 1 congruence congruence PROPN american_scientific_journal-707 65 2 relations relation NOUN american_scientific_journal-707 65 3 definition definition NOUN american_scientific_journal-707 65 4 3.3.1 3.3.1 NUM american_scientific_journal-707 65 5 : : PUNCT american_scientific_journal-707 65 6 let let VERB american_scientific_journal-707 65 7 a a PRON american_scientific_journal-707 65 8 be be AUX american_scientific_journal-707 65 9 an an DET american_scientific_journal-707 65 10 algebra algebra NOUN american_scientific_journal-707 65 11 of of ADP american_scientific_journal-707 65 12 type type NOUN american_scientific_journal-707 65 13 β„± β„± PROPN american_scientific_journal-707 65 14 and and CCONJ american_scientific_journal-707 65 15 πœƒ πœƒ PRON american_scientific_journal-707 65 16 be be AUX american_scientific_journal-707 65 17 an an DET american_scientific_journal-707 65 18 equivalence equivalence NOUN american_scientific_journal-707 65 19 relation relation NOUN american_scientific_journal-707 65 20 on on ADP american_scientific_journal-707 65 21 𝐴. 𝐴. PROPN american_scientific_journal-707 65 22 then then ADV american_scientific_journal-707 65 23 ΞΈ ΞΈ PROPN american_scientific_journal-707 65 24 is be AUX american_scientific_journal-707 65 25 said say VERB american_scientific_journal-707 65 26 to to PART american_scientific_journal-707 65 27 be be AUX american_scientific_journal-707 65 28 a a DET american_scientific_journal-707 65 29 congruence congruence NOUN american_scientific_journal-707 65 30 relation relation NOUN american_scientific_journal-707 65 31 on on ADP american_scientific_journal-707 65 32 a a PRON american_scientific_journal-707 65 33 , , PUNCT american_scientific_journal-707 65 34 if if SCONJ american_scientific_journal-707 65 35 the the DET american_scientific_journal-707 65 36 following follow VERB american_scientific_journal-707 65 37 is be AUX american_scientific_journal-707 65 38 satisfied satisfied ADJ american_scientific_journal-707 65 39 for for ADP american_scientific_journal-707 65 40 any any DET american_scientific_journal-707 65 41 π‘Ž1 π‘Ž1 NOUN american_scientific_journal-707 65 42 , , PUNCT american_scientific_journal-707 65 43 π‘Ž2 π‘Ž2 PROPN american_scientific_journal-707 65 44 , , PUNCT american_scientific_journal-707 65 45 … … PUNCT american_scientific_journal-707 65 46 . . PUNCT american_scientific_journal-707 65 47 . . PUNCT american_scientific_journal-707 66 1 π‘Žπ‘› π‘Žπ‘› NOUN american_scientific_journal-707 66 2 , , PUNCT american_scientific_journal-707 66 3 𝑏1 𝑏1 NOUN american_scientific_journal-707 66 4 , , PUNCT american_scientific_journal-707 66 5 𝑏2 𝑏2 NOUN american_scientific_journal-707 66 6 , , PUNCT american_scientific_journal-707 66 7 … … PUNCT american_scientific_journal-707 66 8 . . PUNCT american_scientific_journal-707 66 9 . . PUNCT american_scientific_journal-707 67 1 𝑏𝑛 𝑏𝑛 ADP american_scientific_journal-707 67 2 ∈ ∈ PROPN american_scientific_journal-707 67 3 𝐴 𝐴 PROPN american_scientific_journal-707 67 4 , , PUNCT american_scientific_journal-707 67 5 if if SCONJ american_scientific_journal-707 67 6 οΏ½ οΏ½ PROPN american_scientific_journal-707 67 7 π‘Ž1,𝑏1 π‘Ž1,𝑏1 PROPN american_scientific_journal-707 67 8 οΏ½ οΏ½ PROPN american_scientific_journal-707 67 9 , , PUNCT american_scientific_journal-707 67 10 οΏ½ οΏ½ PROPN american_scientific_journal-707 67 11 π‘Ž2,𝑏2 π‘Ž2,𝑏2 PROPN american_scientific_journal-707 67 12 οΏ½ οΏ½ PROPN american_scientific_journal-707 67 13 , , PUNCT american_scientific_journal-707 67 14 … … PUNCT american_scientific_journal-707 67 15 … … PUNCT american_scientific_journal-707 67 16 … … PUNCT american_scientific_journal-707 67 17 … … PUNCT american_scientific_journal-707 67 18 … … PUNCT american_scientific_journal-707 67 19 … … PUNCT american_scientific_journal-707 67 20 . . PUNCT american_scientific_journal-707 67 21 . . PUNCT american_scientific_journal-707 68 1 , , PUNCT american_scientific_journal-707 68 2 οΏ½ οΏ½ PROPN american_scientific_journal-707 68 3 π‘Žπ‘›,𝑏𝑛 π‘Žπ‘›,𝑏𝑛 NOUN american_scientific_journal-707 68 4 οΏ½ οΏ½ PROPN american_scientific_journal-707 68 5 ∈ ∈ PROPN american_scientific_journal-707 68 6 πœƒ,𝑓 πœƒ,𝑓 NOUN american_scientific_journal-707 68 7 ∈ ∈ PROPN american_scientific_journal-707 69 1 β„±is β„±is NOUN american_scientific_journal-707 69 2 an an DET american_scientific_journal-707 69 3 𝑛-ary 𝑛-ary ADJ american_scientific_journal-707 69 4 operation operation NOUN american_scientific_journal-707 69 5 symbol symbol NOUN american_scientific_journal-707 69 6 and and CCONJ american_scientific_journal-707 69 7 𝑛 𝑛 ADP american_scientific_journal-707 69 8 > > X american_scientific_journal-707 69 9 0,then 0,then PROPN american_scientific_journal-707 69 10 ( ( PUNCT american_scientific_journal-707 69 11 𝑓𝐴(π‘Ž1 𝑓𝐴(π‘Ž1 PROPN american_scientific_journal-707 69 12 , , PUNCT american_scientific_journal-707 69 13 π‘Ž2 π‘Ž2 PROPN american_scientific_journal-707 69 14 , , PUNCT american_scientific_journal-707 69 15 … … PUNCT american_scientific_journal-707 69 16 . . PUNCT american_scientific_journal-707 69 17 . . PUNCT american_scientific_journal-707 70 1 π‘Žπ‘› π‘Žπ‘› PROPN american_scientific_journal-707 70 2 ) ) PUNCT american_scientific_journal-707 70 3 , , PUNCT american_scientific_journal-707 70 4 𝑓𝐴(𝑏1 𝑓𝐴(𝑏1 PROPN american_scientific_journal-707 70 5 , , PUNCT american_scientific_journal-707 70 6 𝑏2 𝑏2 PROPN american_scientific_journal-707 70 7 , , PUNCT american_scientific_journal-707 70 8 … … PUNCT american_scientific_journal-707 70 9 . . PUNCT american_scientific_journal-707 70 10 . . PUNCT american_scientific_journal-707 71 1 𝑏𝑛 𝑏𝑛 NOUN american_scientific_journal-707 71 2 ) ) PUNCT american_scientific_journal-707 71 3 ) ) PUNCT american_scientific_journal-707 72 1 ∈ ∈ PROPN american_scientific_journal-707 72 2 πœƒ. πœƒ. NOUN american_scientific_journal-707 73 1 [ [ X american_scientific_journal-707 73 2 1 1 NUM american_scientific_journal-707 73 3 , , PUNCT american_scientific_journal-707 73 4 2 2 NUM american_scientific_journal-707 73 5 , , PUNCT american_scientific_journal-707 73 6 6 6 NUM american_scientific_journal-707 73 7 & & CCONJ american_scientific_journal-707 73 8 7 7 NUM american_scientific_journal-707 73 9 ] ] PUNCT american_scientific_journal-707 73 10 in in ADP american_scientific_journal-707 73 11 other other ADJ american_scientific_journal-707 73 12 words word NOUN american_scientific_journal-707 73 13 , , PUNCT american_scientific_journal-707 73 14 a a DET american_scientific_journal-707 73 15 congruence congruence NOUN american_scientific_journal-707 73 16 relation relation NOUN american_scientific_journal-707 73 17 is be AUX american_scientific_journal-707 73 18 an an DET american_scientific_journal-707 73 19 equivalence equivalence NOUN american_scientific_journal-707 73 20 relation relation NOUN american_scientific_journal-707 73 21 on on ADP american_scientific_journal-707 73 22 an an DET american_scientific_journal-707 73 23 algebraic algebraic ADJ american_scientific_journal-707 73 24 structure structure NOUN american_scientific_journal-707 73 25 ( ( PUNCT american_scientific_journal-707 73 26 such such ADJ american_scientific_journal-707 73 27 as as ADP american_scientific_journal-707 73 28 groups group NOUN american_scientific_journal-707 73 29 , , PUNCT american_scientific_journal-707 73 30 rings ring NOUN american_scientific_journal-707 73 31 or or CCONJ american_scientific_journal-707 73 32 vector vector NOUN american_scientific_journal-707 73 33 spaces space NOUN american_scientific_journal-707 73 34 and and CCONJ american_scientific_journal-707 73 35 of of ADP american_scientific_journal-707 73 36 general general ADJ american_scientific_journal-707 73 37 universal universal ADJ american_scientific_journal-707 73 38 algebras algebra NOUN american_scientific_journal-707 73 39 ) ) PUNCT american_scientific_journal-707 73 40 that that PRON american_scientific_journal-707 73 41 is be AUX american_scientific_journal-707 73 42 compatible compatible ADJ american_scientific_journal-707 73 43 with with ADP american_scientific_journal-707 73 44 the the DET american_scientific_journal-707 73 45 structure structure NOUN american_scientific_journal-707 73 46 . . PUNCT american_scientific_journal-707 74 1 in in ADP american_scientific_journal-707 74 2 the the DET american_scientific_journal-707 74 3 next next ADJ american_scientific_journal-707 74 4 two two NUM american_scientific_journal-707 74 5 theorems theorem NOUN american_scientific_journal-707 74 6 we we PRON american_scientific_journal-707 74 7 will will AUX american_scientific_journal-707 74 8 observe observe VERB american_scientific_journal-707 74 9 that that SCONJ american_scientific_journal-707 74 10 kernels kernel NOUN american_scientific_journal-707 74 11 of of ADP american_scientific_journal-707 74 12 homomorphisms homomorphism NOUN american_scientific_journal-707 74 13 and and CCONJ american_scientific_journal-707 74 14 congruence congruence PROPN american_scientific_journal-707 74 15 relations relation NOUN american_scientific_journal-707 74 16 are be AUX american_scientific_journal-707 74 17 same same ADJ american_scientific_journal-707 74 18 . . PUNCT american_scientific_journal-707 75 1 theorem theorem VERB american_scientific_journal-707 75 2 3.3.1 3.3.1 NUM american_scientific_journal-707 75 3 : : PUNCT american_scientific_journal-707 75 4 let let VERB american_scientific_journal-707 75 5 a a PRON american_scientific_journal-707 75 6 and and CCONJ american_scientific_journal-707 75 7 b b NOUN american_scientific_journal-707 75 8 be be AUX american_scientific_journal-707 75 9 algebras algebra NOUN american_scientific_journal-707 75 10 of of ADP american_scientific_journal-707 75 11 a a DET american_scientific_journal-707 75 12 given give VERB american_scientific_journal-707 75 13 type type NOUN american_scientific_journal-707 75 14 β„±. β„±. PROPN american_scientific_journal-707 75 15 let let VERB american_scientific_journal-707 75 16 𝛼:𝐴 𝛼:𝐴 NOUN american_scientific_journal-707 75 17 β†’ β†’ SYM american_scientific_journal-707 75 18 𝐡 𝐡 NOUN american_scientific_journal-707 75 19 be be AUX american_scientific_journal-707 75 20 a a DET american_scientific_journal-707 75 21 homomorphism homomorphism NOUN american_scientific_journal-707 75 22 . . PUNCT american_scientific_journal-707 76 1 then then ADV american_scientific_journal-707 76 2 kernel kernel PROPN american_scientific_journal-707 76 3 of of ADP american_scientific_journal-707 76 4 𝛼 𝛼 PROPN american_scientific_journal-707 76 5 defined define VERB american_scientific_journal-707 76 6 by;ker𝛼 by;ker𝛼 X american_scientific_journal-707 77 1 = = SYM american_scientific_journal-707 78 1 { { PUNCT american_scientific_journal-707 78 2 ( ( PUNCT american_scientific_journal-707 78 3 π‘Ž π‘Ž X american_scientific_journal-707 78 4 , , PUNCT american_scientific_journal-707 78 5 𝑏 𝑏 NOUN american_scientific_journal-707 78 6 ) ) PUNCT american_scientific_journal-707 78 7 ∈ ∈ PROPN american_scientific_journal-707 78 8 𝐴 𝐴 PROPN american_scientific_journal-707 78 9 Γ— Γ— PROPN american_scientific_journal-707 78 10 𝐴 𝐴 PROPN american_scientific_journal-707 78 11 ∢ ∢ PROPN american_scientific_journal-707 78 12 𝛼(π‘Ž 𝛼(π‘Ž NUM american_scientific_journal-707 78 13 ) ) PUNCT american_scientific_journal-707 78 14 = = SYM american_scientific_journal-707 78 15 𝛼(𝑏 𝛼(𝑏 PROPN american_scientific_journal-707 78 16 ) ) PUNCT american_scientific_journal-707 78 17 } } PUNCT american_scientific_journal-707 78 18 is be AUX american_scientific_journal-707 78 19 a a DET american_scientific_journal-707 78 20 congruence congruence NOUN american_scientific_journal-707 78 21 relation relation NOUN american_scientific_journal-707 78 22 on on ADP american_scientific_journal-707 78 23 a. a. NOUN american_scientific_journal-707 79 1 [ [ X american_scientific_journal-707 79 2 1 1 NUM american_scientific_journal-707 79 3 , , PUNCT american_scientific_journal-707 79 4 2 2 NUM american_scientific_journal-707 79 5 , , PUNCT american_scientific_journal-707 79 6 6 6 NUM american_scientific_journal-707 79 7 ] ] SYM american_scientific_journal-707 79 8 64 64 NUM american_scientific_journal-707 79 9 american american ADJ american_scientific_journal-707 79 10 scientific scientific ADJ american_scientific_journal-707 79 11 research research NOUN american_scientific_journal-707 79 12 journal journal NOUN american_scientific_journal-707 79 13 for for ADP american_scientific_journal-707 79 14 engineering engineering NOUN american_scientific_journal-707 79 15 , , PUNCT american_scientific_journal-707 79 16 technology technology NOUN american_scientific_journal-707 79 17 , , PUNCT american_scientific_journal-707 79 18 and and CCONJ american_scientific_journal-707 79 19 sciences science NOUN american_scientific_journal-707 79 20 ( ( PUNCT american_scientific_journal-707 79 21 asrjets asrjet NOUN american_scientific_journal-707 79 22 ) ) PUNCT american_scientific_journal-707 79 23 ( ( PUNCT american_scientific_journal-707 79 24 2014 2014 NUM american_scientific_journal-707 79 25 ) ) PUNCT american_scientific_journal-707 79 26 volume volume NOUN american_scientific_journal-707 79 27 10 10 NUM american_scientific_journal-707 79 28 , , PUNCT american_scientific_journal-707 79 29 no no DET american_scientific_journal-707 79 30 1 1 NUM american_scientific_journal-707 79 31 , , PUNCT american_scientific_journal-707 79 32 pp pp ADV american_scientific_journal-707 79 33 61 61 NUM american_scientific_journal-707 79 34 - - SYM american_scientific_journal-707 79 35 73 73 NUM american_scientific_journal-707 79 36 proof proof NOUN american_scientific_journal-707 79 37 : : PUNCT american_scientific_journal-707 79 38 put put VERB american_scientific_journal-707 79 39 πœƒ πœƒ NOUN american_scientific_journal-707 80 1 = = SYM american_scientific_journal-707 81 1 π‘˜π‘’π‘Ÿπ›Ό. π‘˜π‘’π‘Ÿπ›Ό. NOUN american_scientific_journal-707 82 1 then then ADV american_scientific_journal-707 82 2 it it PRON american_scientific_journal-707 82 3 is be AUX american_scientific_journal-707 82 4 clear clear ADJ american_scientific_journal-707 82 5 that that SCONJ american_scientific_journal-707 82 6 , , PUNCT american_scientific_journal-707 82 7 ΞΈ ΞΈ PROPN american_scientific_journal-707 82 8 is be AUX american_scientific_journal-707 82 9 an an DET american_scientific_journal-707 82 10 equivalence equivalence NOUN american_scientific_journal-707 82 11 relation relation NOUN american_scientific_journal-707 82 12 on on ADP american_scientific_journal-707 82 13 a. a. NOUN american_scientific_journal-707 82 14 now now ADV american_scientific_journal-707 82 15 , , PUNCT american_scientific_journal-707 82 16 for for ADP american_scientific_journal-707 82 17 any any DET american_scientific_journal-707 82 18 π‘Ž1 π‘Ž1 NOUN american_scientific_journal-707 82 19 , , PUNCT american_scientific_journal-707 82 20 π‘Ž2 π‘Ž2 PROPN american_scientific_journal-707 82 21 , , PUNCT american_scientific_journal-707 82 22 … … PUNCT american_scientific_journal-707 82 23 . . PUNCT american_scientific_journal-707 82 24 . . PUNCT american_scientific_journal-707 83 1 π‘Žπ‘› π‘Žπ‘› PRON american_scientific_journal-707 83 2 , , PUNCT american_scientific_journal-707 83 3 𝑏1 𝑏1 NOUN american_scientific_journal-707 83 4 , , PUNCT american_scientific_journal-707 83 5 𝑏2 𝑏2 NOUN american_scientific_journal-707 83 6 , , PUNCT american_scientific_journal-707 83 7 … … PUNCT american_scientific_journal-707 83 8 . . PUNCT american_scientific_journal-707 83 9 . . PUNCT american_scientific_journal-707 84 1 𝑏𝑛 𝑏𝑛 ADP american_scientific_journal-707 84 2 ∈ ∈ PROPN american_scientific_journal-707 84 3 𝐴 𝐴 PROPN american_scientific_journal-707 84 4 , , PUNCT american_scientific_journal-707 84 5 let let VERB american_scientific_journal-707 84 6 οΏ½ οΏ½ PRON american_scientific_journal-707 84 7 π‘Ž1,𝑏1 π‘Ž1,𝑏1 PROPN american_scientific_journal-707 84 8 οΏ½ οΏ½ PROPN american_scientific_journal-707 84 9 , , PUNCT american_scientific_journal-707 84 10 οΏ½ οΏ½ PROPN american_scientific_journal-707 84 11 π‘Ž2,𝑏2 π‘Ž2,𝑏2 PROPN american_scientific_journal-707 84 12 οΏ½ οΏ½ PROPN american_scientific_journal-707 84 13 , , PUNCT american_scientific_journal-707 84 14 … … PUNCT american_scientific_journal-707 84 15 … … PUNCT american_scientific_journal-707 84 16 . . PUNCT american_scientific_journal-707 84 17 . . PUNCT american_scientific_journal-707 85 1 , , PUNCT american_scientific_journal-707 85 2 οΏ½ οΏ½ PROPN american_scientific_journal-707 85 3 π‘Žπ‘›,𝑏𝑛 π‘Žπ‘›,𝑏𝑛 NOUN american_scientific_journal-707 85 4 οΏ½ οΏ½ PROPN american_scientific_journal-707 85 5 ∈ ∈ PROPN american_scientific_journal-707 85 6 πœƒ πœƒ NOUN american_scientific_journal-707 85 7 and and CCONJ american_scientific_journal-707 85 8 let let VERB american_scientific_journal-707 85 9 𝑓 𝑓 PRON american_scientific_journal-707 85 10 ∈ ∈ NOUN american_scientific_journal-707 85 11 β„± β„± PROPN american_scientific_journal-707 85 12 be be VERB american_scientific_journal-707 85 13 an an DET american_scientific_journal-707 85 14 𝑛ary 𝑛ary ADJ american_scientific_journal-707 85 15 operation operation NOUN american_scientific_journal-707 85 16 symbol symbol NOUN american_scientific_journal-707 85 17 and and CCONJ american_scientific_journal-707 85 18 𝑛 𝑛 ADJ american_scientific_journal-707 85 19 > > X american_scientific_journal-707 85 20 0 0 NUM american_scientific_journal-707 85 21 , , PUNCT american_scientific_journal-707 85 22 then then ADV american_scientific_journal-707 85 23 𝛼(π‘Žπ‘– 𝛼(π‘Žπ‘– PROPN american_scientific_journal-707 85 24 ) ) PUNCT american_scientific_journal-707 85 25 = = SYM american_scientific_journal-707 85 26 𝛼(𝑏𝑖 𝛼(𝑏𝑖 NOUN american_scientific_journal-707 85 27 ) ) PUNCT american_scientific_journal-707 85 28 for for ADP american_scientific_journal-707 85 29 all all PRON american_scientific_journal-707 85 30 1 1 NUM american_scientific_journal-707 85 31 ≀ ≀ NUM american_scientific_journal-707 85 32 𝑖 𝑖 PRON american_scientific_journal-707 85 33 ≀ ≀ NOUN american_scientific_journal-707 85 34 𝑛. 𝑛. NOUN american_scientific_journal-707 85 35 therefore therefore ADV american_scientific_journal-707 85 36 , , PUNCT american_scientific_journal-707 85 37 𝛼 𝛼 PROPN american_scientific_journal-707 85 38 ( ( PUNCT american_scientific_journal-707 85 39 𝑓𝐴(π‘Ž1 𝑓𝐴(π‘Ž1 PROPN american_scientific_journal-707 85 40 , , PUNCT american_scientific_journal-707 85 41 π‘Ž2 π‘Ž2 PROPN american_scientific_journal-707 85 42 , , PUNCT american_scientific_journal-707 85 43 … … PUNCT american_scientific_journal-707 85 44 . . PUNCT american_scientific_journal-707 85 45 . . PUNCT american_scientific_journal-707 86 1 π‘Žπ‘› π‘Žπ‘› X american_scientific_journal-707 86 2 ) ) PUNCT american_scientific_journal-707 86 3 ) ) PUNCT american_scientific_journal-707 87 1 = = SYM american_scientific_journal-707 87 2 𝑓𝐡(𝛼(π‘Ž1),𝛼(π‘Ž2 𝑓𝐡(𝛼(π‘Ž1),𝛼(π‘Ž2 PROPN american_scientific_journal-707 87 3 ) ) PUNCT american_scientific_journal-707 87 4 , , PUNCT american_scientific_journal-707 87 5 … … PUNCT american_scientific_journal-707 87 6 … … PUNCT american_scientific_journal-707 87 7 , , PUNCT american_scientific_journal-707 87 8 𝛼(π‘Žπ‘› 𝛼(π‘Žπ‘› NOUN american_scientific_journal-707 87 9 ) ) PUNCT american_scientific_journal-707 87 10 ) ) PUNCT american_scientific_journal-707 88 1 ( ( PUNCT american_scientific_journal-707 88 2 ∡ ∡ X american_scientific_journal-707 88 3 𝛼is 𝛼is PROPN american_scientific_journal-707 88 4 a a DET american_scientific_journal-707 88 5 homomorphism homomorphism NOUN american_scientific_journal-707 88 6 ) ) PUNCT american_scientific_journal-707 88 7 = = SYM american_scientific_journal-707 88 8 𝑓𝐡(𝛼(𝑏1),𝛼(𝑏2 𝑓𝐡(𝛼(𝑏1),𝛼(𝑏2 PROPN american_scientific_journal-707 88 9 ) ) PUNCT american_scientific_journal-707 88 10 , , PUNCT american_scientific_journal-707 88 11 … … PUNCT american_scientific_journal-707 88 12 … … PUNCT american_scientific_journal-707 88 13 , , PUNCT american_scientific_journal-707 88 14 𝛼(𝑏𝑛 𝛼(𝑏𝑛 NOUN american_scientific_journal-707 88 15 ) ) PUNCT american_scientific_journal-707 88 16 ) ) PUNCT american_scientific_journal-707 89 1 = = PUNCT american_scientific_journal-707 89 2 𝛼(𝑓𝐴(𝑏1 𝛼(𝑓𝐴(𝑏1 NOUN american_scientific_journal-707 89 3 , , PUNCT american_scientific_journal-707 89 4 𝑏2 𝑏2 PROPN american_scientific_journal-707 89 5 , , PUNCT american_scientific_journal-707 89 6 … … PUNCT american_scientific_journal-707 89 7 . . PUNCT american_scientific_journal-707 89 8 . . PUNCT american_scientific_journal-707 90 1 𝑏𝑛 𝑏𝑛 NOUN american_scientific_journal-707 90 2 ) ) PUNCT american_scientific_journal-707 90 3 ) ) PUNCT american_scientific_journal-707 91 1 ( ( PUNCT american_scientific_journal-707 91 2 ∡ ∡ X american_scientific_journal-707 91 3 𝛼is 𝛼is PROPN american_scientific_journal-707 91 4 a a DET american_scientific_journal-707 91 5 homomorphism homomorphism NOUN american_scientific_journal-707 91 6 ) ) PUNCT american_scientific_journal-707 91 7 thus thus ADV american_scientific_journal-707 91 8 , , PUNCT american_scientific_journal-707 91 9 οΏ½ οΏ½ PROPN american_scientific_journal-707 91 10 𝑓𝐴(π‘Ž1 𝑓𝐴(π‘Ž1 PROPN american_scientific_journal-707 91 11 , , PUNCT american_scientific_journal-707 91 12 π‘Ž2 π‘Ž2 PROPN american_scientific_journal-707 91 13 , , PUNCT american_scientific_journal-707 91 14 … … PUNCT american_scientific_journal-707 91 15 . . PUNCT american_scientific_journal-707 91 16 . . PUNCT american_scientific_journal-707 92 1 π‘Žπ‘› π‘Žπ‘› PROPN american_scientific_journal-707 92 2 ) ) PUNCT american_scientific_journal-707 92 3 , , PUNCT american_scientific_journal-707 92 4 𝑓𝐴(𝑏1 𝑓𝐴(𝑏1 PROPN american_scientific_journal-707 92 5 , , PUNCT american_scientific_journal-707 92 6 𝑏2 𝑏2 PROPN american_scientific_journal-707 92 7 , , PUNCT american_scientific_journal-707 92 8 … … PUNCT american_scientific_journal-707 92 9 . . PUNCT american_scientific_journal-707 92 10 . . PUNCT american_scientific_journal-707 93 1 𝑏𝑛 𝑏𝑛 ADP american_scientific_journal-707 93 2 ) ) PUNCT american_scientific_journal-707 93 3 οΏ½ οΏ½ PROPN american_scientific_journal-707 93 4 ∈ ∈ PROPN american_scientific_journal-707 93 5 ker𝛼 ker𝛼 NOUN american_scientific_journal-707 93 6 = = PROPN american_scientific_journal-707 93 7 πœƒ πœƒ PROPN american_scientific_journal-707 93 8 and and CCONJ american_scientific_journal-707 93 9 hence hence ADV american_scientific_journal-707 93 10 , , PUNCT american_scientific_journal-707 93 11 ker𝛼 ker𝛼 PROPN american_scientific_journal-707 93 12 is be AUX american_scientific_journal-707 93 13 a a DET american_scientific_journal-707 93 14 congruence congruence NOUN american_scientific_journal-707 93 15 relation relation NOUN american_scientific_journal-707 93 16 on on ADP american_scientific_journal-707 93 17 a. a. NOUN american_scientific_journal-707 93 18 theorem theorem NOUN american_scientific_journal-707 93 19 3.3.2 3.3.2 NUM american_scientific_journal-707 93 20 : : PUNCT american_scientific_journal-707 93 21 let let VERB american_scientific_journal-707 93 22 a a PRON american_scientific_journal-707 93 23 be be AUX american_scientific_journal-707 93 24 an an DET american_scientific_journal-707 93 25 algebra algebra NOUN american_scientific_journal-707 93 26 of of ADP american_scientific_journal-707 93 27 typeβ„± typeβ„± PROPN american_scientific_journal-707 93 28 and and CCONJ american_scientific_journal-707 93 29 πœƒ πœƒ PRON american_scientific_journal-707 93 30 be be AUX american_scientific_journal-707 93 31 a a DET american_scientific_journal-707 93 32 congruence congruence NOUN american_scientific_journal-707 93 33 relation relation NOUN american_scientific_journal-707 93 34 on on ADP american_scientific_journal-707 93 35 a. a. NOUN american_scientific_journal-707 93 36 then then ADV american_scientific_journal-707 93 37 there there PRON american_scientific_journal-707 93 38 exists exist VERB american_scientific_journal-707 93 39 an an DET american_scientific_journal-707 93 40 algebra algebra NOUN american_scientific_journal-707 93 41 𝐡 𝐡 NOUN american_scientific_journal-707 93 42 of of ADP american_scientific_journal-707 93 43 type type NOUN american_scientific_journal-707 93 44 β„± β„± PROPN american_scientific_journal-707 93 45 and and CCONJ american_scientific_journal-707 93 46 a a DET american_scientific_journal-707 93 47 homomorphism homomorphism NOUN american_scientific_journal-707 93 48 𝛼:𝐴 𝛼:𝐴 NOUN american_scientific_journal-707 93 49 β†’ β†’ SYM american_scientific_journal-707 93 50 𝐡 𝐡 NOUN american_scientific_journal-707 93 51 such such ADJ american_scientific_journal-707 94 1 that that DET american_scientific_journal-707 94 2 πœƒ πœƒ NOUN american_scientific_journal-707 94 3 = = NOUN american_scientific_journal-707 94 4 ker𝛼. ker𝛼. ADJ american_scientific_journal-707 95 1 [ [ X american_scientific_journal-707 95 2 2 2 NUM american_scientific_journal-707 95 3 ] ] PUNCT american_scientific_journal-707 95 4 proof proof NOUN american_scientific_journal-707 95 5 : : PUNCT american_scientific_journal-707 95 6 put put VERB american_scientific_journal-707 95 7 πœƒ πœƒ NUM american_scientific_journal-707 95 8 = = SYM american_scientific_journal-707 95 9 π‘˜π‘’π‘Ÿπ›Ό π‘˜π‘’π‘Ÿπ›Ό PROPN american_scientific_journal-707 95 10 = = SYM american_scientific_journal-707 95 11 { { PUNCT american_scientific_journal-707 95 12 ( ( PUNCT american_scientific_journal-707 95 13 π‘Ž π‘Ž X american_scientific_journal-707 95 14 , , PUNCT american_scientific_journal-707 95 15 𝑏 𝑏 NOUN american_scientific_journal-707 95 16 ) ) PUNCT american_scientific_journal-707 95 17 ∈ ∈ PROPN american_scientific_journal-707 95 18 𝐴 𝐴 PROPN american_scientific_journal-707 95 19 Γ— Γ— PROPN american_scientific_journal-707 95 20 𝐴 𝐴 PROPN american_scientific_journal-707 95 21 ∢ ∢ PROPN american_scientific_journal-707 95 22 𝛼(π‘Ž 𝛼(π‘Ž NUM american_scientific_journal-707 95 23 ) ) PUNCT american_scientific_journal-707 96 1 = = SYM american_scientific_journal-707 96 2 𝛼(𝑏 𝛼(𝑏 PROPN american_scientific_journal-707 96 3 ) ) PUNCT american_scientific_journal-707 96 4 } } PUNCT american_scientific_journal-707 96 5 and and CCONJ american_scientific_journal-707 96 6 let let VERB american_scientific_journal-707 96 7 𝐴 𝐴 PROPN american_scientific_journal-707 96 8 πœƒ πœƒ PRON american_scientific_journal-707 96 9 οΏ½ οΏ½ PROPN american_scientific_journal-707 96 10 = = SYM american_scientific_journal-707 96 11 { { PUNCT american_scientific_journal-707 96 12 πœƒ(π‘Ž πœƒ(π‘Ž NOUN american_scientific_journal-707 96 13 ): ): PUNCT american_scientific_journal-707 96 14 π‘Ž π‘Ž PROPN american_scientific_journal-707 96 15 ∈ ∈ PROPN american_scientific_journal-707 96 16 𝐴 𝐴 PROPN american_scientific_journal-707 96 17 } } PUNCT american_scientific_journal-707 96 18 be be AUX american_scientific_journal-707 96 19 the the DET american_scientific_journal-707 96 20 quotient quotient NOUN american_scientific_journal-707 96 21 of of ADP american_scientific_journal-707 96 22 a a DET american_scientific_journal-707 96 23 over over ADP american_scientific_journal-707 96 24 πœƒ πœƒ NOUN american_scientific_journal-707 96 25 , , PUNCT american_scientific_journal-707 96 26 where where SCONJ american_scientific_journal-707 96 27 πœƒ πœƒ X american_scientific_journal-707 96 28 ( ( PUNCT american_scientific_journal-707 96 29 π‘Ž π‘Ž NOUN american_scientific_journal-707 96 30 ) ) PUNCT american_scientific_journal-707 96 31 = = SYM american_scientific_journal-707 96 32 { { PUNCT american_scientific_journal-707 96 33 𝑏 𝑏 PROPN american_scientific_journal-707 96 34 ∈ ∈ PROPN american_scientific_journal-707 96 35 𝐴 𝐴 PROPN american_scientific_journal-707 96 36 : : PUNCT american_scientific_journal-707 96 37 ( ( PUNCT american_scientific_journal-707 96 38 π‘Ž π‘Ž X american_scientific_journal-707 96 39 , , PUNCT american_scientific_journal-707 96 40 𝑏 𝑏 NOUN american_scientific_journal-707 96 41 ) ) PUNCT american_scientific_journal-707 96 42 ∈ ∈ PROPN american_scientific_journal-707 96 43 πœƒ πœƒ X american_scientific_journal-707 96 44 } } PUNCT american_scientific_journal-707 96 45 . . PUNCT american_scientific_journal-707 97 1 then then ADV american_scientific_journal-707 97 2 𝐴 𝐴 PROPN american_scientific_journal-707 97 3 πœƒ πœƒ PRON american_scientific_journal-707 97 4 οΏ½ οΏ½ PROPN american_scientific_journal-707 97 5 is be AUX american_scientific_journal-707 97 6 an an DET american_scientific_journal-707 97 7 algebra algebra NOUN american_scientific_journal-707 97 8 of of ADP american_scientific_journal-707 97 9 the the DET american_scientific_journal-707 97 10 given give VERB american_scientific_journal-707 97 11 type type NOUN american_scientific_journal-707 97 12 β„±. β„±. PROPN american_scientific_journal-707 97 13 now now ADV american_scientific_journal-707 97 14 consider consider VERB american_scientific_journal-707 97 15 a a DET american_scientific_journal-707 97 16 natural natural ADJ american_scientific_journal-707 97 17 epimorphism𝛼:𝐴 epimorphism𝛼:𝐴 NOUN american_scientific_journal-707 97 18 β†’ β†’ SYM american_scientific_journal-707 97 19 𝐴 𝐴 PROPN american_scientific_journal-707 97 20 πœƒ πœƒ NUM american_scientific_journal-707 97 21 οΏ½ οΏ½ PROPN american_scientific_journal-707 97 22 defined define VERB american_scientific_journal-707 97 23 by by ADP american_scientific_journal-707 97 24 𝛼(π‘Ž 𝛼(π‘Ž NOUN american_scientific_journal-707 97 25 ) ) PUNCT american_scientific_journal-707 97 26 = = PUNCT american_scientific_journal-707 97 27 πœƒ(π‘Ž πœƒ(π‘Ž NOUN american_scientific_journal-707 97 28 ) ) PUNCT american_scientific_journal-707 97 29 for for ADP american_scientific_journal-707 97 30 all all PRON american_scientific_journal-707 97 31 π‘Ž π‘Ž PROPN american_scientific_journal-707 97 32 ∈ ∈ NOUN american_scientific_journal-707 97 33 𝐴. 𝐴. NOUN american_scientific_journal-707 97 34 then then ADV american_scientific_journal-707 97 35 , , PUNCT american_scientific_journal-707 97 36 π‘˜π‘’π‘Ÿ π‘˜π‘’π‘Ÿ NOUN american_scientific_journal-707 97 37 𝛼 𝛼 NOUN american_scientific_journal-707 97 38 = = PRON american_scientific_journal-707 97 39 { { PUNCT american_scientific_journal-707 97 40 ( ( PUNCT american_scientific_journal-707 97 41 π‘Ž π‘Ž NOUN american_scientific_journal-707 97 42 , , PUNCT american_scientific_journal-707 97 43 𝑏 𝑏 NOUN american_scientific_journal-707 97 44 ) ) PUNCT american_scientific_journal-707 97 45 ∈ ∈ PROPN american_scientific_journal-707 97 46 𝐴 𝐴 PROPN american_scientific_journal-707 97 47 Γ— Γ— PROPN american_scientific_journal-707 97 48 𝐴 𝐴 PROPN american_scientific_journal-707 97 49 ∢ ∢ PROPN american_scientific_journal-707 97 50 𝛼(π‘Ž 𝛼(π‘Ž NUM american_scientific_journal-707 97 51 ) ) PUNCT american_scientific_journal-707 98 1 = = SYM american_scientific_journal-707 98 2 𝛼(𝑏 𝛼(𝑏 NOUN american_scientific_journal-707 98 3 ) ) PUNCT american_scientific_journal-707 98 4 } } PUNCT american_scientific_journal-707 98 5 = = SYM american_scientific_journal-707 98 6 { { PUNCT american_scientific_journal-707 98 7 ( ( PUNCT american_scientific_journal-707 98 8 π‘Ž π‘Ž X american_scientific_journal-707 98 9 , , PUNCT american_scientific_journal-707 98 10 𝑏 𝑏 NOUN american_scientific_journal-707 98 11 ) ) PUNCT american_scientific_journal-707 98 12 ∈ ∈ PROPN american_scientific_journal-707 98 13 𝐴 𝐴 PROPN american_scientific_journal-707 98 14 Γ— Γ— PROPN american_scientific_journal-707 98 15 𝐴 𝐴 PROPN american_scientific_journal-707 98 16 ∢ ∢ NOUN american_scientific_journal-707 98 17 πœƒ(π‘Ž πœƒ(π‘Ž VERB american_scientific_journal-707 98 18 ) ) PUNCT american_scientific_journal-707 98 19 = = SYM american_scientific_journal-707 98 20 πœƒ(𝑏 πœƒ(𝑏 PROPN american_scientific_journal-707 98 21 ) ) PUNCT american_scientific_journal-707 98 22 } } PUNCT american_scientific_journal-707 98 23 = = SYM american_scientific_journal-707 98 24 { { PUNCT american_scientific_journal-707 98 25 ( ( PUNCT american_scientific_journal-707 98 26 π‘Ž π‘Ž X american_scientific_journal-707 98 27 , , PUNCT american_scientific_journal-707 98 28 𝑏 𝑏 NOUN american_scientific_journal-707 98 29 ) ) PUNCT american_scientific_journal-707 98 30 ∈ ∈ PROPN american_scientific_journal-707 98 31 𝐴 𝐴 PROPN american_scientific_journal-707 98 32 Γ— Γ— PROPN american_scientific_journal-707 98 33 𝐴 𝐴 PROPN american_scientific_journal-707 98 34 ∢ ∢ NOUN american_scientific_journal-707 98 35 ( ( PUNCT american_scientific_journal-707 98 36 π‘Ž π‘Ž NOUN american_scientific_journal-707 98 37 , , PUNCT american_scientific_journal-707 98 38 𝑏 𝑏 NOUN american_scientific_journal-707 98 39 ) ) PUNCT american_scientific_journal-707 98 40 ∈ ∈ PROPN american_scientific_journal-707 98 41 πœƒ πœƒ NOUN american_scientific_journal-707 98 42 } } PUNCT american_scientific_journal-707 98 43 = = SYM american_scientific_journal-707 98 44 πœƒ πœƒ X american_scientific_journal-707 98 45 therefore therefore ADV american_scientific_journal-707 98 46 , , PUNCT american_scientific_journal-707 98 47 the the DET american_scientific_journal-707 98 48 required require VERB american_scientific_journal-707 98 49 algebra algebra NOUN american_scientific_journal-707 98 50 and and CCONJ american_scientific_journal-707 98 51 a a DET american_scientific_journal-707 98 52 homomorphism homomorphism NOUN american_scientific_journal-707 98 53 are be AUX american_scientific_journal-707 98 54 𝐴 𝐴 PROPN american_scientific_journal-707 98 55 πœƒ πœƒ SYM american_scientific_journal-707 98 56 οΏ½ οΏ½ PROPN american_scientific_journal-707 98 57 and and CCONJ american_scientific_journal-707 98 58 𝛼 𝛼 NOUN american_scientific_journal-707 98 59 such such ADJ american_scientific_journal-707 98 60 that that SCONJ american_scientific_journal-707 98 61 ker𝛼 ker𝛼 NOUN american_scientific_journal-707 98 62 = = PUNCT american_scientific_journal-707 98 63 πœƒ. πœƒ. PROPN american_scientific_journal-707 98 64 ∎ ∎ PROPN american_scientific_journal-707 98 65 now now ADV american_scientific_journal-707 98 66 , , PUNCT american_scientific_journal-707 98 67 from from ADP american_scientific_journal-707 98 68 the the DET american_scientific_journal-707 98 69 above above ADJ american_scientific_journal-707 98 70 two two NUM american_scientific_journal-707 98 71 theorems theorem NOUN american_scientific_journal-707 98 72 we we PRON american_scientific_journal-707 98 73 observe observe VERB american_scientific_journal-707 98 74 that that SCONJ american_scientific_journal-707 98 75 , , PUNCT american_scientific_journal-707 98 76 kernels kernel NOUN american_scientific_journal-707 98 77 of of ADP american_scientific_journal-707 98 78 homomorphisms homomorphism NOUN american_scientific_journal-707 98 79 from from ADP american_scientific_journal-707 98 80 an an DET american_scientific_journal-707 98 81 algebra algebra NOUN american_scientific_journal-707 98 82 𝐴 𝐴 PROPN american_scientific_journal-707 98 83 and and CCONJ american_scientific_journal-707 98 84 congruence congruence PROPN american_scientific_journal-707 98 85 relations relation NOUN american_scientific_journal-707 98 86 on on ADP american_scientific_journal-707 98 87 𝐴 𝐴 PROPN american_scientific_journal-707 98 88 are be AUX american_scientific_journal-707 98 89 the the DET american_scientific_journal-707 98 90 same same ADJ american_scientific_journal-707 98 91 . . PUNCT american_scientific_journal-707 99 1 4 4 X american_scientific_journal-707 99 2 . . X american_scientific_journal-707 100 1 the the DET american_scientific_journal-707 100 2 updated update VERB american_scientific_journal-707 100 3 fundamental fundamental ADJ american_scientific_journal-707 100 4 theorem theorem NOUN american_scientific_journal-707 100 5 of of ADP american_scientific_journal-707 100 6 homomomorphisms homomomorphism NOUN american_scientific_journal-707 100 7 of of ADP american_scientific_journal-707 100 8 general general ADJ american_scientific_journal-707 100 9 universal universal ADJ american_scientific_journal-707 100 10 algebras algebras PROPN american_scientific_journal-707 100 11 4.1 4.1 NUM american_scientific_journal-707 100 12 . . PUNCT american_scientific_journal-707 101 1 an an DET american_scientific_journal-707 101 2 associate associate NOUN american_scientific_journal-707 101 3 of of ADP american_scientific_journal-707 101 4 a a DET american_scientific_journal-707 101 5 homomorphism homomorphism NOUN american_scientific_journal-707 101 6 of of ADP american_scientific_journal-707 101 7 general general ADJ american_scientific_journal-707 101 8 universal universal ADJ american_scientific_journal-707 101 9 algebras algebras PROPN american_scientific_journal-707 101 10 definition definition NOUN american_scientific_journal-707 101 11 4.1.1 4.1.1 NUM american_scientific_journal-707 101 12 : : PUNCT american_scientific_journal-707 101 13 let let VERB american_scientific_journal-707 101 14 𝐴,𝐡,𝐢 𝐴,𝐡,𝐢 NOUN american_scientific_journal-707 101 15 and and CCONJ american_scientific_journal-707 101 16 𝐷 𝐷 PROPN american_scientific_journal-707 101 17 be be AUX american_scientific_journal-707 101 18 algebras algebra NOUN american_scientific_journal-707 101 19 of of ADP american_scientific_journal-707 101 20 a a DET american_scientific_journal-707 101 21 given give VERB american_scientific_journal-707 101 22 type type NOUN american_scientific_journal-707 101 23 β„±. β„±. PROPN american_scientific_journal-707 101 24 let𝑔 let𝑔 NOUN american_scientific_journal-707 101 25 : : PUNCT american_scientific_journal-707 101 26 𝐴 𝐴 PROPN american_scientific_journal-707 101 27 β†’ β†’ SYM american_scientific_journal-707 101 28 𝐡 𝐡 PROPN american_scientific_journal-707 101 29 and and CCONJ american_scientific_journal-707 101 30 β„Ž:𝐢 β„Ž:𝐢 PROPN american_scientific_journal-707 101 31 β†’ β†’ SYM american_scientific_journal-707 101 32 𝐷 𝐷 NOUN american_scientific_journal-707 101 33 be be VERB american_scientific_journal-707 101 34 65 65 NUM american_scientific_journal-707 101 35 american american ADJ american_scientific_journal-707 101 36 scientific scientific ADJ american_scientific_journal-707 101 37 research research NOUN american_scientific_journal-707 101 38 journal journal NOUN american_scientific_journal-707 101 39 for for ADP american_scientific_journal-707 101 40 engineering engineering NOUN american_scientific_journal-707 101 41 , , PUNCT american_scientific_journal-707 101 42 technology technology NOUN american_scientific_journal-707 101 43 , , PUNCT american_scientific_journal-707 101 44 and and CCONJ american_scientific_journal-707 101 45 sciences science NOUN american_scientific_journal-707 101 46 ( ( PUNCT american_scientific_journal-707 101 47 asrjets asrjet NOUN american_scientific_journal-707 101 48 ) ) PUNCT american_scientific_journal-707 101 49 ( ( PUNCT american_scientific_journal-707 101 50 2014 2014 NUM american_scientific_journal-707 101 51 ) ) PUNCT american_scientific_journal-707 101 52 volume volume NOUN american_scientific_journal-707 101 53 10 10 NUM american_scientific_journal-707 101 54 , , PUNCT american_scientific_journal-707 101 55 no no DET american_scientific_journal-707 101 56 1 1 NUM american_scientific_journal-707 101 57 , , PUNCT american_scientific_journal-707 101 58 pp pp ADV american_scientific_journal-707 101 59 61 61 NUM american_scientific_journal-707 101 60 - - SYM american_scientific_journal-707 101 61 73 73 NUM american_scientific_journal-707 101 62 homomorphisms homomorphism NOUN american_scientific_journal-707 101 63 . . PUNCT american_scientific_journal-707 102 1 then then ADV american_scientific_journal-707 102 2 𝑔is 𝑔is NOUN american_scientific_journal-707 102 3 said say VERB american_scientific_journal-707 102 4 to to PART american_scientific_journal-707 102 5 be be AUX american_scientific_journal-707 102 6 an an DET american_scientific_journal-707 102 7 associate associate NOUN american_scientific_journal-707 102 8 of of ADP american_scientific_journal-707 102 9 β„Ž β„Ž NOUN american_scientific_journal-707 102 10 , , PUNCT american_scientific_journal-707 102 11 if if SCONJ american_scientific_journal-707 102 12 there there PRON american_scientific_journal-707 102 13 exist exist VERB american_scientific_journal-707 102 14 two two NUM american_scientific_journal-707 102 15 isomorphisms isomorphism NOUN american_scientific_journal-707 102 16 𝛼 𝛼 X american_scientific_journal-707 102 17 : : PUNCT american_scientific_journal-707 102 18 𝐢 𝐢 PROPN american_scientific_journal-707 102 19 β†’ β†’ SYM american_scientific_journal-707 102 20 𝐴 𝐴 PROPN american_scientific_journal-707 102 21 and and CCONJ american_scientific_journal-707 102 22 𝛽 𝛽 NOUN american_scientific_journal-707 102 23 : : PUNCT american_scientific_journal-707 102 24 𝐷 𝐷 PROPN american_scientific_journal-707 102 25 β†’ β†’ SYM american_scientific_journal-707 102 26 𝐡such 𝐡such PROPN american_scientific_journal-707 102 27 that that SCONJ american_scientific_journal-707 102 28 , , PUNCT american_scientific_journal-707 102 29 the the DET american_scientific_journal-707 102 30 diagram diagram NOUN american_scientific_journal-707 102 31 figure figure NOUN american_scientific_journal-707 102 32 2 2 NUM american_scientific_journal-707 102 33 : : PUNCT american_scientific_journal-707 102 34 an an DET american_scientific_journal-707 102 35 associate associate NOUN american_scientific_journal-707 102 36 of of ADP american_scientific_journal-707 102 37 a a DET american_scientific_journal-707 102 38 homomorphism homomorphism NOUN american_scientific_journal-707 102 39 is be AUX american_scientific_journal-707 102 40 commutative commutative ADJ american_scientific_journal-707 102 41 ; ; PUNCT american_scientific_journal-707 102 42 in in ADP american_scientific_journal-707 102 43 the the DET american_scientific_journal-707 102 44 sense sense NOUN american_scientific_journal-707 102 45 that that SCONJ american_scientific_journal-707 102 46 , , PUNCT american_scientific_journal-707 102 47 𝑔 𝑔 PROPN american_scientific_journal-707 102 48 ∘ ∘ PROPN american_scientific_journal-707 102 49 𝛼 𝛼 NOUN american_scientific_journal-707 102 50 = = NOUN american_scientific_journal-707 102 51 π›½π‘œβ„Ž. π›½π‘œβ„Ž. NOUN american_scientific_journal-707 102 52 remark remark NOUN american_scientific_journal-707 102 53 : : PUNCT american_scientific_journal-707 102 54 following follow VERB american_scientific_journal-707 102 55 the the DET american_scientific_journal-707 102 56 above above ADJ american_scientific_journal-707 102 57 definition definition NOUN american_scientific_journal-707 102 58 we we PRON american_scientific_journal-707 102 59 can can AUX american_scientific_journal-707 102 60 observe observe VERB american_scientific_journal-707 102 61 that that SCONJ american_scientific_journal-707 102 62 the the DET american_scientific_journal-707 102 63 binary binary PROPN american_scientific_journal-707 102 64 relation relation NOUN american_scientific_journal-707 102 65 β€œ " PUNCT american_scientific_journal-707 102 66 ~ ~ NOUN american_scientific_journal-707 102 67 ” " PUNCT american_scientific_journal-707 102 68 forms form VERB american_scientific_journal-707 102 69 an an DET american_scientific_journal-707 102 70 equivalence equivalence NOUN american_scientific_journal-707 102 71 relation relation NOUN american_scientific_journal-707 102 72 on on ADP american_scientific_journal-707 102 73 the the DET american_scientific_journal-707 102 74 class class NOUN american_scientific_journal-707 102 75 of of ADP american_scientific_journal-707 102 76 all all DET american_scientific_journal-707 102 77 algebras algebra NOUN american_scientific_journal-707 102 78 of of ADP american_scientific_journal-707 102 79 a a DET american_scientific_journal-707 102 80 given give VERB american_scientific_journal-707 102 81 type type NOUN american_scientific_journal-707 102 82 . . PUNCT american_scientific_journal-707 103 1 furthermore furthermore ADV american_scientific_journal-707 103 2 , , PUNCT american_scientific_journal-707 103 3 if if SCONJ american_scientific_journal-707 103 4 any any DET american_scientific_journal-707 103 5 two two NUM american_scientific_journal-707 103 6 homomorphisms homomorphism NOUN american_scientific_journal-707 103 7 are be AUX american_scientific_journal-707 103 8 given give VERB american_scientific_journal-707 103 9 to to PART american_scientific_journal-707 103 10 be be AUX american_scientific_journal-707 103 11 an an DET american_scientific_journal-707 103 12 associate associate NOUN american_scientific_journal-707 103 13 to to ADP american_scientific_journal-707 103 14 each each DET american_scientific_journal-707 103 15 other other ADJ american_scientific_journal-707 103 16 , , PUNCT american_scientific_journal-707 103 17 then then ADV american_scientific_journal-707 103 18 we we PRON american_scientific_journal-707 103 19 can can AUX american_scientific_journal-707 103 20 draw draw VERB american_scientific_journal-707 103 21 the the DET american_scientific_journal-707 103 22 properties property NOUN american_scientific_journal-707 103 23 of of ADP american_scientific_journal-707 103 24 a a DET american_scientific_journal-707 103 25 homomorphism homomorphism NOUN american_scientific_journal-707 103 26 like like ADP american_scientific_journal-707 103 27 injectivity injectivity NOUN american_scientific_journal-707 103 28 , , PUNCT american_scientific_journal-707 103 29 surjectivity surjectivity NOUN american_scientific_journal-707 103 30 and and CCONJ american_scientific_journal-707 103 31 bijectivity bijectivity NOUN american_scientific_journal-707 103 32 which which PRON american_scientific_journal-707 103 33 is be AUX american_scientific_journal-707 103 34 held hold VERB american_scientific_journal-707 103 35 in in ADP american_scientific_journal-707 103 36 one one NUM american_scientific_journal-707 103 37 of of ADP american_scientific_journal-707 103 38 the the DET american_scientific_journal-707 103 39 two two NUM american_scientific_journal-707 103 40 associate associate NOUN american_scientific_journal-707 103 41 homomorphisms homomorphism NOUN american_scientific_journal-707 103 42 to to ADP american_scientific_journal-707 103 43 the the DET american_scientific_journal-707 103 44 other other ADJ american_scientific_journal-707 103 45 . . PUNCT american_scientific_journal-707 104 1 4.2 4.2 NUM american_scientific_journal-707 104 2 . . PUNCT american_scientific_journal-707 105 1 the the DET american_scientific_journal-707 105 2 updated update VERB american_scientific_journal-707 105 3 fundamental fundamental ADJ american_scientific_journal-707 105 4 theorem theorem NOUN american_scientific_journal-707 105 5 of of ADP american_scientific_journal-707 105 6 homomorphisms homomorphism NOUN american_scientific_journal-707 105 7 of of ADP american_scientific_journal-707 105 8 algebras algebras PROPN american_scientific_journal-707 105 9 theorem theorem VERB american_scientific_journal-707 105 10 4.2.1(the 4.2.1(the DET american_scientific_journal-707 105 11 updated update VERB american_scientific_journal-707 105 12 fundamental fundamental ADJ american_scientific_journal-707 105 13 theorem theorem NOUN american_scientific_journal-707 105 14 of of ADP american_scientific_journal-707 105 15 homomorphisms homomorphism NOUN american_scientific_journal-707 105 16 of of ADP american_scientific_journal-707 105 17 algebras algebra NOUN american_scientific_journal-707 105 18 ): ): PUNCT american_scientific_journal-707 105 19 let let VERB american_scientific_journal-707 105 20 𝐴 𝐴 PROPN american_scientific_journal-707 105 21 and and CCONJ american_scientific_journal-707 105 22 𝐡 𝐡 NOUN american_scientific_journal-707 105 23 be be AUX american_scientific_journal-707 105 24 algebras algebra NOUN american_scientific_journal-707 105 25 of of ADP american_scientific_journal-707 105 26 a a DET american_scientific_journal-707 105 27 given give VERB american_scientific_journal-707 105 28 type type NOUN american_scientific_journal-707 105 29 β„± β„± PROPN american_scientific_journal-707 105 30 and and CCONJ american_scientific_journal-707 105 31 πœ‹ πœ‹ NOUN american_scientific_journal-707 105 32 : : PUNCT american_scientific_journal-707 105 33 𝐴 𝐴 PROPN american_scientific_journal-707 105 34 β†’ β†’ SYM american_scientific_journal-707 105 35 𝐡 𝐡 NOUN american_scientific_journal-707 105 36 be be AUX american_scientific_journal-707 105 37 a a DET american_scientific_journal-707 105 38 homomorphism homomorphism NOUN american_scientific_journal-707 105 39 . . PUNCT american_scientific_journal-707 106 1 then then ADV american_scientific_journal-707 106 2 , , PUNCT american_scientific_journal-707 106 3 1 1 X american_scientific_journal-707 106 4 . . X american_scientific_journal-707 106 5 there there PRON american_scientific_journal-707 106 6 exists exist VERB american_scientific_journal-707 106 7 an an DET american_scientific_journal-707 106 8 algebra algebra NOUN american_scientific_journal-707 106 9 𝑋 𝑋 NOUN american_scientific_journal-707 106 10 of of ADP american_scientific_journal-707 106 11 the the DET american_scientific_journal-707 106 12 given give VERB american_scientific_journal-707 106 13 type type NOUN american_scientific_journal-707 106 14 β„± β„± PROPN american_scientific_journal-707 106 15 , , PUNCT american_scientific_journal-707 106 16 an an DET american_scientific_journal-707 106 17 epimorphism epimorphism NOUN american_scientific_journal-707 106 18 β„Ž:𝐴 β„Ž:𝐴 PROPN american_scientific_journal-707 106 19 β†’ β†’ PUNCT american_scientific_journal-707 106 20 𝑋 𝑋 PROPN american_scientific_journal-707 106 21 and and CCONJ american_scientific_journal-707 106 22 a a DET american_scientific_journal-707 106 23 monomorphism monomorphism NOUN american_scientific_journal-707 106 24 𝑔:𝑋 𝑔:𝑋 PROPN american_scientific_journal-707 106 25 β†’ β†’ SYM american_scientific_journal-707 106 26 𝐡 𝐡 NOUN american_scientific_journal-707 106 27 such such ADJ american_scientific_journal-707 106 28 that that PRON american_scientific_journal-707 106 29 ; ; PUNCT american_scientific_journal-707 106 30 πœ‹ πœ‹ NOUN american_scientific_journal-707 106 31 = = PUNCT american_scientific_journal-707 106 32 𝑔 𝑔 NOUN american_scientific_journal-707 106 33 ∘ ∘ NUM american_scientific_journal-707 106 34 β„Ž β„Ž PROPN american_scientific_journal-707 106 35 ( ( PUNCT american_scientific_journal-707 106 36 𝑖. 𝑖. PROPN american_scientific_journal-707 106 37 𝑒. 𝑒. PROPN american_scientific_journal-707 107 1 πœ‹ πœ‹ X american_scientific_journal-707 107 2 ( ( PUNCT american_scientific_journal-707 107 3 π‘Ž π‘Ž NOUN american_scientific_journal-707 107 4 ) ) PUNCT american_scientific_journal-707 107 5 = = SYM american_scientific_journal-707 108 1 𝑔 𝑔 PROPN american_scientific_journal-707 108 2 ∘ ∘ X american_scientific_journal-707 108 3 β„Ž(π‘Ž)for β„Ž(π‘Ž)for ADP american_scientific_journal-707 108 4 all all PRON american_scientific_journal-707 108 5 π‘Ž π‘Ž PRON american_scientific_journal-707 108 6 ∈ ∈ PROPN american_scientific_journal-707 108 7 𝐴 𝐴 PROPN american_scientific_journal-707 108 8 ) ) PUNCT american_scientific_journal-707 108 9 this this DET american_scientific_journal-707 108 10 property property NOUN american_scientific_journal-707 108 11 is be AUX american_scientific_journal-707 108 12 known know VERB american_scientific_journal-707 108 13 as as ADP american_scientific_journal-707 108 14 decomposition decomposition NOUN american_scientific_journal-707 108 15 of of ADP american_scientific_journal-707 108 16 homomorphisms homomorphism NOUN american_scientific_journal-707 108 17 and and CCONJ american_scientific_journal-707 108 18 2 2 NUM american_scientific_journal-707 108 19 . . PUNCT american_scientific_journal-707 109 1 this this DET american_scientific_journal-707 109 2 decomposition decomposition NOUN american_scientific_journal-707 109 3 is be AUX american_scientific_journal-707 109 4 unique unique ADJ american_scientific_journal-707 109 5 upto upto NOUN american_scientific_journal-707 109 6 associate associate NOUN american_scientific_journal-707 109 7 ; ; PUNCT american_scientific_journal-707 109 8 in in ADP american_scientific_journal-707 109 9 the the DET american_scientific_journal-707 109 10 sense sense NOUN american_scientific_journal-707 109 11 that that SCONJ american_scientific_journal-707 109 12 , , PUNCT american_scientific_journal-707 109 13 if if SCONJ american_scientific_journal-707 109 14 πœ‹ πœ‹ PRON american_scientific_journal-707 109 15 = = X american_scientific_journal-707 109 16 𝑔 𝑔 ADP american_scientific_journal-707 109 17 ∘ ∘ ADJ american_scientific_journal-707 109 18 β„Ž β„Ž NOUN american_scientific_journal-707 109 19 and and CCONJ american_scientific_journal-707 109 20 πœ‹ πœ‹ NOUN american_scientific_journal-707 109 21 = = ADJ american_scientific_journal-707 109 22 𝑔 𝑔 NOUN american_scientific_journal-707 109 23 ’ ' PUNCT american_scientific_journal-707 109 24 ∘ ∘ NUM american_scientific_journal-707 109 25 β„Ž β„Ž NOUN american_scientific_journal-707 109 26 ’ ' PUNCT american_scientific_journal-707 109 27 are be AUX american_scientific_journal-707 109 28 two two NUM american_scientific_journal-707 109 29 decompositions decomposition NOUN american_scientific_journal-707 109 30 of of ADP american_scientific_journal-707 109 31 πœ‹ πœ‹ PRON american_scientific_journal-707 109 32 where where SCONJ american_scientific_journal-707 109 33 𝑋 𝑋 NOUN american_scientific_journal-707 109 34 and and CCONJ american_scientific_journal-707 109 35 π‘Œ π‘Œ PROPN american_scientific_journal-707 109 36 are be AUX american_scientific_journal-707 109 37 algebras algebra NOUN american_scientific_journal-707 109 38 of of ADP american_scientific_journal-707 109 39 the the DET american_scientific_journal-707 109 40 given give VERB american_scientific_journal-707 109 41 type type NOUN american_scientific_journal-707 109 42 β„± β„± PROPN american_scientific_journal-707 109 43 , , PUNCT american_scientific_journal-707 109 44 β„Ž:𝐴 β„Ž:𝐴 PROPN american_scientific_journal-707 109 45 β†’ β†’ SYM american_scientific_journal-707 109 46 𝑋and 𝑋and PROPN american_scientific_journal-707 109 47 β„Žβ€²:𝐴 β„Žβ€²:𝐴 NOUN american_scientific_journal-707 109 48 β†’ β†’ SYM american_scientific_journal-707 109 49 π‘Œ π‘Œ PROPN american_scientific_journal-707 109 50 are be AUX american_scientific_journal-707 109 51 epimorphisms epimorphism NOUN american_scientific_journal-707 109 52 and and CCONJ american_scientific_journal-707 109 53 𝑔:𝑋 𝑔:𝑋 PROPN american_scientific_journal-707 109 54 β†’ β†’ SYM american_scientific_journal-707 109 55 𝐡 𝐡 NOUN american_scientific_journal-707 109 56 and and CCONJ american_scientific_journal-707 109 57 𝑔’:π‘Œ 𝑔’:π‘Œ PROPN american_scientific_journal-707 109 58 β†’ β†’ SYM american_scientific_journal-707 109 59 𝐡 𝐡 NOUN american_scientific_journal-707 109 60 are be AUX american_scientific_journal-707 109 61 monomorphisms monomorphism NOUN american_scientific_journal-707 109 62 , , PUNCT american_scientific_journal-707 109 63 then then ADV american_scientific_journal-707 109 64 𝑔~𝑔′and 𝑔~𝑔′and PROPN american_scientific_journal-707 109 65 β„Ž~β„Žβ€². β„Ž~β„Žβ€². PROPN american_scientific_journal-707 109 66 66 66 NUM american_scientific_journal-707 109 67 american american PROPN american_scientific_journal-707 109 68 scientific scientific ADJ american_scientific_journal-707 109 69 research research NOUN american_scientific_journal-707 109 70 journal journal NOUN american_scientific_journal-707 109 71 for for ADP american_scientific_journal-707 109 72 engineering engineering NOUN american_scientific_journal-707 109 73 , , PUNCT american_scientific_journal-707 109 74 technology technology NOUN american_scientific_journal-707 109 75 , , PUNCT american_scientific_journal-707 109 76 and and CCONJ american_scientific_journal-707 109 77 sciences science NOUN american_scientific_journal-707 109 78 ( ( PUNCT american_scientific_journal-707 109 79 asrjets asrjet NOUN american_scientific_journal-707 109 80 ) ) PUNCT american_scientific_journal-707 109 81 ( ( PUNCT american_scientific_journal-707 109 82 2014 2014 NUM american_scientific_journal-707 109 83 ) ) PUNCT american_scientific_journal-707 109 84 volume volume NOUN american_scientific_journal-707 109 85 10 10 NUM american_scientific_journal-707 109 86 , , PUNCT american_scientific_journal-707 109 87 no no DET american_scientific_journal-707 109 88 1 1 NUM american_scientific_journal-707 109 89 , , PUNCT american_scientific_journal-707 109 90 pp pp ADV american_scientific_journal-707 109 91 61 61 NUM american_scientific_journal-707 109 92 - - SYM american_scientific_journal-707 109 93 73 73 NUM american_scientific_journal-707 109 94 proof proof NOUN american_scientific_journal-707 109 95 : : PUNCT american_scientific_journal-707 109 96 1 1 X american_scientific_journal-707 109 97 . . X american_scientific_journal-707 109 98 put put VERB american_scientific_journal-707 109 99 πœƒ πœƒ NOUN american_scientific_journal-707 109 100 = = PUNCT american_scientific_journal-707 109 101 kerπœ‹ kerπœ‹ NOUN american_scientific_journal-707 109 102 = = SYM american_scientific_journal-707 109 103 { { PUNCT american_scientific_journal-707 109 104 ( ( PUNCT american_scientific_journal-707 109 105 π‘Ž π‘Ž X american_scientific_journal-707 109 106 , , PUNCT american_scientific_journal-707 109 107 𝑏 𝑏 NOUN american_scientific_journal-707 109 108 ) ) PUNCT american_scientific_journal-707 109 109 ∈ ∈ PROPN american_scientific_journal-707 109 110 𝐴 𝐴 PROPN american_scientific_journal-707 109 111 Γ— Γ— PROPN american_scientific_journal-707 109 112 𝐴 𝐴 PROPN american_scientific_journal-707 109 113 ∢ ∢ PROPN american_scientific_journal-707 109 114 πœ‹(π‘Ž πœ‹(π‘Ž PROPN american_scientific_journal-707 109 115 ) ) PUNCT american_scientific_journal-707 110 1 = = SYM american_scientific_journal-707 110 2 πœ‹(𝑏 πœ‹(𝑏 NOUN american_scientific_journal-707 110 3 ) ) PUNCT american_scientific_journal-707 110 4 } } PUNCT american_scientific_journal-707 110 5 . . PUNCT american_scientific_journal-707 111 1 then then ADV american_scientific_journal-707 111 2 it it PRON american_scientific_journal-707 111 3 is be AUX american_scientific_journal-707 111 4 clear clear ADJ american_scientific_journal-707 111 5 that that SCONJ american_scientific_journal-707 111 6 𝐴 𝐴 PROPN american_scientific_journal-707 111 7 πœƒ πœƒ PRON american_scientific_journal-707 111 8 οΏ½ οΏ½ PROPN american_scientific_journal-707 111 9 = = SYM american_scientific_journal-707 111 10 { { PUNCT american_scientific_journal-707 111 11 πœƒ(π‘Ž πœƒ(π‘Ž NOUN american_scientific_journal-707 111 12 ): ): PUNCT american_scientific_journal-707 111 13 π‘Ž π‘Ž PROPN american_scientific_journal-707 111 14 ∈ ∈ PROPN american_scientific_journal-707 111 15 𝐴 𝐴 PROPN american_scientific_journal-707 111 16 } } PUNCT american_scientific_journal-707 111 17 , , PUNCT american_scientific_journal-707 111 18 the the DET american_scientific_journal-707 111 19 quotient quotient NOUN american_scientific_journal-707 111 20 of of ADP american_scientific_journal-707 111 21 a a DET american_scientific_journal-707 111 22 over over ADP american_scientific_journal-707 111 23 πœƒ πœƒ PROPN american_scientific_journal-707 111 24 , , PUNCT american_scientific_journal-707 111 25 is be AUX american_scientific_journal-707 111 26 an an DET american_scientific_journal-707 111 27 algebra algebra NOUN american_scientific_journal-707 111 28 of of ADP american_scientific_journal-707 111 29 the the DET american_scientific_journal-707 111 30 given give VERB american_scientific_journal-707 111 31 type type NOUN american_scientific_journal-707 111 32 β„±. β„±. PROPN american_scientific_journal-707 111 33 now now ADV american_scientific_journal-707 111 34 consider consider VERB american_scientific_journal-707 111 35 a a DET american_scientific_journal-707 111 36 natural natural ADJ american_scientific_journal-707 111 37 epimorphism epimorphism NOUN american_scientific_journal-707 111 38 𝛼:𝐴 𝛼:𝐴 NOUN american_scientific_journal-707 111 39 β†’ β†’ SYM american_scientific_journal-707 111 40 𝐴 𝐴 PROPN american_scientific_journal-707 111 41 πœƒ πœƒ NUM american_scientific_journal-707 111 42 οΏ½ οΏ½ PROPN american_scientific_journal-707 111 43 defined define VERB american_scientific_journal-707 111 44 by by ADP american_scientific_journal-707 111 45 𝛼(π‘Ž 𝛼(π‘Ž NOUN american_scientific_journal-707 111 46 ) ) PUNCT american_scientific_journal-707 111 47 = = PUNCT american_scientific_journal-707 111 48 πœƒ(π‘Ž πœƒ(π‘Ž NOUN american_scientific_journal-707 111 49 ) ) PUNCT american_scientific_journal-707 111 50 for for ADP american_scientific_journal-707 111 51 all all PRON american_scientific_journal-707 111 52 π‘Ž π‘Ž PROPN american_scientific_journal-707 111 53 ∈ ∈ PROPN american_scientific_journal-707 111 54 𝐴 𝐴 NOUN american_scientific_journal-707 111 55 and and CCONJ american_scientific_journal-707 111 56 a a DET american_scientific_journal-707 111 57 monomorphism monomorphism NOUN american_scientific_journal-707 111 58 𝑔:𝐴 𝑔:𝐴 PROPN american_scientific_journal-707 111 59 πœƒ πœƒ NUM american_scientific_journal-707 111 60 οΏ½ οΏ½ PROPN american_scientific_journal-707 111 61 β†’ β†’ SYM american_scientific_journal-707 111 62 𝐡 𝐡 NOUN american_scientific_journal-707 111 63 defined define VERB american_scientific_journal-707 111 64 by by ADP american_scientific_journal-707 111 65 𝑔 𝑔 PROPN american_scientific_journal-707 111 66 οΏ½ οΏ½ PROPN american_scientific_journal-707 111 67 πœƒ(π‘Ž πœƒ(π‘Ž NOUN american_scientific_journal-707 111 68 ) ) PUNCT american_scientific_journal-707 111 69 οΏ½ οΏ½ PROPN american_scientific_journal-707 111 70 = = SYM american_scientific_journal-707 111 71 πœ‹(π‘Ž πœ‹(π‘Ž PROPN american_scientific_journal-707 111 72 ) ) PUNCT american_scientific_journal-707 111 73 for for ADP american_scientific_journal-707 111 74 all all PRON american_scientific_journal-707 111 75 πœƒ πœƒ PART american_scientific_journal-707 111 76 ∈ ∈ PROPN american_scientific_journal-707 111 77 𝐴 𝐴 PROPN american_scientific_journal-707 111 78 πœƒ πœƒ PROPN american_scientific_journal-707 111 79 οΏ½ οΏ½ PROPN american_scientific_journal-707 111 80 , , PUNCT american_scientific_journal-707 111 81 π‘Ž π‘Ž PROPN american_scientific_journal-707 111 82 ∈ ∈ NOUN american_scientific_journal-707 111 83 𝐴. 𝐴. NOUN american_scientific_journal-707 111 84 then then ADV american_scientific_journal-707 111 85 , , PUNCT american_scientific_journal-707 111 86 for for SCONJ american_scientific_journal-707 111 87 any any DET american_scientific_journal-707 111 88 π‘Ž π‘Ž PROPN american_scientific_journal-707 111 89 ∈ ∈ PROPN american_scientific_journal-707 111 90 𝐴 𝐴 PROPN american_scientific_journal-707 111 91 consider consider VERB american_scientific_journal-707 111 92 , , PUNCT american_scientific_journal-707 111 93 𝑔 𝑔 PROPN american_scientific_journal-707 111 94 ∘ ∘ PROPN american_scientific_journal-707 111 95 β„Ž(π‘Ž β„Ž(π‘Ž PROPN american_scientific_journal-707 111 96 ) ) PUNCT american_scientific_journal-707 111 97 = = PUNCT american_scientific_journal-707 111 98 𝑔 𝑔 NOUN american_scientific_journal-707 111 99 οΏ½ οΏ½ PROPN american_scientific_journal-707 111 100 β„Ž(π‘Ž β„Ž(π‘Ž NOUN american_scientific_journal-707 111 101 ) ) PUNCT american_scientific_journal-707 111 102 οΏ½ οΏ½ PROPN american_scientific_journal-707 111 103 = = SYM american_scientific_journal-707 111 104 𝑔 𝑔 NOUN american_scientific_journal-707 111 105 οΏ½ οΏ½ NOUN american_scientific_journal-707 111 106 πœƒ(π‘Ž πœƒ(π‘Ž NOUN american_scientific_journal-707 111 107 ) ) PUNCT american_scientific_journal-707 111 108 οΏ½ οΏ½ PROPN american_scientific_journal-707 111 109 ( ( PUNCT american_scientific_journal-707 111 110 by by ADP american_scientific_journal-707 111 111 the the DET american_scientific_journal-707 111 112 definition definition NOUN american_scientific_journal-707 111 113 of of ADP american_scientific_journal-707 111 114 β„Ž β„Ž PROPN american_scientific_journal-707 111 115 ) ) PUNCT american_scientific_journal-707 111 116 = = SYM american_scientific_journal-707 111 117 πœ‹(π‘Ž πœ‹(π‘Ž PROPN american_scientific_journal-707 111 118 ) ) PUNCT american_scientific_journal-707 111 119 ( ( PUNCT american_scientific_journal-707 111 120 by by ADP american_scientific_journal-707 111 121 the the DET american_scientific_journal-707 111 122 definition definition NOUN american_scientific_journal-707 111 123 of of ADP american_scientific_journal-707 111 124 𝑔 𝑔 NOUN american_scientific_journal-707 111 125 ) ) PUNCT american_scientific_journal-707 111 126 thus,𝑔 thus,𝑔 PUNCT american_scientific_journal-707 111 127 ∘ ∘ NOUN american_scientific_journal-707 111 128 β„Ž β„Ž X american_scientific_journal-707 112 1 = = PUNCT american_scientific_journal-707 112 2 πœ‹ πœ‹ NOUN american_scientific_journal-707 112 3 is be AUX american_scientific_journal-707 112 4 the the DET american_scientific_journal-707 112 5 required require VERB american_scientific_journal-707 112 6 decomposition decomposition NOUN american_scientific_journal-707 112 7 of of ADP american_scientific_journal-707 112 8 πœ‹. πœ‹. ADJ american_scientific_journal-707 112 9 2 2 NUM american_scientific_journal-707 112 10 . . PUNCT american_scientific_journal-707 113 1 now now ADV american_scientific_journal-707 113 2 we we PRON american_scientific_journal-707 113 3 prove prove VERB american_scientific_journal-707 113 4 the the DET american_scientific_journal-707 113 5 uniqueness uniqueness NOUN american_scientific_journal-707 113 6 ( ( PUNCT american_scientific_journal-707 113 7 upto upto NOUN american_scientific_journal-707 113 8 associate associate NOUN american_scientific_journal-707 113 9 ) ) PUNCT american_scientific_journal-707 113 10 of of ADP american_scientific_journal-707 113 11 the the DET american_scientific_journal-707 113 12 above above ADJ american_scientific_journal-707 113 13 decomposition.suppose decomposition.suppose NUM american_scientific_journal-707 113 14 that that PRON american_scientific_journal-707 113 15 πœ‹ πœ‹ NOUN american_scientific_journal-707 113 16 = = NOUN american_scientific_journal-707 113 17 𝑔 𝑔 NOUN american_scientific_journal-707 113 18 ∘ ∘ ADJ american_scientific_journal-707 113 19 β„Ž β„Ž NOUN american_scientific_journal-707 113 20 and and CCONJ american_scientific_journal-707 113 21 πœ‹ πœ‹ NOUN american_scientific_journal-707 113 22 = = ADJ american_scientific_journal-707 113 23 𝑔 𝑔 NOUN american_scientific_journal-707 113 24 ’ ' PUNCT american_scientific_journal-707 113 25 ∘ ∘ NUM american_scientific_journal-707 113 26 β„Ž β„Ž NOUN american_scientific_journal-707 113 27 ’ ' PUNCT american_scientific_journal-707 113 28 are be AUX american_scientific_journal-707 113 29 two two NUM american_scientific_journal-707 113 30 decompositions decomposition NOUN american_scientific_journal-707 113 31 of of ADP american_scientific_journal-707 113 32 πœ‹ πœ‹ PRON american_scientific_journal-707 113 33 where where SCONJ american_scientific_journal-707 113 34 𝑋 𝑋 NOUN american_scientific_journal-707 113 35 and and CCONJ american_scientific_journal-707 113 36 π‘Œ π‘Œ PROPN american_scientific_journal-707 113 37 are be AUX american_scientific_journal-707 113 38 algebras algebra NOUN american_scientific_journal-707 113 39 of of ADP american_scientific_journal-707 113 40 the the DET american_scientific_journal-707 113 41 given give VERB american_scientific_journal-707 113 42 type type NOUN american_scientific_journal-707 113 43 β„± β„± PROPN american_scientific_journal-707 113 44 , , PUNCT american_scientific_journal-707 113 45 β„Ž:𝐴 β„Ž:𝐴 PROPN american_scientific_journal-707 113 46 β†’ β†’ SYM american_scientific_journal-707 113 47 𝑋and 𝑋and PROPN american_scientific_journal-707 113 48 β„Žβ€²:𝐴 β„Žβ€²:𝐴 NOUN american_scientific_journal-707 113 49 β†’ β†’ SYM american_scientific_journal-707 113 50 π‘Œ π‘Œ PROPN american_scientific_journal-707 113 51 are be AUX american_scientific_journal-707 113 52 epimorphisms epimorphism NOUN american_scientific_journal-707 113 53 and and CCONJ american_scientific_journal-707 113 54 𝑔:𝑋 𝑔:𝑋 PROPN american_scientific_journal-707 113 55 β†’ β†’ SYM american_scientific_journal-707 113 56 𝐡 𝐡 NOUN american_scientific_journal-707 113 57 and𝑔’:π‘Œ and𝑔’:π‘Œ PROPN american_scientific_journal-707 113 58 β†’ β†’ SYM american_scientific_journal-707 113 59 𝐡 𝐡 PROPN american_scientific_journal-707 113 60 are be AUX american_scientific_journal-707 113 61 monomorphisms monomorphism NOUN american_scientific_journal-707 113 62 . . PUNCT american_scientific_journal-707 114 1 claim claim NOUN american_scientific_journal-707 114 2 1:β„Ž~β„Žβ€² 1:β„Ž~β„Žβ€² NUM american_scientific_journal-707 114 3 ( ( PUNCT american_scientific_journal-707 114 4 h h NOUN american_scientific_journal-707 114 5 and and CCONJ american_scientific_journal-707 114 6 h h NOUN american_scientific_journal-707 114 7 ’ ' PUNCT american_scientific_journal-707 114 8 are be AUX american_scientific_journal-707 114 9 associate associate ADJ american_scientific_journal-707 114 10 to to ADP american_scientific_journal-707 114 11 each each DET american_scientific_journal-707 114 12 other other ADJ american_scientific_journal-707 114 13 ) ) PUNCT american_scientific_journal-707 114 14 and and CCONJ american_scientific_journal-707 114 15 𝑔~𝑔′ 𝑔~𝑔′ NOUN american_scientific_journal-707 114 16 figure figure VERB american_scientific_journal-707 114 17 3 3 NUM american_scientific_journal-707 114 18 : : PUNCT american_scientific_journal-707 114 19 two two NUM american_scientific_journal-707 114 20 decompositions decomposition NOUN american_scientific_journal-707 114 21 of of ADP american_scientific_journal-707 114 22 πœ‹ πœ‹ PRON american_scientific_journal-707 114 23 now now ADV american_scientific_journal-707 114 24 consider consider VERB american_scientific_journal-707 114 25 the the DET american_scientific_journal-707 114 26 identity identity NOUN american_scientific_journal-707 114 27 map map NOUN american_scientific_journal-707 114 28 𝐼𝐴 𝐼𝐴 NOUN american_scientific_journal-707 114 29 on on ADP american_scientific_journal-707 114 30 an an DET american_scientific_journal-707 114 31 algebra algebra PROPN american_scientific_journal-707 114 32 𝐴 𝐴 PROPN american_scientific_journal-707 114 33 , , PUNCT american_scientific_journal-707 114 34 which which PRON american_scientific_journal-707 114 35 is be AUX american_scientific_journal-707 114 36 an an DET american_scientific_journal-707 114 37 isomorphism isomorphism NOUN american_scientific_journal-707 114 38 . . PUNCT american_scientific_journal-707 115 1 on on ADP american_scientific_journal-707 115 2 the the DET american_scientific_journal-707 115 3 other other ADJ american_scientific_journal-707 115 4 hand hand NOUN american_scientific_journal-707 115 5 since since SCONJ american_scientific_journal-707 115 6 β„Žβ€²:𝐴 β„Žβ€²:𝐴 PROPN american_scientific_journal-707 115 7 ⟢ ⟢ PROPN american_scientific_journal-707 115 8 π‘Œ π‘Œ PROPN american_scientific_journal-707 115 9 is be AUX american_scientific_journal-707 115 10 an an DET american_scientific_journal-707 115 11 epimorphism epimorphism NOUN american_scientific_journal-707 115 12 , , PUNCT american_scientific_journal-707 115 13 every every DET american_scientific_journal-707 115 14 element element NOUN american_scientific_journal-707 115 15 of of ADP american_scientific_journal-707 115 16 y y PROPN american_scientific_journal-707 115 17 can can AUX american_scientific_journal-707 115 18 be be AUX american_scientific_journal-707 115 19 expressed express VERB american_scientific_journal-707 115 20 as as ADP american_scientific_journal-707 115 21 β„Žβ€™(π‘Ž β„Žβ€™(π‘Ž PROPN american_scientific_journal-707 115 22 ) ) PUNCT american_scientific_journal-707 115 23 for for ADP american_scientific_journal-707 115 24 some some DET american_scientific_journal-707 115 25 π‘Ž π‘Ž PROPN american_scientific_journal-707 115 26 ∈ ∈ NOUN american_scientific_journal-707 115 27 𝐴. 𝐴. NOUN american_scientific_journal-707 115 28 define define VERB american_scientific_journal-707 115 29 𝛼:π‘Œ 𝛼:π‘Œ PROPN american_scientific_journal-707 115 30 β†’ β†’ SYM american_scientific_journal-707 115 31 𝑋 𝑋 NOUN american_scientific_journal-707 115 32 by by ADP american_scientific_journal-707 115 33 : : PUNCT american_scientific_journal-707 115 34 𝛼 𝛼 X american_scientific_journal-707 115 35 οΏ½ οΏ½ PROPN american_scientific_journal-707 115 36 β„Žβ€²(π‘Ž β„Žβ€²(π‘Ž NOUN american_scientific_journal-707 115 37 ) ) PUNCT american_scientific_journal-707 115 38 οΏ½ οΏ½ PROPN american_scientific_journal-707 116 1 = = PRON american_scientific_journal-707 116 2 β„Ž(π‘Ž)for β„Ž(π‘Ž)for VERB american_scientific_journal-707 116 3 all all DET american_scientific_journal-707 116 4 β„Žβ€²(π‘Ž β„Žβ€²(π‘Ž NOUN american_scientific_journal-707 116 5 ) ) PUNCT american_scientific_journal-707 116 6 ∈ ∈ PROPN american_scientific_journal-707 116 7 π‘Œ π‘Œ PROPN american_scientific_journal-707 116 8 where where SCONJ american_scientific_journal-707 116 9 π‘Ž π‘Ž DET american_scientific_journal-707 116 10 ∈ ∈ NOUN american_scientific_journal-707 116 11 𝐴. 𝐴. NOUN american_scientific_journal-707 116 12 then then ADV american_scientific_journal-707 116 13 we we PRON american_scientific_journal-707 116 14 prove prove VERB american_scientific_journal-707 116 15 that that SCONJ american_scientific_journal-707 116 16 : : PUNCT american_scientific_journal-707 116 17 a a X american_scientific_journal-707 116 18 ) ) PUNCT american_scientific_journal-707 116 19 𝛼 𝛼 NOUN american_scientific_journal-707 116 20 is be AUX american_scientific_journal-707 116 21 well well ADV american_scientific_journal-707 116 22 defined define VERB american_scientific_journal-707 116 23 ; ; PUNCT american_scientific_journal-707 116 24 for for ADP american_scientific_journal-707 116 25 , , PUNCT american_scientific_journal-707 116 26 67 67 NUM american_scientific_journal-707 116 27 american american ADJ american_scientific_journal-707 116 28 scientific scientific ADJ american_scientific_journal-707 116 29 research research NOUN american_scientific_journal-707 116 30 journal journal NOUN american_scientific_journal-707 116 31 for for ADP american_scientific_journal-707 116 32 engineering engineering NOUN american_scientific_journal-707 116 33 , , PUNCT american_scientific_journal-707 116 34 technology technology NOUN american_scientific_journal-707 116 35 , , PUNCT american_scientific_journal-707 116 36 and and CCONJ american_scientific_journal-707 116 37 sciences science NOUN american_scientific_journal-707 116 38 ( ( PUNCT american_scientific_journal-707 116 39 asrjets asrjet NOUN american_scientific_journal-707 116 40 ) ) PUNCT american_scientific_journal-707 116 41 ( ( PUNCT american_scientific_journal-707 116 42 2014 2014 NUM american_scientific_journal-707 116 43 ) ) PUNCT american_scientific_journal-707 116 44 volume volume NOUN american_scientific_journal-707 116 45 10 10 NUM american_scientific_journal-707 116 46 , , PUNCT american_scientific_journal-707 116 47 no no DET american_scientific_journal-707 116 48 1 1 NUM american_scientific_journal-707 116 49 , , PUNCT american_scientific_journal-707 116 50 pp pp ADV american_scientific_journal-707 116 51 61 61 NUM american_scientific_journal-707 116 52 - - SYM american_scientific_journal-707 116 53 73 73 NUM american_scientific_journal-707 116 54 for for ADP american_scientific_journal-707 116 55 any any DET american_scientific_journal-707 116 56 𝑦1 𝑦1 NOUN american_scientific_journal-707 116 57 , , PUNCT american_scientific_journal-707 116 58 𝑦2 𝑦2 PROPN american_scientific_journal-707 116 59 ∈ ∈ PROPN american_scientific_journal-707 116 60 π‘Œ π‘Œ PROPN american_scientific_journal-707 116 61 there there PRON american_scientific_journal-707 116 62 exists exist VERB american_scientific_journal-707 116 63 π‘Ž1 π‘Ž1 INTJ american_scientific_journal-707 116 64 andπ‘Ž2∈ andπ‘Ž2∈ PROPN american_scientific_journal-707 116 65 𝐴 𝐴 PROPN american_scientific_journal-707 116 66 such such ADJ american_scientific_journal-707 116 67 that𝑦1 that𝑦1 NOUN american_scientific_journal-707 116 68 = = SYM american_scientific_journal-707 116 69 β„Žβ€²(π‘Ž1 β„Žβ€²(π‘Ž1 NOUN american_scientific_journal-707 116 70 ) ) PUNCT american_scientific_journal-707 116 71 and and CCONJ american_scientific_journal-707 116 72 𝑦2 𝑦2 PROPN american_scientific_journal-707 116 73 = = SYM american_scientific_journal-707 116 74 β„Žβ€²(π‘Ž2 β„Žβ€²(π‘Ž2 PROPN american_scientific_journal-707 116 75 ) ) PUNCT american_scientific_journal-707 116 76 therefore therefore ADV american_scientific_journal-707 116 77 , , PUNCT american_scientific_journal-707 116 78 𝑦1 𝑦1 PROPN american_scientific_journal-707 116 79 = = PUNCT american_scientific_journal-707 116 80 𝑦2 𝑦2 PROPN american_scientific_journal-707 116 81 ⟹ ⟹ NUM american_scientific_journal-707 116 82 β„Žβ€²(π‘Ž1 β„Žβ€²(π‘Ž1 PROPN american_scientific_journal-707 116 83 ) ) PUNCT american_scientific_journal-707 116 84 = = SYM american_scientific_journal-707 116 85 β„Žβ€²(π‘Ž2 β„Žβ€²(π‘Ž2 NOUN american_scientific_journal-707 116 86 ) ) PUNCT american_scientific_journal-707 116 87 inπ‘Œ inπ‘Œ VERB american_scientific_journal-707 116 88 ⟹ ⟹ NUM american_scientific_journal-707 116 89 𝑔′(β„Žβ€²(π‘Ž1 𝑔′(β„Žβ€²(π‘Ž1 PROPN american_scientific_journal-707 116 90 ) ) PUNCT american_scientific_journal-707 116 91 ) ) PUNCT american_scientific_journal-707 117 1 = = SYM american_scientific_journal-707 117 2 𝑔′ 𝑔′ NUM american_scientific_journal-707 117 3 οΏ½ οΏ½ PROPN american_scientific_journal-707 117 4 β„Žβ€²(π‘Ž2 β„Žβ€²(π‘Ž2 NOUN american_scientific_journal-707 117 5 ) ) PUNCT american_scientific_journal-707 117 6 οΏ½ οΏ½ PROPN american_scientific_journal-707 117 7 in𝐡 in𝐡 PROPN american_scientific_journal-707 117 8 ⟹ ⟹ NUM american_scientific_journal-707 117 9 𝑔′ 𝑔′ ADV american_scientific_journal-707 117 10 ∘ ∘ PROPN american_scientific_journal-707 117 11 β„Žβ€²(π‘Ž1 β„Žβ€²(π‘Ž1 NOUN american_scientific_journal-707 117 12 ) ) PUNCT american_scientific_journal-707 117 13 = = PUNCT american_scientific_journal-707 117 14 𝑔′ 𝑔′ X american_scientific_journal-707 117 15 ∘ ∘ NUM american_scientific_journal-707 117 16 β„Žβ€²(π‘Ž2 β„Žβ€²(π‘Ž2 NOUN american_scientific_journal-707 117 17 ) ) PUNCT american_scientific_journal-707 117 18 in𝐡 in𝐡 NOUN american_scientific_journal-707 117 19 ⟹ ⟹ NUM american_scientific_journal-707 117 20 𝑓(π‘Ž1 𝑓(π‘Ž1 NOUN american_scientific_journal-707 117 21 ) ) PUNCT american_scientific_journal-707 117 22 = = SYM american_scientific_journal-707 117 23 𝑓(π‘Ž2 𝑓(π‘Ž2 NOUN american_scientific_journal-707 117 24 ) ) PUNCT american_scientific_journal-707 117 25 in𝐡 in𝐡 NOUN american_scientific_journal-707 117 26 ( ( PUNCT american_scientific_journal-707 117 27 ∡ ∡ NOUN american_scientific_journal-707 117 28 𝑓 𝑓 X american_scientific_journal-707 117 29 = = PUNCT american_scientific_journal-707 117 30 𝑔′ 𝑔′ PRON american_scientific_journal-707 117 31 ∘ ∘ NUM american_scientific_journal-707 117 32 β„Žβ€² β„Žβ€² NOUN american_scientific_journal-707 117 33 ) ) PUNCT american_scientific_journal-707 117 34 ⟹ ⟹ PROPN american_scientific_journal-707 118 1 𝑔 𝑔 PROPN american_scientific_journal-707 118 2 ∘ ∘ NUM american_scientific_journal-707 118 3 β„Ž(π‘Ž1 β„Ž(π‘Ž1 NUM american_scientific_journal-707 118 4 ) ) PUNCT american_scientific_journal-707 118 5 = = SYM american_scientific_journal-707 118 6 𝑔 𝑔 PROPN american_scientific_journal-707 118 7 ∘ ∘ ADJ american_scientific_journal-707 118 8 β„Ž(π‘Ž2 β„Ž(π‘Ž2 ADJ american_scientific_journal-707 118 9 ) ) PUNCT american_scientific_journal-707 118 10 in𝐡 in𝐡 NOUN american_scientific_journal-707 118 11 ( ( PUNCT american_scientific_journal-707 118 12 ∡ ∡ NOUN american_scientific_journal-707 118 13 𝑓 𝑓 PRON american_scientific_journal-707 118 14 = = SYM american_scientific_journal-707 118 15 𝑔 𝑔 PROPN american_scientific_journal-707 118 16 ∘ ∘ NUM american_scientific_journal-707 118 17 β„Ž β„Ž NOUN american_scientific_journal-707 118 18 ) ) PUNCT american_scientific_journal-707 118 19 ⟹ ⟹ PUNCT american_scientific_journal-707 118 20 𝑔(β„Ž(π‘Ž1 𝑔(β„Ž(π‘Ž1 ADJ american_scientific_journal-707 118 21 ) ) PUNCT american_scientific_journal-707 118 22 ) ) PUNCT american_scientific_journal-707 119 1 = = SYM american_scientific_journal-707 119 2 𝑔(β„Ž(π‘Ž2 𝑔(β„Ž(π‘Ž2 NOUN american_scientific_journal-707 119 3 ) ) PUNCT american_scientific_journal-707 119 4 ) ) PUNCT american_scientific_journal-707 119 5 in𝐡 in𝐡 NOUN american_scientific_journal-707 119 6 ⟹ ⟹ NUM american_scientific_journal-707 119 7 β„Ž(π‘Ž1 β„Ž(π‘Ž1 NOUN american_scientific_journal-707 119 8 ) ) PUNCT american_scientific_journal-707 119 9 = = SYM american_scientific_journal-707 119 10 β„Ž(π‘Ž2 β„Ž(π‘Ž2 ADJ american_scientific_journal-707 119 11 ) ) PUNCT american_scientific_journal-707 119 12 in𝑋 in𝑋 NOUN american_scientific_journal-707 119 13 ( ( PUNCT american_scientific_journal-707 119 14 ∡ ∡ ADJ american_scientific_journal-707 119 15 𝑔 𝑔 PROPN american_scientific_journal-707 119 16 is be AUX american_scientific_journal-707 119 17 an an DET american_scientific_journal-707 119 18 injection injection NOUN american_scientific_journal-707 119 19 ) ) PUNCT american_scientific_journal-707 119 20 therefore therefore ADV american_scientific_journal-707 119 21 , , PUNCT american_scientific_journal-707 119 22 𝛼 𝛼 X american_scientific_journal-707 119 23 is be AUX american_scientific_journal-707 119 24 well well ADV american_scientific_journal-707 119 25 defined define VERB american_scientific_journal-707 119 26 . . PUNCT american_scientific_journal-707 120 1 b b X american_scientific_journal-707 120 2 ) ) PUNCT american_scientific_journal-707 120 3 𝛼 𝛼 NOUN american_scientific_journal-707 120 4 is be AUX american_scientific_journal-707 120 5 an an DET american_scientific_journal-707 120 6 injection injection NOUN american_scientific_journal-707 120 7 ; ; PUNCT american_scientific_journal-707 120 8 for for SCONJ american_scientific_journal-707 120 9 , , PUNCT american_scientific_journal-707 120 10 let let VERB american_scientific_journal-707 120 11 𝑦1 𝑦1 PROPN american_scientific_journal-707 120 12 = = SYM american_scientific_journal-707 120 13 β„Žβ€²(π‘Ž1 β„Žβ€²(π‘Ž1 PROPN american_scientific_journal-707 120 14 ) ) PUNCT american_scientific_journal-707 120 15 and and CCONJ american_scientific_journal-707 120 16 𝑦2 𝑦2 PROPN american_scientific_journal-707 120 17 = = SYM american_scientific_journal-707 120 18 β„Žβ€²(π‘Ž2 β„Žβ€²(π‘Ž2 PROPN american_scientific_journal-707 120 19 ) ) PUNCT american_scientific_journal-707 120 20 ∈ ∈ PROPN american_scientific_journal-707 120 21 π‘Œ π‘Œ PROPN american_scientific_journal-707 120 22 for for ADP american_scientific_journal-707 120 23 some some DET american_scientific_journal-707 120 24 π‘Ž1 π‘Ž1 NOUN american_scientific_journal-707 120 25 andπ‘Ž2 andπ‘Ž2 NOUN american_scientific_journal-707 120 26 ∈ ∈ PROPN american_scientific_journal-707 120 27 𝐴. 𝐴. PROPN american_scientific_journal-707 120 28 then then ADV american_scientific_journal-707 120 29 𝛼(𝑦1 𝛼(𝑦1 NUM american_scientific_journal-707 120 30 ) ) PUNCT american_scientific_journal-707 121 1 = = SYM american_scientific_journal-707 121 2 𝛼(𝑦2)𝑖𝑛𝑋 𝛼(𝑦2)𝑖𝑛𝑋 VERB american_scientific_journal-707 121 3 ⟹ ⟹ NUM american_scientific_journal-707 121 4 𝛼 𝛼 NOUN american_scientific_journal-707 121 5 οΏ½ οΏ½ NOUN american_scientific_journal-707 121 6 β„Žβ€²(π‘Ž1 β„Žβ€²(π‘Ž1 NOUN american_scientific_journal-707 121 7 ) ) PUNCT american_scientific_journal-707 121 8 οΏ½ οΏ½ PROPN american_scientific_journal-707 121 9 = = PUNCT american_scientific_journal-707 121 10 𝛼 𝛼 VERB american_scientific_journal-707 121 11 οΏ½ οΏ½ NOUN american_scientific_journal-707 121 12 β„Žβ€²(π‘Ž2 β„Žβ€²(π‘Ž2 NOUN american_scientific_journal-707 121 13 ) ) PUNCT american_scientific_journal-707 121 14 οΏ½ οΏ½ PROPN american_scientific_journal-707 121 15 in𝑋 in𝑋 PROPN american_scientific_journal-707 121 16 ⟹ ⟹ NUM american_scientific_journal-707 121 17 β„Ž(π‘Ž1 β„Ž(π‘Ž1 NOUN american_scientific_journal-707 121 18 ) ) PUNCT american_scientific_journal-707 121 19 = = SYM american_scientific_journal-707 121 20 β„Ž(π‘Ž2 β„Ž(π‘Ž2 ADJ american_scientific_journal-707 121 21 ) ) PUNCT american_scientific_journal-707 121 22 𝑖𝑛𝑋 𝑖𝑛𝑋 NOUN american_scientific_journal-707 121 23 ( ( PUNCT american_scientific_journal-707 121 24 by by ADP american_scientific_journal-707 121 25 the the DET american_scientific_journal-707 121 26 definition definition NOUN american_scientific_journal-707 121 27 of of ADP american_scientific_journal-707 121 28 𝛼 𝛼 NOUN american_scientific_journal-707 121 29 ) ) PUNCT american_scientific_journal-707 121 30 ⟹ ⟹ PUNCT american_scientific_journal-707 121 31 𝑔 𝑔 PROPN american_scientific_journal-707 121 32 οΏ½ οΏ½ NOUN american_scientific_journal-707 121 33 β„Ž(π‘Ž1 β„Ž(π‘Ž1 NUM american_scientific_journal-707 121 34 ) ) PUNCT american_scientific_journal-707 121 35 οΏ½ οΏ½ PROPN american_scientific_journal-707 121 36 = = SYM american_scientific_journal-707 121 37 𝑔 𝑔 NOUN american_scientific_journal-707 121 38 οΏ½ οΏ½ NOUN american_scientific_journal-707 121 39 β„Ž(π‘Ž2 β„Ž(π‘Ž2 ADJ american_scientific_journal-707 121 40 ) ) PUNCT american_scientific_journal-707 121 41 οΏ½ οΏ½ PROPN american_scientific_journal-707 121 42 in𝐡 in𝐡 NOUN american_scientific_journal-707 121 43 ⟹ ⟹ ADV american_scientific_journal-707 121 44 𝑔 𝑔 PROPN american_scientific_journal-707 121 45 ∘ ∘ NUM american_scientific_journal-707 121 46 β„Ž(π‘Ž1 β„Ž(π‘Ž1 NUM american_scientific_journal-707 121 47 ) ) PUNCT american_scientific_journal-707 121 48 = = SYM american_scientific_journal-707 121 49 𝑔 𝑔 PROPN american_scientific_journal-707 121 50 ∘ ∘ PROPN american_scientific_journal-707 121 51 β„Ž(π‘Ž2 β„Ž(π‘Ž2 ADJ american_scientific_journal-707 121 52 ) ) PUNCT american_scientific_journal-707 121 53 in𝐡 in𝐡 NOUN american_scientific_journal-707 121 54 ⟹ ⟹ NUM american_scientific_journal-707 121 55 𝑓(π‘Ž1 𝑓(π‘Ž1 NOUN american_scientific_journal-707 121 56 ) ) PUNCT american_scientific_journal-707 121 57 = = SYM american_scientific_journal-707 121 58 𝑓(π‘Ž2 𝑓(π‘Ž2 NOUN american_scientific_journal-707 121 59 ) ) PUNCT american_scientific_journal-707 121 60 in𝐡 in𝐡 NOUN american_scientific_journal-707 121 61 ( ( PUNCT american_scientific_journal-707 121 62 ∡ ∡ NOUN american_scientific_journal-707 121 63 𝑓 𝑓 PRON american_scientific_journal-707 121 64 = = SYM american_scientific_journal-707 121 65 𝑔 𝑔 PROPN american_scientific_journal-707 121 66 ∘ ∘ NUM american_scientific_journal-707 121 67 β„Ž β„Ž NOUN american_scientific_journal-707 121 68 ) ) PUNCT american_scientific_journal-707 121 69 ⟹ ⟹ NUM american_scientific_journal-707 121 70 𝑔′ 𝑔′ X american_scientific_journal-707 121 71 ∘ ∘ PROPN american_scientific_journal-707 121 72 β„Žβ€²(π‘Ž1 β„Žβ€²(π‘Ž1 NOUN american_scientific_journal-707 121 73 ) ) PUNCT american_scientific_journal-707 121 74 = = PUNCT american_scientific_journal-707 121 75 𝑔′ 𝑔′ NUM american_scientific_journal-707 121 76 ∘ ∘ PROPN american_scientific_journal-707 121 77 β„Žβ€²(π‘Ž2)in𝐡 β„Žβ€²(π‘Ž2)in𝐡 PROPN american_scientific_journal-707 121 78 ( ( PUNCT american_scientific_journal-707 121 79 ∡ ∡ NOUN american_scientific_journal-707 121 80 𝑓 𝑓 PRON american_scientific_journal-707 121 81 = = PUNCT american_scientific_journal-707 121 82 𝑔′ 𝑔′ PRON american_scientific_journal-707 121 83 ∘ ∘ NUM american_scientific_journal-707 121 84 β„Žβ€² β„Žβ€² NOUN american_scientific_journal-707 121 85 ) ) PUNCT american_scientific_journal-707 121 86 ⟹ ⟹ NUM american_scientific_journal-707 121 87 𝑔′ 𝑔′ NUM american_scientific_journal-707 121 88 οΏ½ οΏ½ NOUN american_scientific_journal-707 121 89 β„Žβ€²(π‘Ž1 β„Žβ€²(π‘Ž1 NOUN american_scientific_journal-707 121 90 ) ) PUNCT american_scientific_journal-707 121 91 οΏ½ οΏ½ PROPN american_scientific_journal-707 121 92 = = PUNCT american_scientific_journal-707 121 93 𝑔′(β„Žβ€²(π‘Ž2 𝑔′(β„Žβ€²(π‘Ž2 NOUN american_scientific_journal-707 121 94 ) ) PUNCT american_scientific_journal-707 121 95 in𝐡 in𝐡 NOUN american_scientific_journal-707 121 96 ⟹ ⟹ NUM american_scientific_journal-707 121 97 β„Žβ€²(π‘Ž1 β„Žβ€²(π‘Ž1 NOUN american_scientific_journal-707 121 98 ) ) PUNCT american_scientific_journal-707 121 99 = = SYM american_scientific_journal-707 121 100 β„Žβ€²(π‘Ž2 β„Žβ€²(π‘Ž2 NOUN american_scientific_journal-707 121 101 ) ) PUNCT american_scientific_journal-707 121 102 inπ‘Œ inπ‘Œ PROPN american_scientific_journal-707 121 103 ( ( PUNCT american_scientific_journal-707 121 104 ∡ ∡ NOUN american_scientific_journal-707 121 105 𝑔′ 𝑔′ PRON american_scientific_journal-707 121 106 is be AUX american_scientific_journal-707 121 107 an an DET american_scientific_journal-707 121 108 injection injection NOUN american_scientific_journal-707 121 109 ) ) PUNCT american_scientific_journal-707 121 110 ⟹ ⟹ NUM american_scientific_journal-707 122 1 𝑦1 𝑦1 NOUN american_scientific_journal-707 122 2 = = PROPN american_scientific_journal-707 122 3 𝑦2 𝑦2 PROPN american_scientific_journal-707 122 4 inπ‘Œ inπ‘Œ VERB american_scientific_journal-707 122 5 therefore therefore ADV american_scientific_journal-707 122 6 , , PUNCT american_scientific_journal-707 122 7 𝛼 𝛼 PROPN american_scientific_journal-707 122 8 is be AUX american_scientific_journal-707 122 9 an an DET american_scientific_journal-707 122 10 injection injection NOUN american_scientific_journal-707 122 11 . . PUNCT american_scientific_journal-707 123 1 68 68 NUM american_scientific_journal-707 123 2 american american ADJ american_scientific_journal-707 123 3 scientific scientific ADJ american_scientific_journal-707 123 4 research research NOUN american_scientific_journal-707 123 5 journal journal NOUN american_scientific_journal-707 123 6 for for ADP american_scientific_journal-707 123 7 engineering engineering NOUN american_scientific_journal-707 123 8 , , PUNCT american_scientific_journal-707 123 9 technology technology NOUN american_scientific_journal-707 123 10 , , PUNCT american_scientific_journal-707 123 11 and and CCONJ american_scientific_journal-707 123 12 sciences science NOUN american_scientific_journal-707 123 13 ( ( PUNCT american_scientific_journal-707 123 14 asrjets asrjet NOUN american_scientific_journal-707 123 15 ) ) PUNCT american_scientific_journal-707 123 16 ( ( PUNCT american_scientific_journal-707 123 17 2014 2014 NUM american_scientific_journal-707 123 18 ) ) PUNCT american_scientific_journal-707 123 19 volume volume NOUN american_scientific_journal-707 123 20 10 10 NUM american_scientific_journal-707 123 21 , , PUNCT american_scientific_journal-707 123 22 no no DET american_scientific_journal-707 123 23 1 1 NUM american_scientific_journal-707 123 24 , , PUNCT american_scientific_journal-707 123 25 pp pp ADV american_scientific_journal-707 123 26 61 61 NUM american_scientific_journal-707 123 27 - - SYM american_scientific_journal-707 123 28 73 73 NUM american_scientific_journal-707 123 29 c c NOUN american_scientific_journal-707 123 30 ) ) PUNCT american_scientific_journal-707 123 31 𝛼 𝛼 PROPN american_scientific_journal-707 123 32 is be AUX american_scientific_journal-707 123 33 a a DET american_scientific_journal-707 123 34 surjection surjection NOUN american_scientific_journal-707 123 35 ; ; PUNCT american_scientific_journal-707 123 36 for for ADP american_scientific_journal-707 123 37 , , PUNCT american_scientific_journal-707 123 38 π‘₯ π‘₯ DET american_scientific_journal-707 123 39 ∈ ∈ NOUN american_scientific_journal-707 123 40 𝑋 𝑋 NOUN american_scientific_journal-707 123 41 β‡’ β‡’ VERB american_scientific_journal-707 123 42 π‘₯ π‘₯ PROPN american_scientific_journal-707 124 1 = = SYM american_scientific_journal-707 124 2 β„Ž(π‘Ž)for β„Ž(π‘Ž)for VERB american_scientific_journal-707 124 3 some some DET american_scientific_journal-707 124 4 π‘Ž π‘Ž PROPN american_scientific_journal-707 124 5 ∈ ∈ PROPN american_scientific_journal-707 124 6 𝐴 𝐴 NOUN american_scientific_journal-707 124 7 ( ( PUNCT american_scientific_journal-707 124 8 ∡ ∡ X american_scientific_journal-707 124 9 β„Žis β„Žis VERB american_scientific_journal-707 124 10 a a DET american_scientific_journal-707 124 11 surjection surjection NOUN american_scientific_journal-707 124 12 ) ) PUNCT american_scientific_journal-707 124 13 β‡’ β‡’ NOUN american_scientific_journal-707 124 14 π‘₯ π‘₯ X american_scientific_journal-707 125 1 = = PUNCT american_scientific_journal-707 125 2 𝛼(β„Žβ€²(π‘Ž))for 𝛼(β„Žβ€²(π‘Ž))for VERB american_scientific_journal-707 125 3 some some DET american_scientific_journal-707 125 4 π‘Ž π‘Ž PRON american_scientific_journal-707 125 5 ∈ ∈ PROPN american_scientific_journal-707 125 6 𝐴 𝐴 PROPN american_scientific_journal-707 125 7 , , PUNCT american_scientific_journal-707 125 8 where where SCONJ american_scientific_journal-707 125 9 β„Žβ€²(π‘Ž β„Žβ€²(π‘Ž ADP american_scientific_journal-707 125 10 ) ) PUNCT american_scientific_journal-707 125 11 ∈ ∈ PROPN american_scientific_journal-707 125 12 π‘Œ π‘Œ PROPN american_scientific_journal-707 125 13 β‡’ β‡’ VERB american_scientific_journal-707 125 14 π‘₯ π‘₯ X american_scientific_journal-707 126 1 = = PUNCT american_scientific_journal-707 126 2 𝛼(𝑦)for 𝛼(𝑦)for VERB american_scientific_journal-707 126 3 some some DET american_scientific_journal-707 126 4 𝑦 𝑦 NOUN american_scientific_journal-707 126 5 ∈ ∈ PROPN american_scientific_journal-707 126 6 π‘Œ π‘Œ PROPN american_scientific_journal-707 126 7 , , PUNCT american_scientific_journal-707 126 8 where where SCONJ american_scientific_journal-707 126 9 𝑦 𝑦 NOUN american_scientific_journal-707 126 10 = = SYM american_scientific_journal-707 126 11 β„Žβ€²(π‘Ž β„Žβ€²(π‘Ž NOUN american_scientific_journal-707 126 12 ) ) PUNCT american_scientific_journal-707 126 13 therefore therefore ADV american_scientific_journal-707 126 14 , , PUNCT american_scientific_journal-707 126 15 𝛼 𝛼 PROPN american_scientific_journal-707 126 16 is be AUX american_scientific_journal-707 126 17 a a DET american_scientific_journal-707 126 18 surjection surjection NOUN american_scientific_journal-707 126 19 . . PUNCT american_scientific_journal-707 127 1 d d X american_scientific_journal-707 127 2 ) ) PUNCT american_scientific_journal-707 127 3 𝛼 𝛼 NOUN american_scientific_journal-707 127 4 is be AUX american_scientific_journal-707 127 5 a a DET american_scientific_journal-707 127 6 homomorphism homomorphism NOUN american_scientific_journal-707 127 7 ; ; PUNCT american_scientific_journal-707 127 8 for for SCONJ american_scientific_journal-707 127 9 , , PUNCT american_scientific_journal-707 127 10 let let VERB american_scientific_journal-707 127 11 𝑓 𝑓 PRON american_scientific_journal-707 127 12 ∈ ∈ NOUN american_scientific_journal-707 127 13 β„± β„± PROPN american_scientific_journal-707 127 14 be be VERB american_scientific_journal-707 127 15 a a DET american_scientific_journal-707 127 16 nullary nullary ADJ american_scientific_journal-707 127 17 operation operation NOUN american_scientific_journal-707 127 18 symbol symbol NOUN american_scientific_journal-707 127 19 . . PUNCT american_scientific_journal-707 128 1 then then ADV american_scientific_journal-707 128 2 π‘“π‘Œ π‘“π‘Œ PROPN american_scientific_journal-707 128 3 ∈ ∈ PROPN american_scientific_journal-707 128 4 π‘Œ π‘Œ PROPN american_scientific_journal-707 128 5 and and CCONJ american_scientific_journal-707 128 6 𝑓𝐴 𝑓𝐴 NOUN american_scientific_journal-707 128 7 ∈ ∈ NOUN american_scientific_journal-707 128 8 𝐴. 𝐴. NOUN american_scientific_journal-707 128 9 since since SCONJ american_scientific_journal-707 128 10 β„Žβ€² β„Žβ€² ADJ american_scientific_journal-707 128 11 is be AUX american_scientific_journal-707 128 12 a a DET american_scientific_journal-707 128 13 homomorphism homomorphism NOUN american_scientific_journal-707 128 14 , , PUNCT american_scientific_journal-707 128 15 we we PRON american_scientific_journal-707 128 16 have have VERB american_scientific_journal-707 128 17 β„Žβ€²(𝑓𝐴 β„Žβ€²(𝑓𝐴 NOUN american_scientific_journal-707 128 18 ) ) PUNCT american_scientific_journal-707 129 1 = = PUNCT american_scientific_journal-707 129 2 π‘“π‘Œ π‘“π‘Œ NOUN american_scientific_journal-707 129 3 . . PUNCT american_scientific_journal-707 130 1 therefore therefore ADV american_scientific_journal-707 130 2 , , PUNCT american_scientific_journal-707 130 3 𝛼(π‘“π‘Œ 𝛼(π‘“π‘Œ NUM american_scientific_journal-707 130 4 ) ) PUNCT american_scientific_journal-707 130 5 = = SYM american_scientific_journal-707 130 6 𝛼(β„Žβ€²(𝑓𝐴 𝛼(β„Žβ€²(𝑓𝐴 NOUN american_scientific_journal-707 130 7 ) ) PUNCT american_scientific_journal-707 130 8 ) ) PUNCT american_scientific_journal-707 131 1 = = SYM american_scientific_journal-707 131 2 β„Ž(𝑓𝐴 β„Ž(𝑓𝐴 PROPN american_scientific_journal-707 131 3 ) ) PUNCT american_scientific_journal-707 131 4 ( ( PUNCT american_scientific_journal-707 131 5 by by ADP american_scientific_journal-707 131 6 the the DET american_scientific_journal-707 131 7 definition definition NOUN american_scientific_journal-707 131 8 of of ADP american_scientific_journal-707 131 9 𝛼 𝛼 NOUN american_scientific_journal-707 131 10 ) ) PUNCT american_scientific_journal-707 131 11 = = SYM american_scientific_journal-707 131 12 𝑓𝑋 𝑓𝑋 NOUN american_scientific_journal-707 131 13 ( ( PUNCT american_scientific_journal-707 131 14 ∡ ∡ X american_scientific_journal-707 131 15 β„Žis β„Žis VERB american_scientific_journal-707 131 16 a a DET american_scientific_journal-707 131 17 homomorphism homomorphism NOUN american_scientific_journal-707 131 18 ) ) PUNCT american_scientific_journal-707 131 19 also also ADV american_scientific_journal-707 131 20 , , PUNCT american_scientific_journal-707 131 21 let let VERB american_scientific_journal-707 131 22 𝑓 𝑓 PRON american_scientific_journal-707 131 23 ∈ ∈ NOUN american_scientific_journal-707 131 24 β„± β„± PROPN american_scientific_journal-707 131 25 be be VERB american_scientific_journal-707 131 26 an an DET american_scientific_journal-707 131 27 𝑛-ary 𝑛-ary ADJ american_scientific_journal-707 131 28 operation operation NOUN american_scientific_journal-707 131 29 symbol,𝑛 symbol,𝑛 ADP american_scientific_journal-707 131 30 > > X american_scientific_journal-707 131 31 0and 0and PROPN american_scientific_journal-707 131 32 𝑦1,𝑦2 𝑦1,𝑦2 PROPN american_scientific_journal-707 131 33 , , PUNCT american_scientific_journal-707 131 34 … … PUNCT american_scientific_journal-707 131 35 … … PUNCT american_scientific_journal-707 131 36 . . PUNCT american_scientific_journal-707 131 37 . . PUNCT american_scientific_journal-707 132 1 , , PUNCT american_scientific_journal-707 132 2 𝑦𝑛 𝑦𝑛 NOUN american_scientific_journal-707 132 3 ∈ ∈ PROPN american_scientific_journal-707 132 4 π‘Œ. π‘Œ. PROPN american_scientific_journal-707 132 5 then then ADV american_scientific_journal-707 132 6 since since SCONJ american_scientific_journal-707 132 7 β„Žβ€² β„Žβ€² NOUN american_scientific_journal-707 132 8 is be AUX american_scientific_journal-707 132 9 an an DET american_scientific_journal-707 132 10 epimorphism epimorphism NOUN american_scientific_journal-707 132 11 , , PUNCT american_scientific_journal-707 132 12 there there ADV american_scientific_journal-707 132 13 existsπ‘Ž1 existsπ‘Ž1 VERB american_scientific_journal-707 132 14 , , PUNCT american_scientific_journal-707 132 15 π‘Ž2 π‘Ž2 PROPN american_scientific_journal-707 132 16 , , PUNCT american_scientific_journal-707 132 17 … … PUNCT american_scientific_journal-707 132 18 . . PUNCT american_scientific_journal-707 132 19 . . PUNCT american_scientific_journal-707 133 1 π‘Žπ‘› π‘Žπ‘› PROPN american_scientific_journal-707 133 2 ∈ ∈ PROPN american_scientific_journal-707 133 3 𝐴 𝐴 PROPN american_scientific_journal-707 133 4 , , PUNCT american_scientific_journal-707 133 5 such such ADJ american_scientific_journal-707 133 6 that that SCONJ american_scientific_journal-707 133 7 𝑦𝑖 𝑦𝑖 PROPN american_scientific_journal-707 133 8 = = PUNCT american_scientific_journal-707 133 9 β„Žβ€²(π‘Žπ‘– β„Žβ€²(π‘Žπ‘– PROPN american_scientific_journal-707 133 10 ) ) PUNCT american_scientific_journal-707 133 11 for for ADP american_scientific_journal-707 133 12 all all PRON american_scientific_journal-707 133 13 1 1 NUM american_scientific_journal-707 133 14 ≀ ≀ NUM american_scientific_journal-707 133 15 𝑖 𝑖 SYM american_scientific_journal-707 133 16 ≀ ≀ NOUN american_scientific_journal-707 133 17 𝑛 𝑛 PRON american_scientific_journal-707 133 18 and and CCONJ american_scientific_journal-707 133 19 hence hence ADV american_scientific_journal-707 133 20 , , PUNCT american_scientific_journal-707 133 21 𝛼 𝛼 PROPN american_scientific_journal-707 133 22 οΏ½ οΏ½ PROPN american_scientific_journal-707 133 23 π‘“π‘Œ(𝑦1 π‘“π‘Œ(𝑦1 PROPN american_scientific_journal-707 133 24 , , PUNCT american_scientific_journal-707 133 25 𝑦2 𝑦2 PROPN american_scientific_journal-707 133 26 , , PUNCT american_scientific_journal-707 133 27 … … PUNCT american_scientific_journal-707 133 28 … … PUNCT american_scientific_journal-707 133 29 . . PUNCT american_scientific_journal-707 133 30 . . PUNCT american_scientific_journal-707 134 1 , , PUNCT american_scientific_journal-707 134 2 𝑦𝑛 𝑦𝑛 NOUN american_scientific_journal-707 134 3 ) ) PUNCT american_scientific_journal-707 134 4 οΏ½ οΏ½ PROPN american_scientific_journal-707 134 5 = = PUNCT american_scientific_journal-707 134 6 𝛼 𝛼 NOUN american_scientific_journal-707 134 7 οΏ½ οΏ½ NOUN american_scientific_journal-707 134 8 π‘“π‘Œ(β„Žβ€²(π‘Ž1 π‘“π‘Œ(β„Žβ€²(π‘Ž1 NOUN american_scientific_journal-707 134 9 ) ) PUNCT american_scientific_journal-707 134 10 , , PUNCT american_scientific_journal-707 134 11 β„Žβ€²(π‘Ž2 β„Žβ€²(π‘Ž2 NOUN american_scientific_journal-707 134 12 ) ) PUNCT american_scientific_journal-707 134 13 , , PUNCT american_scientific_journal-707 134 14 … … PUNCT american_scientific_journal-707 134 15 … … PUNCT american_scientific_journal-707 134 16 . . PUNCT american_scientific_journal-707 134 17 . . PUNCT american_scientific_journal-707 135 1 , , PUNCT american_scientific_journal-707 135 2 β„Žβ€²(π‘Žπ‘› β„Žβ€²(π‘Žπ‘› PROPN american_scientific_journal-707 135 3 ) ) PUNCT american_scientific_journal-707 135 4 ) ) PUNCT american_scientific_journal-707 135 5 οΏ½ οΏ½ PROPN american_scientific_journal-707 135 6 = = SYM american_scientific_journal-707 135 7 𝛼(β„Žβ€²(𝑓𝐴(π‘Ž1 𝛼(β„Žβ€²(𝑓𝐴(π‘Ž1 PROPN american_scientific_journal-707 135 8 , , PUNCT american_scientific_journal-707 135 9 π‘Ž2 π‘Ž2 PROPN american_scientific_journal-707 135 10 , , PUNCT american_scientific_journal-707 135 11 … … PUNCT american_scientific_journal-707 135 12 . . PUNCT american_scientific_journal-707 135 13 . . PUNCT american_scientific_journal-707 136 1 π‘Žπ‘› π‘Žπ‘› PROPN american_scientific_journal-707 136 2 ) ) PUNCT american_scientific_journal-707 136 3 ) ) PUNCT american_scientific_journal-707 136 4 ) ) PUNCT american_scientific_journal-707 137 1 ( ( PUNCT american_scientific_journal-707 137 2 ∡ ∡ NOUN american_scientific_journal-707 137 3 β„Žβ€²is β„Žβ€²is ADP american_scientific_journal-707 137 4 a a DET american_scientific_journal-707 137 5 homomorphism homomorphism NOUN american_scientific_journal-707 137 6 ) ) PUNCT american_scientific_journal-707 137 7 = = SYM american_scientific_journal-707 137 8 β„Ž(𝑓𝐴(π‘Ž1 β„Ž(𝑓𝐴(π‘Ž1 PROPN american_scientific_journal-707 137 9 , , PUNCT american_scientific_journal-707 137 10 π‘Ž2 π‘Ž2 PROPN american_scientific_journal-707 137 11 , , PUNCT american_scientific_journal-707 137 12 … … PUNCT american_scientific_journal-707 137 13 . . PUNCT american_scientific_journal-707 137 14 . . PUNCT american_scientific_journal-707 138 1 π‘Žπ‘› π‘Žπ‘› X american_scientific_journal-707 138 2 ) ) PUNCT american_scientific_journal-707 138 3 ) ) PUNCT american_scientific_journal-707 139 1 = = SYM american_scientific_journal-707 139 2 𝑓𝑋(β„Ž(π‘Ž1 𝑓𝑋(β„Ž(π‘Ž1 NUM american_scientific_journal-707 139 3 ) ) PUNCT american_scientific_journal-707 139 4 , , PUNCT american_scientific_journal-707 139 5 β„Ž(π‘Ž2 β„Ž(π‘Ž2 NOUN american_scientific_journal-707 139 6 ) ) PUNCT american_scientific_journal-707 139 7 , , PUNCT american_scientific_journal-707 139 8 … … PUNCT american_scientific_journal-707 139 9 . . PUNCT american_scientific_journal-707 139 10 . . PUNCT american_scientific_journal-707 140 1 β„Ž(π‘Žπ‘› β„Ž(π‘Žπ‘› NOUN american_scientific_journal-707 140 2 ) ) PUNCT american_scientific_journal-707 140 3 ) ) PUNCT american_scientific_journal-707 141 1 ( ( PUNCT american_scientific_journal-707 141 2 ∡ ∡ X american_scientific_journal-707 141 3 β„Žis β„Žis VERB american_scientific_journal-707 141 4 a a DET american_scientific_journal-707 141 5 homomorphism homomorphism NOUN american_scientific_journal-707 141 6 ) ) PUNCT american_scientific_journal-707 141 7 = = SYM american_scientific_journal-707 141 8 𝑓𝑋(𝛼(β„Žβ€²(π‘Ž1)),𝛼(β„Žβ€²(π‘Ž2 𝑓𝑋(𝛼(β„Žβ€²(π‘Ž1)),𝛼(β„Žβ€²(π‘Ž2 NOUN american_scientific_journal-707 141 9 ) ) PUNCT american_scientific_journal-707 141 10 ) ) PUNCT american_scientific_journal-707 141 11 , , PUNCT american_scientific_journal-707 141 12 … … PUNCT american_scientific_journal-707 141 13 . . PUNCT american_scientific_journal-707 142 1 .𝛼(β„Žβ€²(π‘Žπ‘› .𝛼(β„Žβ€²(π‘Žπ‘› PUNCT american_scientific_journal-707 142 2 ) ) PUNCT american_scientific_journal-707 142 3 ) ) PUNCT american_scientific_journal-707 142 4 ) ) PUNCT american_scientific_journal-707 143 1 = = PUNCT american_scientific_journal-707 143 2 𝑓𝑋(𝛼(𝑦1),𝛼(𝑦2 𝑓𝑋(𝛼(𝑦1),𝛼(𝑦2 NOUN american_scientific_journal-707 143 3 ) ) PUNCT american_scientific_journal-707 143 4 , , PUNCT american_scientific_journal-707 143 5 … … PUNCT american_scientific_journal-707 143 6 . . PUNCT american_scientific_journal-707 144 1 .𝛼(𝑦𝑛 .𝛼(𝑦𝑛 PUNCT american_scientific_journal-707 144 2 ) ) PUNCT american_scientific_journal-707 144 3 ) ) PUNCT american_scientific_journal-707 145 1 therefore therefore ADV american_scientific_journal-707 145 2 , , PUNCT american_scientific_journal-707 145 3 𝛼 𝛼 X american_scientific_journal-707 145 4 is be AUX american_scientific_journal-707 145 5 a a DET american_scientific_journal-707 145 6 homomorphism homomorphism NOUN american_scientific_journal-707 145 7 . . PUNCT american_scientific_journal-707 146 1 thus thus ADV american_scientific_journal-707 146 2 , , PUNCT american_scientific_journal-707 146 3 by by ADP american_scientific_journal-707 146 4 the the DET american_scientific_journal-707 146 5 results result NOUN american_scientific_journal-707 146 6 in in ADP american_scientific_journal-707 146 7 ( ( PUNCT american_scientific_journal-707 146 8 a a NOUN american_scientific_journal-707 146 9 ) ) PUNCT american_scientific_journal-707 146 10 , , PUNCT american_scientific_journal-707 146 11 ( ( PUNCT american_scientific_journal-707 146 12 b b NOUN american_scientific_journal-707 146 13 ) ) PUNCT american_scientific_journal-707 146 14 , , PUNCT american_scientific_journal-707 146 15 ( ( PUNCT american_scientific_journal-707 146 16 c c X american_scientific_journal-707 146 17 ) ) PUNCT american_scientific_journal-707 146 18 and and CCONJ american_scientific_journal-707 146 19 ( ( PUNCT american_scientific_journal-707 146 20 d d X american_scientific_journal-707 146 21 ) ) PUNCT american_scientific_journal-707 146 22 we we PRON american_scientific_journal-707 146 23 get get VERB american_scientific_journal-707 146 24 that that SCONJ american_scientific_journal-707 146 25 𝛼 𝛼 PROPN american_scientific_journal-707 146 26 is be AUX american_scientific_journal-707 146 27 an an DET american_scientific_journal-707 146 28 isomorphism isomorphism NOUN american_scientific_journal-707 146 29 . . PUNCT american_scientific_journal-707 147 1 now now ADV american_scientific_journal-707 147 2 , , PUNCT american_scientific_journal-707 147 3 for for ADP american_scientific_journal-707 147 4 any any DET american_scientific_journal-707 147 5 π‘Ž π‘Ž PROPN american_scientific_journal-707 147 6 ∈ ∈ PROPN american_scientific_journal-707 147 7 𝐴 𝐴 PROPN american_scientific_journal-707 147 8 , , PUNCT american_scientific_journal-707 147 9 consider consider VERB american_scientific_journal-707 147 10 , , PUNCT american_scientific_journal-707 147 11 69 69 NUM american_scientific_journal-707 147 12 american american ADJ american_scientific_journal-707 147 13 scientific scientific ADJ american_scientific_journal-707 147 14 research research NOUN american_scientific_journal-707 147 15 journal journal NOUN american_scientific_journal-707 147 16 for for ADP american_scientific_journal-707 147 17 engineering engineering NOUN american_scientific_journal-707 147 18 , , PUNCT american_scientific_journal-707 147 19 technology technology NOUN american_scientific_journal-707 147 20 , , PUNCT american_scientific_journal-707 147 21 and and CCONJ american_scientific_journal-707 147 22 sciences science NOUN american_scientific_journal-707 147 23 ( ( PUNCT american_scientific_journal-707 147 24 asrjets asrjet NOUN american_scientific_journal-707 147 25 ) ) PUNCT american_scientific_journal-707 147 26 ( ( PUNCT american_scientific_journal-707 147 27 2014 2014 NUM american_scientific_journal-707 147 28 ) ) PUNCT american_scientific_journal-707 147 29 volume volume NOUN american_scientific_journal-707 147 30 10 10 NUM american_scientific_journal-707 147 31 , , PUNCT american_scientific_journal-707 147 32 no no DET american_scientific_journal-707 147 33 1 1 NUM american_scientific_journal-707 147 34 , , PUNCT american_scientific_journal-707 147 35 pp pp ADV american_scientific_journal-707 147 36 61 61 NUM american_scientific_journal-707 147 37 - - SYM american_scientific_journal-707 147 38 73 73 NUM american_scientific_journal-707 147 39 π›Όπ‘œβ„Žβ€²(π‘Ž π›Όπ‘œβ„Žβ€²(π‘Ž NOUN american_scientific_journal-707 147 40 ) ) PUNCT american_scientific_journal-707 147 41 = = SYM american_scientific_journal-707 148 1 𝛼(β„Žβ€²(π‘Ž 𝛼(β„Žβ€²(π‘Ž NOUN american_scientific_journal-707 148 2 ) ) PUNCT american_scientific_journal-707 148 3 ) ) PUNCT american_scientific_journal-707 148 4 = = SYM american_scientific_journal-707 148 5 β„Ž(π‘Ž β„Ž(π‘Ž PROPN american_scientific_journal-707 148 6 ) ) PUNCT american_scientific_journal-707 148 7 ( ( PUNCT american_scientific_journal-707 148 8 by by ADP american_scientific_journal-707 148 9 the the DET american_scientific_journal-707 148 10 definition definition NOUN american_scientific_journal-707 148 11 of of ADP american_scientific_journal-707 148 12 𝛼 𝛼 NOUN american_scientific_journal-707 148 13 ) ) PUNCT american_scientific_journal-707 148 14 = = SYM american_scientific_journal-707 148 15 β„Ž β„Ž PROPN american_scientific_journal-707 148 16 ( ( PUNCT american_scientific_journal-707 148 17 𝐼𝐴(π‘Ž 𝐼𝐴(π‘Ž PROPN american_scientific_journal-707 148 18 ) ) PUNCT american_scientific_journal-707 148 19 ) ) PUNCT american_scientific_journal-707 149 1 ( ( PUNCT american_scientific_journal-707 149 2 ∡ ∡ X american_scientific_journal-707 149 3 𝐼𝐴is 𝐼𝐴is PROPN american_scientific_journal-707 149 4 an an DET american_scientific_journal-707 149 5 identity identity NOUN american_scientific_journal-707 149 6 map map NOUN american_scientific_journal-707 149 7 on on ADP american_scientific_journal-707 149 8 a a PRON american_scientific_journal-707 149 9 ) ) PUNCT american_scientific_journal-707 150 1 = = SYM american_scientific_journal-707 150 2 β„Žπ‘œπΌπ΄ β„Žπ‘œπΌπ΄ NOUN american_scientific_journal-707 150 3 ( ( PUNCT american_scientific_journal-707 150 4 π‘Ž π‘Ž NOUN american_scientific_journal-707 150 5 ) ) PUNCT american_scientific_journal-707 150 6 therefore therefore ADV american_scientific_journal-707 150 7 , , PUNCT american_scientific_journal-707 150 8 π›Όπ‘œβ„Žβ€² π›Όπ‘œβ„Žβ€² NOUN american_scientific_journal-707 150 9 = = SYM american_scientific_journal-707 150 10 β„Žπ‘œπΌπ΄ β„Žπ‘œπΌπ΄ NOUN american_scientific_journal-707 150 11 and and CCONJ american_scientific_journal-707 150 12 hence hence ADV american_scientific_journal-707 150 13 β„Ž β„Ž ADP american_scientific_journal-707 150 14 ∼ ∼ NOUN american_scientific_journal-707 150 15 β„Žβ€² β„Žβ€² ADJ american_scientific_journal-707 150 16 ( ( PUNCT american_scientific_journal-707 150 17 or or CCONJ american_scientific_journal-707 150 18 β„Ž β„Ž PROPN american_scientific_journal-707 150 19 and and CCONJ american_scientific_journal-707 150 20 β„Žβ€² β„Žβ€² NOUN american_scientific_journal-707 150 21 are be AUX american_scientific_journal-707 150 22 associates associate NOUN american_scientific_journal-707 150 23 ) ) PUNCT american_scientific_journal-707 150 24 . . PUNCT american_scientific_journal-707 151 1 claim claim NOUN american_scientific_journal-707 151 2 2 2 NUM american_scientific_journal-707 151 3 : : PUNCT american_scientific_journal-707 151 4 𝑔~𝑔′ 𝑔~𝑔′ X american_scientific_journal-707 151 5 ( ( PUNCT american_scientific_journal-707 151 6 or or CCONJ american_scientific_journal-707 151 7 𝑔 𝑔 PROPN american_scientific_journal-707 151 8 and and CCONJ american_scientific_journal-707 151 9 𝑔′ 𝑔′ PROPN american_scientific_journal-707 151 10 are be AUX american_scientific_journal-707 151 11 associates associate NOUN american_scientific_journal-707 151 12 ) ) PUNCT american_scientific_journal-707 151 13 figure figure NOUN american_scientific_journal-707 151 14 4 4 NUM american_scientific_journal-707 151 15 : : PUNCT american_scientific_journal-707 151 16 two two NUM american_scientific_journal-707 151 17 decompositions decomposition NOUN american_scientific_journal-707 151 18 of of ADP american_scientific_journal-707 151 19 πœ‹ πœ‹ NOUN american_scientific_journal-707 151 20 from from ADP american_scientific_journal-707 151 21 claim claim NOUN american_scientific_journal-707 151 22 ( ( PUNCT american_scientific_journal-707 151 23 1 1 X american_scientific_journal-707 151 24 ) ) PUNCT american_scientific_journal-707 151 25 we we PRON american_scientific_journal-707 151 26 have have VERB american_scientific_journal-707 151 27 , , PUNCT american_scientific_journal-707 151 28 the the DET american_scientific_journal-707 151 29 identity identity NOUN american_scientific_journal-707 151 30 map map NOUN american_scientific_journal-707 151 31 𝐼𝐴 𝐼𝐴 PROPN american_scientific_journal-707 151 32 on on ADP american_scientific_journal-707 151 33 the the DET american_scientific_journal-707 151 34 domain domain NOUN american_scientific_journal-707 151 35 algebra algebra NOUN american_scientific_journal-707 151 36 𝐴and 𝐴and PROPN american_scientific_journal-707 151 37 𝛼:π‘Œ 𝛼:π‘Œ PROPN american_scientific_journal-707 151 38 β†’ β†’ SYM american_scientific_journal-707 151 39 𝑋 𝑋 PROPN american_scientific_journal-707 151 40 , , PUNCT american_scientific_journal-707 151 41 which which PRON american_scientific_journal-707 151 42 is be AUX american_scientific_journal-707 151 43 defined define VERB american_scientific_journal-707 151 44 by by ADP american_scientific_journal-707 151 45 𝛼 𝛼 PROPN american_scientific_journal-707 151 46 οΏ½ οΏ½ PROPN american_scientific_journal-707 151 47 β„Žβ€²(π‘Ž β„Žβ€²(π‘Ž NOUN american_scientific_journal-707 151 48 ) ) PUNCT american_scientific_journal-707 151 49 οΏ½ οΏ½ PROPN american_scientific_journal-707 151 50 = = SYM american_scientific_journal-707 151 51 β„Ž(π‘Ž β„Ž(π‘Ž PROPN american_scientific_journal-707 151 52 ) ) PUNCT american_scientific_journal-707 151 53 for for ADP american_scientific_journal-707 151 54 all all DET american_scientific_journal-707 151 55 β„Žβ€²(π‘Ž β„Žβ€²(π‘Ž NOUN american_scientific_journal-707 151 56 ) ) PUNCT american_scientific_journal-707 151 57 ∈ ∈ PROPN american_scientific_journal-707 151 58 π‘Œ π‘Œ PROPN american_scientific_journal-707 151 59 where where SCONJ american_scientific_journal-707 151 60 π‘Ž π‘Ž PROPN american_scientific_journal-707 151 61 ∈ ∈ PROPN american_scientific_journal-707 151 62 𝐴 𝐴 PROPN american_scientific_journal-707 151 63 , , PUNCT american_scientific_journal-707 151 64 which which PRON american_scientific_journal-707 151 65 are be AUX american_scientific_journal-707 151 66 isomorphisms isomorphism NOUN american_scientific_journal-707 151 67 . . PUNCT american_scientific_journal-707 152 1 also also ADV american_scientific_journal-707 152 2 , , PUNCT american_scientific_journal-707 152 3 consider consider VERB american_scientific_journal-707 152 4 the the DET american_scientific_journal-707 152 5 identity identity NOUN american_scientific_journal-707 152 6 map map NOUN american_scientific_journal-707 152 7 𝐼𝐡 𝐼𝐡 PROPN american_scientific_journal-707 152 8 on on ADP american_scientific_journal-707 152 9 𝐡 𝐡 PROPN american_scientific_journal-707 152 10 which which PRON american_scientific_journal-707 152 11 is be AUX american_scientific_journal-707 152 12 also also ADV american_scientific_journal-707 152 13 an an DET american_scientific_journal-707 152 14 isomorphism isomorphism NOUN american_scientific_journal-707 152 15 . . PUNCT american_scientific_journal-707 153 1 since since SCONJ american_scientific_journal-707 153 2 β„Žβ€²:𝐴 β„Žβ€²:𝐴 PRON american_scientific_journal-707 153 3 β†’ β†’ PUNCT american_scientific_journal-707 153 4 π‘Œ π‘Œ PROPN american_scientific_journal-707 153 5 is be AUX american_scientific_journal-707 153 6 an an DET american_scientific_journal-707 153 7 epimorphism epimorphism NOUN american_scientific_journal-707 153 8 and and CCONJ american_scientific_journal-707 153 9 hence hence ADV american_scientific_journal-707 153 10 a a DET american_scientific_journal-707 153 11 surjection surjection NOUN american_scientific_journal-707 153 12 , , PUNCT american_scientific_journal-707 153 13 for for ADP american_scientific_journal-707 153 14 any any DET american_scientific_journal-707 153 15 𝑦 𝑦 PROPN american_scientific_journal-707 153 16 ∈ ∈ PROPN american_scientific_journal-707 153 17 π‘Œ π‘Œ PROPN american_scientific_journal-707 153 18 , , PUNCT american_scientific_journal-707 153 19 there there PRON american_scientific_journal-707 153 20 exist exist VERB american_scientific_journal-707 153 21 π‘Ž π‘Ž PRON american_scientific_journal-707 153 22 ∈ ∈ PROPN american_scientific_journal-707 153 23 𝐴 𝐴 NOUN american_scientific_journal-707 153 24 such such ADJ american_scientific_journal-707 153 25 that that PRON american_scientific_journal-707 153 26 𝑦 𝑦 NOUN american_scientific_journal-707 153 27 = = SYM american_scientific_journal-707 153 28 β„Žβ€²(π‘Ž β„Žβ€²(π‘Ž ADJ american_scientific_journal-707 153 29 ) ) PUNCT american_scientific_journal-707 153 30 . . PUNCT american_scientific_journal-707 154 1 then then ADV american_scientific_journal-707 154 2 consider consider VERB american_scientific_journal-707 154 3 , , PUNCT american_scientific_journal-707 154 4 𝐼𝐡 𝐼𝐡 PROPN american_scientific_journal-707 154 5 ∘ ∘ PROPN american_scientific_journal-707 154 6 𝑔′(𝑦 𝑔′(𝑦 PROPN american_scientific_journal-707 154 7 ) ) PUNCT american_scientific_journal-707 154 8 = = SYM american_scientific_journal-707 155 1 πΌπ΅π‘œπ‘”β€²(β„Žβ€²(π‘Ž πΌπ΅π‘œπ‘”β€²(β„Žβ€²(π‘Ž NOUN american_scientific_journal-707 155 2 ) ) PUNCT american_scientific_journal-707 155 3 ) ) PUNCT american_scientific_journal-707 156 1 in𝐡 in𝐡 NOUN american_scientific_journal-707 156 2 = = SYM american_scientific_journal-707 156 3 𝐼𝐡(𝑔′ 𝐼𝐡(𝑔′ PROPN american_scientific_journal-707 156 4 οΏ½ οΏ½ PROPN american_scientific_journal-707 156 5 β„Žβ€²(π‘Ž β„Žβ€²(π‘Ž NOUN american_scientific_journal-707 156 6 ) ) PUNCT american_scientific_journal-707 156 7 οΏ½ οΏ½ PROPN american_scientific_journal-707 156 8 ) ) PUNCT american_scientific_journal-707 156 9 in𝐡 in𝐡 NOUN american_scientific_journal-707 156 10 = = SYM american_scientific_journal-707 156 11 𝑔′ 𝑔′ PROPN american_scientific_journal-707 156 12 οΏ½ οΏ½ PROPN american_scientific_journal-707 156 13 β„Žβ€²(π‘Ž) β„Žβ€²(π‘Ž) PROPN american_scientific_journal-707 156 14 οΏ½ οΏ½ PROPN american_scientific_journal-707 156 15 in𝐡 in𝐡 NOUN american_scientific_journal-707 156 16 = = PRON american_scientific_journal-707 156 17 𝑔′ 𝑔′ ADV american_scientific_journal-707 156 18 ∘ ∘ X american_scientific_journal-707 156 19 β„Žβ€²(π‘Ž)in𝐡 β„Žβ€²(π‘Ž)in𝐡 NOUN american_scientific_journal-707 156 20 = = SYM american_scientific_journal-707 156 21 𝑓(π‘Ž 𝑓(π‘Ž NOUN american_scientific_journal-707 156 22 ) ) PUNCT american_scientific_journal-707 156 23 𝑖𝑛𝐡 𝑖𝑛𝐡 VERB american_scientific_journal-707 156 24 70 70 NUM american_scientific_journal-707 156 25 american american ADJ american_scientific_journal-707 156 26 scientific scientific ADJ american_scientific_journal-707 156 27 research research NOUN american_scientific_journal-707 156 28 journal journal NOUN american_scientific_journal-707 156 29 for for ADP american_scientific_journal-707 156 30 engineering engineering NOUN american_scientific_journal-707 156 31 , , PUNCT american_scientific_journal-707 156 32 technology technology NOUN american_scientific_journal-707 156 33 , , PUNCT american_scientific_journal-707 156 34 and and CCONJ american_scientific_journal-707 156 35 sciences science NOUN american_scientific_journal-707 156 36 ( ( PUNCT american_scientific_journal-707 156 37 asrjets asrjet NOUN american_scientific_journal-707 156 38 ) ) PUNCT american_scientific_journal-707 156 39 ( ( PUNCT american_scientific_journal-707 156 40 2014 2014 NUM american_scientific_journal-707 156 41 ) ) PUNCT american_scientific_journal-707 156 42 volume volume NOUN american_scientific_journal-707 156 43 10 10 NUM american_scientific_journal-707 156 44 , , PUNCT american_scientific_journal-707 156 45 no no DET american_scientific_journal-707 156 46 1 1 NUM american_scientific_journal-707 156 47 , , PUNCT american_scientific_journal-707 156 48 pp pp ADV american_scientific_journal-707 156 49 61 61 NUM american_scientific_journal-707 156 50 - - SYM american_scientific_journal-707 156 51 73 73 NUM american_scientific_journal-707 157 1 = = SYM american_scientific_journal-707 157 2 𝑔 𝑔 PROPN american_scientific_journal-707 157 3 ∘ ∘ NUM american_scientific_journal-707 157 4 β„Ž(π‘Ž)in𝐡 β„Ž(π‘Ž)in𝐡 PROPN american_scientific_journal-707 157 5 ( ( PUNCT american_scientific_journal-707 157 6 ∡ ∡ NOUN american_scientific_journal-707 157 7 𝑓 𝑓 PRON american_scientific_journal-707 157 8 = = SYM american_scientific_journal-707 157 9 𝑔 𝑔 PROPN american_scientific_journal-707 157 10 ∘ ∘ NUM american_scientific_journal-707 157 11 β„Ž β„Ž PROPN american_scientific_journal-707 157 12 ) ) PUNCT american_scientific_journal-707 157 13 = = SYM american_scientific_journal-707 157 14 𝑔(β„Ž(π‘Ž))in𝐡 𝑔(β„Ž(π‘Ž))in𝐡 PROPN american_scientific_journal-707 157 15 = = SYM american_scientific_journal-707 157 16 𝑔(𝛼 𝑔(𝛼 PROPN american_scientific_journal-707 157 17 οΏ½ οΏ½ PROPN american_scientific_journal-707 157 18 β„Žβ€²(π‘Ž) β„Žβ€²(π‘Ž) PROPN american_scientific_journal-707 157 19 οΏ½ οΏ½ PROPN american_scientific_journal-707 157 20 )in𝐡 )in𝐡 PUNCT american_scientific_journal-707 157 21 ( ( PUNCT american_scientific_journal-707 157 22 by by ADP american_scientific_journal-707 157 23 the the DET american_scientific_journal-707 157 24 definition definition NOUN american_scientific_journal-707 157 25 of of ADP american_scientific_journal-707 157 26 𝛼 𝛼 NOUN american_scientific_journal-707 157 27 ) ) PUNCT american_scientific_journal-707 157 28 = = SYM american_scientific_journal-707 157 29 𝑔 𝑔 PROPN american_scientific_journal-707 157 30 ∘ ∘ PROPN american_scientific_journal-707 157 31 𝛼 𝛼 X american_scientific_journal-707 157 32 οΏ½ οΏ½ PROPN american_scientific_journal-707 157 33 β„Žβ€²(π‘Ž) β„Žβ€²(π‘Ž) PROPN american_scientific_journal-707 157 34 οΏ½ οΏ½ PROPN american_scientific_journal-707 157 35 in𝐡 in𝐡 NOUN american_scientific_journal-707 157 36 = = PRON american_scientific_journal-707 157 37 𝑔 𝑔 PROPN american_scientific_journal-707 157 38 ∘ ∘ X american_scientific_journal-707 157 39 𝛼(𝑦)in𝐡 𝛼(𝑦)in𝐡 PROPN american_scientific_journal-707 157 40 ( ( PUNCT american_scientific_journal-707 157 41 ∡ ∡ X american_scientific_journal-707 157 42 𝑦 𝑦 X american_scientific_journal-707 157 43 = = SYM american_scientific_journal-707 157 44 β„Žβ€²(π‘Ž β„Žβ€²(π‘Ž ADJ american_scientific_journal-707 157 45 ) ) PUNCT american_scientific_journal-707 157 46 ) ) PUNCT american_scientific_journal-707 158 1 therefore therefore ADV american_scientific_journal-707 158 2 , , PUNCT american_scientific_journal-707 158 3 𝐼𝐡 𝐼𝐡 PROPN american_scientific_journal-707 158 4 ∘ ∘ PROPN american_scientific_journal-707 158 5 𝑔′ 𝑔′ ADV american_scientific_journal-707 158 6 = = SYM american_scientific_journal-707 158 7 𝑔 𝑔 PROPN american_scientific_journal-707 158 8 ∘ ∘ PROPN american_scientific_journal-707 158 9 𝛼. 𝛼. PROPN american_scientific_journal-707 158 10 thus thus ADV american_scientific_journal-707 158 11 , , PUNCT american_scientific_journal-707 158 12 we we PRON american_scientific_journal-707 158 13 have have VERB american_scientific_journal-707 158 14 two two NUM american_scientific_journal-707 158 15 isomorphisms isomorphism NOUN american_scientific_journal-707 158 16 , , PUNCT american_scientific_journal-707 158 17 𝐼𝐡 𝐼𝐡 PROPN american_scientific_journal-707 158 18 on on ADP american_scientific_journal-707 158 19 𝐡 𝐡 PROPN american_scientific_journal-707 158 20 and and CCONJ american_scientific_journal-707 158 21 𝛼:π‘Œ 𝛼:π‘Œ PROPN american_scientific_journal-707 158 22 β†’ β†’ SYM american_scientific_journal-707 158 23 𝑋 𝑋 NOUN american_scientific_journal-707 158 24 satisfying satisfy VERB american_scientific_journal-707 158 25 that that SCONJ american_scientific_journal-707 158 26 , , PUNCT american_scientific_journal-707 158 27 the the DET american_scientific_journal-707 158 28 diagram diagram NOUN american_scientific_journal-707 158 29 ; ; PUNCT american_scientific_journal-707 158 30 figure figure NOUN american_scientific_journal-707 158 31 5 5 NUM american_scientific_journal-707 158 32 : : PUNCT american_scientific_journal-707 158 33 associate associate NOUN american_scientific_journal-707 158 34 homomorphisms homomorphisms PROPN american_scientific_journal-707 158 35 𝑔 𝑔 PROPN american_scientific_journal-707 158 36 and and CCONJ american_scientific_journal-707 158 37 𝑔′ 𝑔′ ADV american_scientific_journal-707 158 38 is be AUX american_scientific_journal-707 158 39 commutative commutative ADJ american_scientific_journal-707 158 40 ; ; PUNCT american_scientific_journal-707 158 41 that that ADV american_scientific_journal-707 158 42 is is ADV american_scientific_journal-707 158 43 , , PUNCT american_scientific_journal-707 158 44 𝐼𝐡 𝐼𝐡 PROPN american_scientific_journal-707 158 45 ∘ ∘ PROPN american_scientific_journal-707 158 46 𝑔′ 𝑔′ ADV american_scientific_journal-707 158 47 = = PUNCT american_scientific_journal-707 158 48 𝑔 𝑔 PROPN american_scientific_journal-707 158 49 ∘ ∘ PROPN american_scientific_journal-707 158 50 𝛼. 𝛼. PROPN american_scientific_journal-707 158 51 therefore therefore ADV american_scientific_journal-707 158 52 , , PUNCT american_scientific_journal-707 158 53 𝑔~𝑔′ 𝑔~𝑔′ PUNCT american_scientific_journal-707 158 54 ( ( PUNCT american_scientific_journal-707 158 55 or or CCONJ american_scientific_journal-707 158 56 𝑔 𝑔 PROPN american_scientific_journal-707 158 57 and and CCONJ american_scientific_journal-707 158 58 𝑔′ 𝑔′ PROPN american_scientific_journal-707 158 59 are be AUX american_scientific_journal-707 158 60 associate associate ADJ american_scientific_journal-707 158 61 ) ) PUNCT american_scientific_journal-707 158 62 . . PUNCT american_scientific_journal-707 159 1 ∎ ∎ PROPN american_scientific_journal-707 159 2 corollary corollary NOUN american_scientific_journal-707 159 3 4.2.1 4.2.1 NOUN american_scientific_journal-707 159 4 : : PUNCT american_scientific_journal-707 159 5 let let VERB american_scientific_journal-707 159 6 a a PRON american_scientific_journal-707 159 7 and and CCONJ american_scientific_journal-707 159 8 b b NOUN american_scientific_journal-707 159 9 be be AUX american_scientific_journal-707 159 10 algebras algebra NOUN american_scientific_journal-707 159 11 of of ADP american_scientific_journal-707 159 12 a a DET american_scientific_journal-707 159 13 given give VERB american_scientific_journal-707 159 14 type type NOUN american_scientific_journal-707 159 15 β„± β„± PROPN american_scientific_journal-707 159 16 and and CCONJ american_scientific_journal-707 159 17 πœ‹ πœ‹ NOUN american_scientific_journal-707 159 18 : : PUNCT american_scientific_journal-707 159 19 𝐴 𝐴 PROPN american_scientific_journal-707 159 20 β†’ β†’ SYM american_scientific_journal-707 159 21 𝐡 𝐡 NOUN american_scientific_journal-707 159 22 be be AUX american_scientific_journal-707 159 23 a a DET american_scientific_journal-707 159 24 homomorphism homomorphism NOUN american_scientific_journal-707 159 25 . . PUNCT american_scientific_journal-707 160 1 let let VERB american_scientific_journal-707 160 2 πœ‹ πœ‹ NOUN american_scientific_journal-707 160 3 = = VERB american_scientific_journal-707 160 4 𝑔 𝑔 ADP american_scientific_journal-707 160 5 ∘ ∘ NOUN american_scientific_journal-707 160 6 β„Ž β„Ž NOUN american_scientific_journal-707 160 7 be be VERB american_scientific_journal-707 160 8 any any DET american_scientific_journal-707 160 9 decomposition decomposition NOUN american_scientific_journal-707 160 10 of of ADP american_scientific_journal-707 160 11 πœ‹ πœ‹ PRON american_scientific_journal-707 160 12 where where SCONJ american_scientific_journal-707 160 13 β„Ž:𝐴 β„Ž:𝐴 PROPN american_scientific_journal-707 160 14 β†’ β†’ PUNCT american_scientific_journal-707 160 15 𝑋 𝑋 PROPN american_scientific_journal-707 160 16 is be AUX american_scientific_journal-707 160 17 an an DET american_scientific_journal-707 160 18 epimorphism epimorphism NOUN american_scientific_journal-707 160 19 and and CCONJ american_scientific_journal-707 160 20 𝑔:𝑋 𝑔:𝑋 NUM american_scientific_journal-707 160 21 β†’ β†’ SYM american_scientific_journal-707 160 22 π΅π‘–π‘ π‘Ž π΅π‘–π‘ π‘Ž PROPN american_scientific_journal-707 160 23 monomorphism monomorphism NOUN american_scientific_journal-707 160 24 for for ADP american_scientific_journal-707 160 25 some some DET american_scientific_journal-707 160 26 algebra algebra NOUN american_scientific_journal-707 160 27 𝑋 𝑋 NOUN american_scientific_journal-707 160 28 of of ADP american_scientific_journal-707 160 29 the the DET american_scientific_journal-707 160 30 given give VERB american_scientific_journal-707 160 31 type type NOUN american_scientific_journal-707 160 32 β„±. β„±. PROPN american_scientific_journal-707 160 33 then then ADV american_scientific_journal-707 160 34 there there PRON american_scientific_journal-707 160 35 exist exist VERB american_scientific_journal-707 160 36 an an DET american_scientific_journal-707 160 37 isomorphism isomorphism NOUN american_scientific_journal-707 160 38 of of ADP american_scientific_journal-707 160 39 𝑋 𝑋 NOUN american_scientific_journal-707 160 40 onto onto ADP american_scientific_journal-707 160 41 the the DET american_scientific_journal-707 160 42 quotient quotient NOUN american_scientific_journal-707 160 43 algebra algebra NOUN american_scientific_journal-707 160 44 𝐴 𝐴 PROPN american_scientific_journal-707 160 45 πΎπ‘’π‘Ÿπœ‹ πΎπ‘’π‘Ÿπœ‹ PROPN american_scientific_journal-707 160 46 οΏ½ οΏ½ PROPN american_scientific_journal-707 160 47 . . PUNCT american_scientific_journal-707 161 1 proof proof NOUN american_scientific_journal-707 161 2 : : PUNCT american_scientific_journal-707 161 3 put put VERB american_scientific_journal-707 161 4 πœƒ πœƒ NOUN american_scientific_journal-707 161 5 = = PUNCT american_scientific_journal-707 161 6 kerπœ‹ kerπœ‹ NOUN american_scientific_journal-707 161 7 = = SYM american_scientific_journal-707 161 8 { { PUNCT american_scientific_journal-707 161 9 ( ( PUNCT american_scientific_journal-707 161 10 π‘Ž π‘Ž X american_scientific_journal-707 161 11 , , PUNCT american_scientific_journal-707 161 12 𝑏 𝑏 NOUN american_scientific_journal-707 161 13 ) ) PUNCT american_scientific_journal-707 161 14 ∈ ∈ PROPN american_scientific_journal-707 161 15 𝐴 𝐴 PROPN american_scientific_journal-707 161 16 Γ— Γ— PROPN american_scientific_journal-707 161 17 𝐴 𝐴 PROPN american_scientific_journal-707 161 18 ∢ ∢ PROPN american_scientific_journal-707 161 19 πœ‹(π‘Ž πœ‹(π‘Ž PROPN american_scientific_journal-707 161 20 ) ) PUNCT american_scientific_journal-707 162 1 = = SYM american_scientific_journal-707 162 2 πœ‹(𝑏 πœ‹(𝑏 NOUN american_scientific_journal-707 162 3 ) ) PUNCT american_scientific_journal-707 162 4 } } PUNCT american_scientific_journal-707 162 5 . . PUNCT american_scientific_journal-707 163 1 considering consider VERB american_scientific_journal-707 163 2 the the DET american_scientific_journal-707 163 3 natural natural ADJ american_scientific_journal-707 163 4 epimorphismβ„Žβ€²:𝐴 epimorphismβ„Žβ€²:𝐴 NOUN american_scientific_journal-707 163 5 β†’ β†’ SYM american_scientific_journal-707 163 6 𝐴 𝐴 PROPN american_scientific_journal-707 163 7 πœƒ πœƒ NUM american_scientific_journal-707 163 8 οΏ½ οΏ½ PROPN american_scientific_journal-707 163 9 defined define VERB american_scientific_journal-707 163 10 by by ADP american_scientific_journal-707 163 11 β„Žβ€²(π‘Ž β„Žβ€²(π‘Ž NOUN american_scientific_journal-707 163 12 ) ) PUNCT american_scientific_journal-707 163 13 = = PUNCT american_scientific_journal-707 163 14 πœƒ(π‘Ž πœƒ(π‘Ž NOUN american_scientific_journal-707 163 15 ) ) PUNCT american_scientific_journal-707 163 16 forall forall VERB american_scientific_journal-707 163 17 π‘Ž π‘Ž PROPN american_scientific_journal-707 163 18 ∈ ∈ PROPN american_scientific_journal-707 163 19 𝐴 𝐴 NOUN american_scientific_journal-707 163 20 and and CCONJ american_scientific_journal-707 163 21 a a DET american_scientific_journal-707 163 22 monomorphism monomorphism NOUN american_scientific_journal-707 163 23 𝑔′:𝐴 𝑔′:𝐴 NOUN american_scientific_journal-707 163 24 πœƒ πœƒ X american_scientific_journal-707 163 25 οΏ½ οΏ½ PROPN american_scientific_journal-707 163 26 β†’ β†’ SYM american_scientific_journal-707 163 27 𝐡defined 𝐡define VERB american_scientific_journal-707 163 28 by by ADP american_scientific_journal-707 163 29 𝑔′ 𝑔′ ADP american_scientific_journal-707 163 30 οΏ½ οΏ½ NOUN american_scientific_journal-707 163 31 πœƒ(π‘Ž πœƒ(π‘Ž NOUN american_scientific_journal-707 163 32 ) ) PUNCT american_scientific_journal-707 163 33 οΏ½ οΏ½ PROPN american_scientific_journal-707 163 34 = = SYM american_scientific_journal-707 163 35 𝑓(π‘Ž 𝑓(π‘Ž PROPN american_scientific_journal-707 163 36 ) ) PUNCT american_scientific_journal-707 163 37 for for ADP american_scientific_journal-707 163 38 all all DET american_scientific_journal-707 163 39 πœƒ(π‘Ž πœƒ(π‘Ž NOUN american_scientific_journal-707 163 40 ) ) PUNCT american_scientific_journal-707 163 41 ∈ ∈ PROPN american_scientific_journal-707 163 42 𝐴 𝐴 PROPN american_scientific_journal-707 163 43 πœƒ πœƒ PROPN american_scientific_journal-707 163 44 οΏ½ οΏ½ PROPN american_scientific_journal-707 163 45 , , PUNCT american_scientific_journal-707 163 46 π‘Ž π‘Ž PROPN american_scientific_journal-707 163 47 ∈ ∈ PROPN american_scientific_journal-707 163 48 𝐴 𝐴 PROPN american_scientific_journal-707 163 49 , , PUNCT american_scientific_journal-707 163 50 we we PRON american_scientific_journal-707 163 51 get get VERB american_scientific_journal-707 163 52 another another DET american_scientific_journal-707 163 53 decomposition decomposition NOUN american_scientific_journal-707 163 54 πœ‹ πœ‹ NOUN american_scientific_journal-707 163 55 = = NOUN american_scientific_journal-707 163 56 𝑔′ 𝑔′ X american_scientific_journal-707 163 57 ∘ ∘ NOUN american_scientific_journal-707 163 58 β„Žβ€². β„Žβ€². NOUN american_scientific_journal-707 163 59 then then ADV american_scientific_journal-707 163 60 , , PUNCT american_scientific_journal-707 163 61 by by ADP american_scientific_journal-707 163 62 the the DET american_scientific_journal-707 163 63 uniqueness uniqueness NOUN american_scientific_journal-707 163 64 of of ADP american_scientific_journal-707 163 65 decomposition decomposition NOUN american_scientific_journal-707 163 66 of of ADP american_scientific_journal-707 163 67 homomorphisms homomorphism NOUN american_scientific_journal-707 163 68 of of ADP american_scientific_journal-707 163 69 algebras algebra NOUN american_scientific_journal-707 163 70 , , PUNCT american_scientific_journal-707 163 71 we we PRON american_scientific_journal-707 163 72 get get VERB american_scientific_journal-707 163 73 an an DET american_scientific_journal-707 163 74 ismorphism ismorphism NOUN american_scientific_journal-707 163 75 ; ; PUNCT american_scientific_journal-707 163 76 𝛼:𝑋 𝛼:𝑋 PROPN american_scientific_journal-707 163 77 β†’ β†’ SYM american_scientific_journal-707 163 78 𝐴 𝐴 PROPN american_scientific_journal-707 163 79 πœƒ πœƒ NUM american_scientific_journal-707 163 80 οΏ½ οΏ½ PROPN american_scientific_journal-707 163 81 71 71 NUM american_scientific_journal-707 163 82 american american ADJ american_scientific_journal-707 163 83 scientific scientific ADJ american_scientific_journal-707 163 84 research research NOUN american_scientific_journal-707 163 85 journal journal NOUN american_scientific_journal-707 163 86 for for ADP american_scientific_journal-707 163 87 engineering engineering NOUN american_scientific_journal-707 163 88 , , PUNCT american_scientific_journal-707 163 89 technology technology NOUN american_scientific_journal-707 163 90 , , PUNCT american_scientific_journal-707 163 91 and and CCONJ american_scientific_journal-707 163 92 sciences science NOUN american_scientific_journal-707 163 93 ( ( PUNCT american_scientific_journal-707 163 94 asrjets asrjet NOUN american_scientific_journal-707 163 95 ) ) PUNCT american_scientific_journal-707 163 96 ( ( PUNCT american_scientific_journal-707 163 97 2014 2014 NUM american_scientific_journal-707 163 98 ) ) PUNCT american_scientific_journal-707 163 99 volume volume NOUN american_scientific_journal-707 163 100 10 10 NUM american_scientific_journal-707 163 101 , , PUNCT american_scientific_journal-707 163 102 no no DET american_scientific_journal-707 163 103 1 1 NUM american_scientific_journal-707 163 104 , , PUNCT american_scientific_journal-707 163 105 pp pp ADV american_scientific_journal-707 163 106 61 61 NUM american_scientific_journal-707 163 107 - - SYM american_scientific_journal-707 163 108 73 73 NUM american_scientific_journal-707 163 109 figure figure NOUN american_scientific_journal-707 163 110 6 6 NUM american_scientific_journal-707 163 111 : : PUNCT american_scientific_journal-707 163 112 two two NUM american_scientific_journal-707 163 113 decompositions decomposition NOUN american_scientific_journal-707 163 114 of of ADP american_scientific_journal-707 163 115 πœ‹ πœ‹ PRON american_scientific_journal-707 163 116 such such ADJ american_scientific_journal-707 163 117 that that PRON american_scientific_journal-707 163 118 , , PUNCT american_scientific_journal-707 163 119 𝛼 𝛼 NOUN american_scientific_journal-707 163 120 ∘ ∘ NOUN american_scientific_journal-707 163 121 β„Ž β„Ž X american_scientific_journal-707 163 122 = = SYM american_scientific_journal-707 163 123 β„Žβ€² β„Žβ€² NOUN american_scientific_journal-707 163 124 and and CCONJ american_scientific_journal-707 163 125 𝑔′ 𝑔′ ADV american_scientific_journal-707 163 126 ∘ ∘ NOUN american_scientific_journal-707 163 127 𝛼 𝛼 X american_scientific_journal-707 163 128 = = NOUN american_scientific_journal-707 163 129 𝑔. 𝑔. NOUN american_scientific_journal-707 163 130 ∎ ∎ PROPN american_scientific_journal-707 164 1 5 5 NUM american_scientific_journal-707 164 2 . . PUNCT american_scientific_journal-707 164 3 conclusion conclusion NOUN american_scientific_journal-707 164 4 and and CCONJ american_scientific_journal-707 164 5 recommendation recommendation NOUN american_scientific_journal-707 164 6 on on ADP american_scientific_journal-707 164 7 the the DET american_scientific_journal-707 164 8 basis basis NOUN american_scientific_journal-707 164 9 of of ADP american_scientific_journal-707 164 10 the the DET american_scientific_journal-707 164 11 updated update VERB american_scientific_journal-707 164 12 fundamental fundamental ADJ american_scientific_journal-707 164 13 theorem theorem NOUN american_scientific_journal-707 164 14 of of ADP american_scientific_journal-707 164 15 homomorphisms homomorphism NOUN american_scientific_journal-707 164 16 of of ADP american_scientific_journal-707 164 17 general general ADJ american_scientific_journal-707 164 18 universal universal ADJ american_scientific_journal-707 164 19 algebras algebra NOUN american_scientific_journal-707 164 20 which which PRON american_scientific_journal-707 164 21 is be AUX american_scientific_journal-707 164 22 formulated formulate VERB american_scientific_journal-707 164 23 and and CCONJ american_scientific_journal-707 164 24 proved prove VERB american_scientific_journal-707 164 25 in in ADP american_scientific_journal-707 164 26 this this DET american_scientific_journal-707 164 27 paper paper NOUN american_scientific_journal-707 164 28 , , PUNCT american_scientific_journal-707 164 29 we we PRON american_scientific_journal-707 164 30 can can AUX american_scientific_journal-707 164 31 conclude conclude VERB american_scientific_journal-707 164 32 that that SCONJ american_scientific_journal-707 164 33 any any DET american_scientific_journal-707 164 34 homomorphism homomorphism NOUN american_scientific_journal-707 164 35 of of ADP american_scientific_journal-707 164 36 algebras algebra NOUN american_scientific_journal-707 164 37 of of ADP american_scientific_journal-707 164 38 a a DET american_scientific_journal-707 164 39 given give VERB american_scientific_journal-707 164 40 type type NOUN american_scientific_journal-707 164 41 can can AUX american_scientific_journal-707 164 42 be be AUX american_scientific_journal-707 164 43 decomposed decompose VERB american_scientific_journal-707 164 44 as as ADP american_scientific_journal-707 164 45 a a DET american_scientific_journal-707 164 46 composition composition NOUN american_scientific_journal-707 164 47 of of ADP american_scientific_journal-707 164 48 a a DET american_scientific_journal-707 164 49 monomorphism monomorphism NOUN american_scientific_journal-707 164 50 and and CCONJ american_scientific_journal-707 164 51 an an DET american_scientific_journal-707 164 52 epimorphism epimorphism NOUN american_scientific_journal-707 164 53 and and CCONJ american_scientific_journal-707 164 54 this this DET american_scientific_journal-707 164 55 decomposition decomposition NOUN american_scientific_journal-707 164 56 is be AUX american_scientific_journal-707 164 57 unique unique ADJ american_scientific_journal-707 164 58 upto upto NOUN american_scientific_journal-707 164 59 associate associate NOUN american_scientific_journal-707 164 60 ; ; PUNCT american_scientific_journal-707 164 61 in in ADP american_scientific_journal-707 164 62 the the DET american_scientific_journal-707 164 63 sense sense NOUN american_scientific_journal-707 164 64 that that SCONJ american_scientific_journal-707 164 65 , , PUNCT american_scientific_journal-707 164 66 if if SCONJ american_scientific_journal-707 164 67 a a DET american_scientific_journal-707 164 68 given give VERB american_scientific_journal-707 164 69 homomorphism homomorphism PROPN american_scientific_journal-707 164 70 πœ‹ πœ‹ NOUN american_scientific_journal-707 164 71 of of ADP american_scientific_journal-707 164 72 algebras algebra NOUN american_scientific_journal-707 164 73 of of ADP american_scientific_journal-707 164 74 a a DET american_scientific_journal-707 164 75 given give VERB american_scientific_journal-707 164 76 type type NOUN american_scientific_journal-707 164 77 can can AUX american_scientific_journal-707 164 78 be be AUX american_scientific_journal-707 164 79 decomposed decompose VERB american_scientific_journal-707 164 80 in in ADP american_scientific_journal-707 164 81 one one NUM american_scientific_journal-707 164 82 way way NOUN american_scientific_journal-707 164 83 as as SCONJ american_scientific_journal-707 164 84 πœ‹ πœ‹ NOUN american_scientific_journal-707 164 85 = = NOUN american_scientific_journal-707 164 86 𝑔 𝑔 ADJ american_scientific_journal-707 164 87 ∘ ∘ ADJ american_scientific_journal-707 164 88 β„Ž β„Ž NOUN american_scientific_journal-707 164 89 where where SCONJ american_scientific_journal-707 164 90 𝑔 𝑔 PROPN american_scientific_journal-707 164 91 is be AUX american_scientific_journal-707 164 92 a a DET american_scientific_journal-707 164 93 monomorphism monomorphism NOUN american_scientific_journal-707 164 94 and and CCONJ american_scientific_journal-707 164 95 β„Ž β„Ž PROPN american_scientific_journal-707 164 96 is be AUX american_scientific_journal-707 164 97 an an DET american_scientific_journal-707 164 98 epimorphism epimorphism NOUN american_scientific_journal-707 164 99 , , PUNCT american_scientific_journal-707 164 100 and and CCONJ american_scientific_journal-707 164 101 if if SCONJ american_scientific_journal-707 164 102 πœ‹ πœ‹ PRON american_scientific_journal-707 164 103 can can AUX american_scientific_journal-707 164 104 also also ADV american_scientific_journal-707 164 105 be be AUX american_scientific_journal-707 164 106 decomposed decompose VERB american_scientific_journal-707 164 107 in in ADP american_scientific_journal-707 164 108 another another DET american_scientific_journal-707 164 109 way way NOUN american_scientific_journal-707 164 110 as as SCONJ american_scientific_journal-707 164 111 πœ‹ πœ‹ NOUN american_scientific_journal-707 164 112 = = NOUN american_scientific_journal-707 164 113 𝑔′ 𝑔′ X american_scientific_journal-707 164 114 ∘ ∘ X american_scientific_journal-707 164 115 β„Žβ€² β„Žβ€² ADJ american_scientific_journal-707 164 116 where where SCONJ american_scientific_journal-707 164 117 𝑔′ 𝑔′ NOUN american_scientific_journal-707 164 118 is be AUX american_scientific_journal-707 164 119 a a DET american_scientific_journal-707 164 120 monomorphism monomorphism NOUN american_scientific_journal-707 164 121 and and CCONJ american_scientific_journal-707 164 122 β„Žβ€² β„Žβ€² NOUN american_scientific_journal-707 164 123 is be AUX american_scientific_journal-707 164 124 an an DET american_scientific_journal-707 164 125 epimorphism epimorphism NOUN american_scientific_journal-707 164 126 then then ADV american_scientific_journal-707 164 127 we we PRON american_scientific_journal-707 164 128 get get VERB american_scientific_journal-707 164 129 that that PRON american_scientific_journal-707 164 130 𝑔~𝑔′ 𝑔~𝑔′ NOUN american_scientific_journal-707 164 131 ( ( PUNCT american_scientific_journal-707 164 132 or or CCONJ american_scientific_journal-707 164 133 𝑔 𝑔 PROPN american_scientific_journal-707 164 134 and and CCONJ american_scientific_journal-707 164 135 𝑔′ 𝑔′ PROPN american_scientific_journal-707 164 136 are be AUX american_scientific_journal-707 164 137 associate associate ADJ american_scientific_journal-707 164 138 to to ADP american_scientific_journal-707 164 139 each each DET american_scientific_journal-707 164 140 other other ADJ american_scientific_journal-707 164 141 ) ) PUNCT american_scientific_journal-707 164 142 and and CCONJ american_scientific_journal-707 164 143 β„Ž~β„Žβ€² β„Ž~β„Žβ€² X american_scientific_journal-707 164 144 ( ( PUNCT american_scientific_journal-707 164 145 or or CCONJ american_scientific_journal-707 164 146 β„Ž β„Ž PROPN american_scientific_journal-707 164 147 and and CCONJ american_scientific_journal-707 164 148 β„Žβ€² β„Žβ€² NOUN american_scientific_journal-707 164 149 are be AUX american_scientific_journal-707 164 150 associate associate ADJ american_scientific_journal-707 164 151 to to ADP american_scientific_journal-707 164 152 each each DET american_scientific_journal-707 164 153 other other ADJ american_scientific_journal-707 164 154 ) ) PUNCT american_scientific_journal-707 164 155 . . PUNCT american_scientific_journal-707 165 1 we we PRON american_scientific_journal-707 165 2 say say VERB american_scientific_journal-707 165 3 that that SCONJ american_scientific_journal-707 165 4 two two NUM american_scientific_journal-707 165 5 homomorphisms𝑔 homomorphisms𝑔 NOUN american_scientific_journal-707 165 6 and and CCONJ american_scientific_journal-707 165 7 β„Ž β„Ž NOUN american_scientific_journal-707 165 8 of of ADP american_scientific_journal-707 165 9 algebras algebra NOUN american_scientific_journal-707 165 10 of of ADP american_scientific_journal-707 165 11 a a DET american_scientific_journal-707 165 12 given give VERB american_scientific_journal-707 165 13 type type NOUN american_scientific_journal-707 165 14 are be AUX american_scientific_journal-707 165 15 associate associate ADJ american_scientific_journal-707 165 16 to to ADP american_scientific_journal-707 165 17 each each DET american_scientific_journal-707 165 18 other other ADJ american_scientific_journal-707 165 19 , , PUNCT american_scientific_journal-707 165 20 if if SCONJ american_scientific_journal-707 165 21 there there PRON american_scientific_journal-707 165 22 exists exist VERB american_scientific_journal-707 165 23 two two NUM american_scientific_journal-707 165 24 isomorphisms𝛼 isomorphisms𝛼 NOUN american_scientific_journal-707 165 25 and and CCONJ american_scientific_journal-707 165 26 𝛽 𝛽 NOUN american_scientific_journal-707 165 27 such such ADJ american_scientific_journal-707 165 28 that that PRON american_scientific_journal-707 165 29 ; ; PUNCT american_scientific_journal-707 165 30 𝑔 𝑔 PROPN american_scientific_journal-707 165 31 ∘ ∘ NOUN american_scientific_journal-707 165 32 𝛼 𝛼 X american_scientific_journal-707 165 33 = = NOUN american_scientific_journal-707 165 34 𝛽 𝛽 NOUN american_scientific_journal-707 165 35 ∘ ∘ NOUN american_scientific_journal-707 165 36 β„Ž. β„Ž. NOUN american_scientific_journal-707 165 37 since since SCONJ american_scientific_journal-707 165 38 𝛼 𝛼 PROPN american_scientific_journal-707 165 39 and and CCONJ american_scientific_journal-707 165 40 𝛽 𝛽 PROPN american_scientific_journal-707 165 41 are be AUX american_scientific_journal-707 165 42 isomorphisms isomorphism NOUN american_scientific_journal-707 165 43 and and CCONJ american_scientific_journal-707 165 44 hence hence ADV american_scientific_journal-707 165 45 invertible invertible ADJ american_scientific_journal-707 165 46 it it PRON american_scientific_journal-707 165 47 is be AUX american_scientific_journal-707 165 48 equivalently equivalently ADV american_scientific_journal-707 165 49 saying say VERB american_scientific_journal-707 165 50 that that SCONJ american_scientific_journal-707 165 51 one one PRON american_scientific_journal-707 165 52 can can AUX american_scientific_journal-707 165 53 be be AUX american_scientific_journal-707 165 54 expressed express VERB american_scientific_journal-707 165 55 as as ADP american_scientific_journal-707 165 56 a a DET american_scientific_journal-707 165 57 composition composition NOUN american_scientific_journal-707 165 58 of of ADP american_scientific_journal-707 165 59 the the DET american_scientific_journal-707 165 60 other other ADJ american_scientific_journal-707 165 61 and and CCONJ american_scientific_journal-707 165 62 two two NUM american_scientific_journal-707 165 63 isomorphisms isomorphism NOUN american_scientific_journal-707 165 64 ; ; PUNCT american_scientific_journal-707 165 65 that that PRON american_scientific_journal-707 165 66 is be AUX american_scientific_journal-707 165 67 , , PUNCT american_scientific_journal-707 165 68 𝑔 𝑔 PROPN american_scientific_journal-707 165 69 ∘ ∘ PROPN american_scientific_journal-707 165 70 𝛼 𝛼 X american_scientific_journal-707 165 71 = = NOUN american_scientific_journal-707 165 72 𝛽 𝛽 NOUN american_scientific_journal-707 165 73 ∘ ∘ NUM american_scientific_journal-707 165 74 β„Ž β„Ž PROPN american_scientific_journal-707 165 75 ⇔ ⇔ PROPN american_scientific_journal-707 165 76 𝑔 𝑔 PROPN american_scientific_journal-707 165 77 = = SYM american_scientific_journal-707 165 78 𝛽 𝛽 PROPN american_scientific_journal-707 165 79 ∘ ∘ NOUN american_scientific_journal-707 165 80 β„Ž β„Ž ADP american_scientific_journal-707 165 81 ∘ ∘ PROPN american_scientific_journal-707 165 82 π›Όβˆ’1 π›Όβˆ’1 PRON american_scientific_journal-707 165 83 ⇔ ⇔ ADJ american_scientific_journal-707 165 84 β„Ž β„Ž PROPN american_scientific_journal-707 165 85 = = SYM american_scientific_journal-707 165 86 π›½βˆ’1 π›½βˆ’1 PROPN american_scientific_journal-707 165 87 ∘ ∘ NOUN american_scientific_journal-707 165 88 𝑔 𝑔 PROPN american_scientific_journal-707 165 89 ∘ ∘ PROPN american_scientific_journal-707 165 90 𝛼. 𝛼. PROPN american_scientific_journal-707 165 91 it it PRON american_scientific_journal-707 165 92 follows follow VERB american_scientific_journal-707 165 93 from from ADP american_scientific_journal-707 165 94 this this DET american_scientific_journal-707 165 95 definition definition NOUN american_scientific_journal-707 165 96 that that SCONJ american_scientific_journal-707 165 97 , , PUNCT american_scientific_journal-707 165 98 two two NUM american_scientific_journal-707 165 99 associate associate NOUN american_scientific_journal-707 165 100 homomorphisms homomorphism NOUN american_scientific_journal-707 165 101 have have VERB american_scientific_journal-707 165 102 different different ADJ american_scientific_journal-707 165 103 properties property NOUN american_scientific_journal-707 165 104 in in ADP american_scientific_journal-707 165 105 common common ADJ american_scientific_journal-707 165 106 such such ADJ american_scientific_journal-707 165 107 as as ADP american_scientific_journal-707 165 108 : : PUNCT american_scientific_journal-707 165 109 one one NUM american_scientific_journal-707 165 110 of of ADP american_scientific_journal-707 165 111 the the DET american_scientific_journal-707 165 112 two two NUM american_scientific_journal-707 165 113 associate associate NOUN american_scientific_journal-707 165 114 homomorphisms homomorphism NOUN american_scientific_journal-707 165 115 is be AUX american_scientific_journal-707 165 116 an an DET american_scientific_journal-707 165 117 injection injection NOUN american_scientific_journal-707 165 118 ( ( PUNCT american_scientific_journal-707 165 119 respectively respectively ADV american_scientific_journal-707 165 120 a a DET american_scientific_journal-707 165 121 surjection surjection NOUN american_scientific_journal-707 165 122 and and CCONJ american_scientific_journal-707 165 123 a a DET american_scientific_journal-707 165 124 bijection bijection NOUN american_scientific_journal-707 165 125 ) ) PUNCT american_scientific_journal-707 165 126 if if SCONJ american_scientific_journal-707 165 127 and and CCONJ american_scientific_journal-707 165 128 only only ADV american_scientific_journal-707 165 129 if if SCONJ american_scientific_journal-707 165 130 the the DET american_scientific_journal-707 165 131 other other ADJ american_scientific_journal-707 165 132 is be AUX american_scientific_journal-707 165 133 an an DET american_scientific_journal-707 165 134 injection injection NOUN american_scientific_journal-707 165 135 ( ( PUNCT american_scientific_journal-707 165 136 respectively respectively ADV american_scientific_journal-707 165 137 a a DET american_scientific_journal-707 165 138 surjection surjection NOUN american_scientific_journal-707 165 139 and and CCONJ american_scientific_journal-707 165 140 a a DET american_scientific_journal-707 165 141 bijection bijection NOUN american_scientific_journal-707 165 142 ) ) PUNCT american_scientific_journal-707 165 143 . . PUNCT american_scientific_journal-707 166 1 moreover moreover ADV american_scientific_journal-707 166 2 , , PUNCT american_scientific_journal-707 166 3 their their PRON american_scientific_journal-707 166 4 domain domain NOUN american_scientific_journal-707 166 5 ( ( PUNCT american_scientific_journal-707 166 6 respectively respectively ADV american_scientific_journal-707 166 7 codomain codomain ADJ american_scientific_journal-707 166 8 ) ) PUNCT american_scientific_journal-707 166 9 are be AUX american_scientific_journal-707 166 10 isomorphic isomorphic ADJ american_scientific_journal-707 166 11 to to ADP american_scientific_journal-707 166 12 each each DET american_scientific_journal-707 166 13 other other ADJ american_scientific_journal-707 166 14 . . PUNCT american_scientific_journal-707 167 1 it it PRON american_scientific_journal-707 167 2 is be AUX american_scientific_journal-707 167 3 finally finally ADV american_scientific_journal-707 167 4 recommended recommend VERB american_scientific_journal-707 167 5 that that SCONJ american_scientific_journal-707 167 6 it it PRON american_scientific_journal-707 167 7 needs need VERB american_scientific_journal-707 167 8 additional additional ADJ american_scientific_journal-707 167 9 researches research NOUN american_scientific_journal-707 167 10 on on ADP american_scientific_journal-707 167 11 the the DET american_scientific_journal-707 167 12 class class NOUN american_scientific_journal-707 167 13 of of ADP american_scientific_journal-707 167 14 associate associate ADJ american_scientific_journal-707 167 15 homomorphisms homomorphism NOUN american_scientific_journal-707 167 16 of of ADP american_scientific_journal-707 167 17 algebras algebra NOUN american_scientific_journal-707 167 18 of of ADP american_scientific_journal-707 167 19 a a DET american_scientific_journal-707 167 20 given give VERB american_scientific_journal-707 167 21 type type NOUN american_scientific_journal-707 167 22 to to PART american_scientific_journal-707 167 23 identify identify VERB american_scientific_journal-707 167 24 and and CCONJ american_scientific_journal-707 167 25 characterize characterize VERB american_scientific_journal-707 167 26 in in ADP american_scientific_journal-707 167 27 a a DET american_scientific_journal-707 167 28 more more ADV american_scientific_journal-707 167 29 general general ADJ american_scientific_journal-707 167 30 way way NOUN american_scientific_journal-707 167 31 . . PUNCT american_scientific_journal-707 168 1 acknowledgements acknowledgement NOUN american_scientific_journal-707 168 2 above above ADP american_scientific_journal-707 168 3 all all PRON american_scientific_journal-707 168 4 , , PUNCT american_scientific_journal-707 168 5 i i PRON american_scientific_journal-707 168 6 would would AUX american_scientific_journal-707 168 7 like like VERB american_scientific_journal-707 168 8 to to PART american_scientific_journal-707 168 9 thank thank VERB american_scientific_journal-707 168 10 the the DET american_scientific_journal-707 168 11 almighty almighty ADJ american_scientific_journal-707 168 12 god god PROPN american_scientific_journal-707 168 13 for for ADP american_scientific_journal-707 168 14 his his PRON american_scientific_journal-707 168 15 mercy mercy NOUN american_scientific_journal-707 168 16 that that PRON american_scientific_journal-707 168 17 endowed endow VERB american_scientific_journal-707 168 18 upon upon SCONJ american_scientific_journal-707 168 19 me i PRON american_scientific_journal-707 168 20 . . PUNCT american_scientific_journal-707 169 1 next next ADV american_scientific_journal-707 169 2 , , PUNCT american_scientific_journal-707 169 3 i i PRON american_scientific_journal-707 169 4 would would AUX american_scientific_journal-707 169 5 like like VERB american_scientific_journal-707 169 6 to to PART american_scientific_journal-707 169 7 offer offer VERB american_scientific_journal-707 169 8 my my PRON american_scientific_journal-707 169 9 sincerest sincere ADJ american_scientific_journal-707 169 10 and and CCONJ american_scientific_journal-707 169 11 deepest deepest VERB american_scientific_journal-707 169 12 gratitude gratitude NOUN american_scientific_journal-707 169 13 to to ADP american_scientific_journal-707 169 14 my my PRON american_scientific_journal-707 169 15 advisor advisor NOUN american_scientific_journal-707 169 16 prof prof NOUN american_scientific_journal-707 169 17 . . PUNCT american_scientific_journal-707 170 1 u. u. PROPN american_scientific_journal-707 170 2 m. m. PROPN american_scientific_journal-707 170 3 swamy swamy PROPN american_scientific_journal-707 170 4 , , PUNCT american_scientific_journal-707 170 5 who who PRON american_scientific_journal-707 170 6 has have AUX american_scientific_journal-707 170 7 supported support VERB american_scientific_journal-707 170 8 me i PRON american_scientific_journal-707 170 9 throughout throughout ADP american_scientific_journal-707 170 10 this this DET american_scientific_journal-707 170 11 research research NOUN american_scientific_journal-707 170 12 work work NOUN american_scientific_journal-707 170 13 with with ADP american_scientific_journal-707 170 14 his his PRON american_scientific_journal-707 170 15 patency patency NOUN american_scientific_journal-707 170 16 and and CCONJ american_scientific_journal-707 170 17 knowledge knowledge NOUN american_scientific_journal-707 170 18 whilst whilst SCONJ american_scientific_journal-707 170 19 allowing allow VERB american_scientific_journal-707 170 20 me i PRON american_scientific_journal-707 170 21 the the DET american_scientific_journal-707 170 22 room room NOUN american_scientific_journal-707 170 23 to to PART american_scientific_journal-707 170 24 work work VERB american_scientific_journal-707 170 25 in in ADP american_scientific_journal-707 170 26 my my PRON american_scientific_journal-707 170 27 own own ADJ american_scientific_journal-707 170 28 way way NOUN american_scientific_journal-707 170 29 . . PUNCT american_scientific_journal-707 171 1 one one PRON american_scientific_journal-707 171 2 could could AUX american_scientific_journal-707 171 3 not not PART american_scientific_journal-707 171 4 wish wish VERB american_scientific_journal-707 171 5 for for ADP american_scientific_journal-707 171 6 a a DET american_scientific_journal-707 171 7 better well ADJ american_scientific_journal-707 171 8 or or CCONJ american_scientific_journal-707 171 9 friendly friendly ADJ american_scientific_journal-707 171 10 advisor advisor NOUN american_scientific_journal-707 171 11 . . PUNCT american_scientific_journal-707 172 1 i i PRON american_scientific_journal-707 172 2 am be AUX american_scientific_journal-707 172 3 also also ADV american_scientific_journal-707 172 4 indebted indebted ADJ american_scientific_journal-707 172 5 to to ADP american_scientific_journal-707 172 6 my my PRON american_scientific_journal-707 172 7 friends friend NOUN american_scientific_journal-707 172 8 helping help VERB american_scientific_journal-707 172 9 me i PRON american_scientific_journal-707 172 10 throughout throughout ADP american_scientific_journal-707 172 11 my my PRON american_scientific_journal-707 172 12 work work NOUN american_scientific_journal-707 172 13 in in ADP american_scientific_journal-707 172 14 different different ADJ american_scientific_journal-707 172 15 ways way NOUN american_scientific_journal-707 172 16 . . PUNCT american_scientific_journal-707 173 1 72 72 NUM american_scientific_journal-707 173 2 american american ADJ american_scientific_journal-707 173 3 scientific scientific ADJ american_scientific_journal-707 173 4 research research NOUN american_scientific_journal-707 173 5 journal journal NOUN american_scientific_journal-707 173 6 for for ADP american_scientific_journal-707 173 7 engineering engineering NOUN american_scientific_journal-707 173 8 , , PUNCT american_scientific_journal-707 173 9 technology technology NOUN american_scientific_journal-707 173 10 , , PUNCT american_scientific_journal-707 173 11 and and CCONJ american_scientific_journal-707 173 12 sciences science NOUN american_scientific_journal-707 173 13 ( ( PUNCT american_scientific_journal-707 173 14 asrjets asrjet NOUN american_scientific_journal-707 173 15 ) ) PUNCT american_scientific_journal-707 173 16 ( ( PUNCT american_scientific_journal-707 173 17 2014 2014 NUM american_scientific_journal-707 173 18 ) ) PUNCT american_scientific_journal-707 173 19 volume volume NOUN american_scientific_journal-707 173 20 10 10 NUM american_scientific_journal-707 173 21 , , PUNCT american_scientific_journal-707 173 22 no no DET american_scientific_journal-707 173 23 1 1 NUM american_scientific_journal-707 173 24 , , PUNCT american_scientific_journal-707 173 25 pp pp ADV american_scientific_journal-707 173 26 61 61 NUM american_scientific_journal-707 173 27 - - SYM american_scientific_journal-707 173 28 73 73 NUM american_scientific_journal-707 173 29 finally finally ADV american_scientific_journal-707 173 30 , , PUNCT american_scientific_journal-707 173 31 i i PRON american_scientific_journal-707 173 32 would would AUX american_scientific_journal-707 173 33 like like VERB american_scientific_journal-707 173 34 to to PART american_scientific_journal-707 173 35 express express VERB american_scientific_journal-707 173 36 my my PRON american_scientific_journal-707 173 37 heartfelt heartfelt ADJ american_scientific_journal-707 173 38 thanks thank NOUN american_scientific_journal-707 173 39 to to ADP american_scientific_journal-707 173 40 my my PRON american_scientific_journal-707 173 41 parents parent NOUN american_scientific_journal-707 173 42 , , PUNCT american_scientific_journal-707 173 43 mom mom NOUN american_scientific_journal-707 173 44 , , PUNCT american_scientific_journal-707 173 45 dad dad NOUN american_scientific_journal-707 173 46 and and CCONJ american_scientific_journal-707 173 47 my my PRON american_scientific_journal-707 173 48 brothers brother NOUN american_scientific_journal-707 173 49 . . PUNCT american_scientific_journal-707 174 1 in in ADP american_scientific_journal-707 174 2 my my PRON american_scientific_journal-707 174 3 daily daily ADJ american_scientific_journal-707 174 4 work work NOUN american_scientific_journal-707 174 5 i i PRON american_scientific_journal-707 174 6 have have AUX american_scientific_journal-707 174 7 been be AUX american_scientific_journal-707 174 8 blessed bless VERB american_scientific_journal-707 174 9 with with ADP american_scientific_journal-707 174 10 their their PRON american_scientific_journal-707 174 11 unconditional unconditional ADJ american_scientific_journal-707 174 12 love love NOUN american_scientific_journal-707 174 13 , , PUNCT american_scientific_journal-707 174 14 kindness kindness NOUN american_scientific_journal-707 174 15 , , PUNCT american_scientific_journal-707 174 16 support support NOUN american_scientific_journal-707 174 17 and and CCONJ american_scientific_journal-707 174 18 encouragement encouragement NOUN american_scientific_journal-707 174 19 . . PUNCT american_scientific_journal-707 175 1 from from ADP american_scientific_journal-707 175 2 the the DET american_scientific_journal-707 175 3 start start NOUN american_scientific_journal-707 175 4 until until ADP american_scientific_journal-707 175 5 the the DET american_scientific_journal-707 175 6 accomplishment accomplishment NOUN american_scientific_journal-707 175 7 of of ADP american_scientific_journal-707 175 8 this this DET american_scientific_journal-707 175 9 manuscript manuscript NOUN american_scientific_journal-707 175 10 , , PUNCT american_scientific_journal-707 175 11 they they PRON american_scientific_journal-707 175 12 have have AUX american_scientific_journal-707 175 13 been be AUX american_scientific_journal-707 175 14 my my PRON american_scientific_journal-707 175 15 source source NOUN american_scientific_journal-707 175 16 of of ADP american_scientific_journal-707 175 17 strength strength NOUN american_scientific_journal-707 175 18 and and CCONJ american_scientific_journal-707 175 19 courage courage NOUN american_scientific_journal-707 175 20 . . PUNCT american_scientific_journal-707 176 1 i i PRON american_scientific_journal-707 176 2 would would AUX american_scientific_journal-707 176 3 not not PART american_scientific_journal-707 176 4 let let VERB american_scientific_journal-707 176 5 this this DET american_scientific_journal-707 176 6 opportunity opportunity NOUN american_scientific_journal-707 176 7 to to PART american_scientific_journal-707 176 8 escape escape VERB american_scientific_journal-707 176 9 without without ADP american_scientific_journal-707 176 10 me i PRON american_scientific_journal-707 176 11 saying say VERB american_scientific_journal-707 176 12 " " PUNCT american_scientific_journal-707 176 13 mom mom NOUN american_scientific_journal-707 176 14 , , PUNCT american_scientific_journal-707 176 15 i i PRON american_scientific_journal-707 176 16 love love VERB american_scientific_journal-707 176 17 you you PRON american_scientific_journal-707 176 18 and and CCONJ american_scientific_journal-707 176 19 always always ADV american_scientific_journal-707 176 20 treasure treasure VERB american_scientific_journal-707 176 21 you you PRON american_scientific_journal-707 176 22 . . PUNCT american_scientific_journal-707 177 1 references reference NOUN american_scientific_journal-707 177 2 [ [ X american_scientific_journal-707 177 3 1 1 NUM american_scientific_journal-707 177 4 ] ] PUNCT american_scientific_journal-707 177 5 burris burris NOUN american_scientific_journal-707 177 6 , , PUNCT american_scientific_journal-707 177 7 s.andsankappanavor s.andsankappanavor NOUN american_scientific_journal-707 177 8 , , PUNCT american_scientific_journal-707 177 9 h.p h.p PROPN american_scientific_journal-707 177 10 . . PROPN american_scientific_journal-707 177 11 , , PUNCT american_scientific_journal-707 177 12 a a DET american_scientific_journal-707 177 13 course course NOUN american_scientific_journal-707 177 14 in in ADP american_scientific_journal-707 177 15 universal universal ADJ american_scientific_journal-707 177 16 algebra algebra NOUN american_scientific_journal-707 177 17 , , PUNCT american_scientific_journal-707 177 18 springer springer NOUN american_scientific_journal-707 177 19 - - PUNCT american_scientific_journal-707 177 20 verlag verlag PROPN american_scientific_journal-707 177 21 , , PUNCT american_scientific_journal-707 177 22 gtm78 gtm78 NOUN american_scientific_journal-707 177 23 , , PUNCT american_scientific_journal-707 177 24 1981 1981 NUM american_scientific_journal-707 177 25 [ [ X american_scientific_journal-707 177 26 2 2 NUM american_scientific_journal-707 177 27 ] ] PUNCT american_scientific_journal-707 177 28 clifford clifford PROPN american_scientific_journal-707 177 29 bergman bergman PROPN american_scientific_journal-707 177 30 , , PUNCT american_scientific_journal-707 177 31 universal universal ADJ american_scientific_journal-707 177 32 algebra algebra NOUN american_scientific_journal-707 177 33 : : PUNCT american_scientific_journal-707 177 34 fundamentals fundamental NOUN american_scientific_journal-707 177 35 and and CCONJ american_scientific_journal-707 177 36 selected select VERB american_scientific_journal-707 177 37 topics topic NOUN american_scientific_journal-707 177 38 , , PUNCT american_scientific_journal-707 177 39 crc crc NOUN american_scientific_journal-707 177 40 press press NOUN american_scientific_journal-707 177 41 , , PUNCT american_scientific_journal-707 177 42 2011 2011 NUM american_scientific_journal-707 177 43 [ [ X american_scientific_journal-707 177 44 3 3 X american_scientific_journal-707 177 45 ] ] PUNCT american_scientific_journal-707 177 46 gertude gertude NOUN american_scientific_journal-707 177 47 ehrlich ehrlich NOUN american_scientific_journal-707 177 48 , , PUNCT american_scientific_journal-707 177 49 fundamental fundamental ADJ american_scientific_journal-707 177 50 concept concept NOUN american_scientific_journal-707 177 51 of of ADP american_scientific_journal-707 177 52 abstract abstract ADJ american_scientific_journal-707 177 53 algebra algebra NOUN american_scientific_journal-707 177 54 , , PUNCT american_scientific_journal-707 177 55 courier courier NOUN american_scientific_journal-707 177 56 corporation corporation NOUN american_scientific_journal-707 177 57 , , PUNCT american_scientific_journal-707 177 58 2013 2013 NUM american_scientific_journal-707 177 59 [ [ X american_scientific_journal-707 177 60 4 4 NUM american_scientific_journal-707 177 61 ] ] X american_scientific_journal-707 177 62 jacobson jacobson PROPN american_scientific_journal-707 177 63 , , PUNCT american_scientific_journal-707 177 64 n. n. NOUN american_scientific_journal-707 177 65 , , PUNCT american_scientific_journal-707 177 66 basic basic ADJ american_scientific_journal-707 177 67 algebra algebra NOUN american_scientific_journal-707 177 68 - - PUNCT american_scientific_journal-707 177 69 vol vol NOUN american_scientific_journal-707 177 70 i i PROPN american_scientific_journal-707 177 71 , , PUNCT american_scientific_journal-707 177 72 w. w. PROPN american_scientific_journal-707 177 73 h. h. PROPN american_scientific_journal-707 177 74 freeman freeman PROPN american_scientific_journal-707 177 75 , , PUNCT american_scientific_journal-707 177 76 san san PROPN american_scientific_journal-707 177 77 francisco francisco PROPN american_scientific_journal-707 177 78 , , PUNCT american_scientific_journal-707 177 79 1980 1980 NUM american_scientific_journal-707 178 1 [ [ X american_scientific_journal-707 178 2 5 5 X american_scientific_journal-707 178 3 ] ] X american_scientific_journal-707 178 4 jonathan jonathan PROPN american_scientific_journal-707 178 5 k. k. PROPN american_scientific_journal-707 178 6 hodge hodge PROPN american_scientific_journal-707 178 7 , , PUNCT american_scientific_journal-707 178 8 steven steven PROPN american_scientific_journal-707 178 9 schlicker schlicker PROPN american_scientific_journal-707 178 10 , , PUNCT american_scientific_journal-707 178 11 ted te VERB american_scientific_journal-707 178 12 sundstrom sundstrom PROPN american_scientific_journal-707 178 13 , , PUNCT american_scientific_journal-707 178 14 abstract abstract ADJ american_scientific_journal-707 178 15 algebra algebra NOUN american_scientific_journal-707 178 16 : : PUNCT american_scientific_journal-707 178 17 an an DET american_scientific_journal-707 178 18 inquiry inquiry NOUN american_scientific_journal-707 178 19 based base VERB american_scientific_journal-707 178 20 approach approach NOUN american_scientific_journal-707 178 21 , , PUNCT american_scientific_journal-707 178 22 crc crc NOUN american_scientific_journal-707 178 23 press press NOUN american_scientific_journal-707 178 24 , , PUNCT american_scientific_journal-707 178 25 2013 2013 NUM american_scientific_journal-707 178 26 [ [ X american_scientific_journal-707 178 27 6 6 NUM american_scientific_journal-707 178 28 ] ] SYM american_scientific_journal-707 178 29 klaus klaus NOUN american_scientific_journal-707 178 30 denecke denecke PROPN american_scientific_journal-707 178 31 , , PUNCT american_scientific_journal-707 178 32 shelly shelly PROPN american_scientific_journal-707 178 33 l. l. PROPN american_scientific_journal-707 178 34 wismath wismath PROPN american_scientific_journal-707 178 35 , , PUNCT american_scientific_journal-707 178 36 universal universal ADJ american_scientific_journal-707 178 37 algebra algebra NOUN american_scientific_journal-707 178 38 and and CCONJ american_scientific_journal-707 178 39 coalgebra coalgebra PROPN american_scientific_journal-707 178 40 : : PUNCT american_scientific_journal-707 178 41 world world NOUN american_scientific_journal-707 178 42 scientific scientific ADJ american_scientific_journal-707 178 43 , , PUNCT american_scientific_journal-707 178 44 2009 2009 NUM american_scientific_journal-707 178 45 [ [ X american_scientific_journal-707 178 46 7 7 NUM american_scientific_journal-707 178 47 ] ] X american_scientific_journal-707 178 48 klaus klaus NOUN american_scientific_journal-707 178 49 denecke denecke PROPN american_scientific_journal-707 178 50 , , PUNCT american_scientific_journal-707 178 51 shelly shelly PROPN american_scientific_journal-707 178 52 l. l. PROPN american_scientific_journal-707 178 53 wismath wismath PROPN american_scientific_journal-707 178 54 , , PUNCT american_scientific_journal-707 178 55 universal universal ADJ american_scientific_journal-707 178 56 algebra algebra NOUN american_scientific_journal-707 178 57 and and CCONJ american_scientific_journal-707 178 58 applications application NOUN american_scientific_journal-707 178 59 in in ADP american_scientific_journal-707 178 60 theoretical theoretical ADJ american_scientific_journal-707 178 61 computer computer NOUN american_scientific_journal-707 178 62 science science NOUN american_scientific_journal-707 178 63 : : PUNCT american_scientific_journal-707 178 64 crc crc NOUN american_scientific_journal-707 178 65 press press PROPN american_scientific_journal-707 178 66 , , PUNCT american_scientific_journal-707 178 67 2002 2002 NUM american_scientific_journal-707 179 1 [ [ X american_scientific_journal-707 179 2 p p X american_scientific_journal-707 179 3 47 47 NUM american_scientific_journal-707 179 4 - - SYM american_scientific_journal-707 179 5 63 63 NUM american_scientific_journal-707 179 6 ] ] PUNCT american_scientific_journal-707 179 7 [ [ X american_scientific_journal-707 179 8 8 8 NUM american_scientific_journal-707 179 9 ] ] X american_scientific_journal-707 179 10 l. l. PROPN american_scientific_journal-707 179 11 szabΓ³ szabΓ³ PROPN american_scientific_journal-707 179 12 , , PUNCT american_scientific_journal-707 179 13 a. a. NOUN american_scientific_journal-707 179 14 szendreilectures szendreilecture VERB american_scientific_journal-707 179 15 in in ADP american_scientific_journal-707 179 16 universal universal ADJ american_scientific_journal-707 179 17 algebra algebra NOUN american_scientific_journal-707 179 18 , , PUNCT american_scientific_journal-707 179 19 volume volume NOUN american_scientific_journal-707 179 20 1 1 NUM american_scientific_journal-707 179 21 of of ADP american_scientific_journal-707 179 22 colloquia colloquium NOUN american_scientific_journal-707 179 23 mathematicasocietatis mathematicasocietatis PROPN american_scientific_journal-707 179 24 janos janos PROPN american_scientific_journal-707 179 25 bolyai bolyai PROPN american_scientific_journal-707 179 26 , , PUNCT american_scientific_journal-707 179 27 elsevier elsevier NOUN american_scientific_journal-707 179 28 , , PUNCT american_scientific_journal-707 179 29 2014 2014 NUM american_scientific_journal-707 180 1 [ [ X american_scientific_journal-707 180 2 9 9 NUM american_scientific_journal-707 180 3 ] ] X american_scientific_journal-707 180 4 paul paul PROPN american_scientific_journal-707 180 5 b. b. PROPN american_scientific_journal-707 180 6 garrett garrett PROPN american_scientific_journal-707 180 7 , , PUNCT american_scientific_journal-707 180 8 abstract abstract ADJ american_scientific_journal-707 180 9 algebra algebra PROPN american_scientific_journal-707 180 10 , , PUNCT american_scientific_journal-707 180 11 crc crc NOUN american_scientific_journal-707 180 12 press press NOUN american_scientific_journal-707 180 13 , , PUNCT american_scientific_journal-707 180 14 2007 2007 NUM american_scientific_journal-707 180 15 ( ( PUNCT american_scientific_journal-707 180 16 page25 page25 NOUN american_scientific_journal-707 180 17 - - PUNCT american_scientific_journal-707 180 18 56 56 NUM american_scientific_journal-707 180 19 ) ) PUNCT american_scientific_journal-707 181 1 [ [ X american_scientific_journal-707 181 2 10 10 NUM american_scientific_journal-707 181 3 ] ] X american_scientific_journal-707 181 4 peter peter PROPN american_scientific_journal-707 181 5 hilton hilton PROPN american_scientific_journal-707 181 6 , , PUNCT american_scientific_journal-707 181 7 yel yel PROPN american_scientific_journal-707 181 8 - - PUNCT american_scientific_journal-707 181 9 chiang chiang PROPN american_scientific_journal-707 181 10 wu wu PROPN american_scientific_journal-707 181 11 , , PUNCT american_scientific_journal-707 181 12 a a DET american_scientific_journal-707 181 13 course course NOUN american_scientific_journal-707 181 14 in in ADP american_scientific_journal-707 181 15 modern modern ADJ american_scientific_journal-707 181 16 algebra algebra NOUN american_scientific_journal-707 181 17 : : PUNCT american_scientific_journal-707 181 18 pure pure ADJ american_scientific_journal-707 181 19 and and CCONJ american_scientific_journal-707 181 20 applied applied ADJ american_scientific_journal-707 181 21 mathematics mathematic NOUN american_scientific_journal-707 181 22 volume volume NOUN american_scientific_journal-707 181 23 12 12 NUM american_scientific_journal-707 181 24 of of ADP american_scientific_journal-707 181 25 wiley wiley NOUN american_scientific_journal-707 181 26 classics classic NOUN american_scientific_journal-707 181 27 library library NOUN american_scientific_journal-707 181 28 , , PUNCT american_scientific_journal-707 181 29 john john PROPN american_scientific_journal-707 181 30 wiley wiley PROPN american_scientific_journal-707 181 31 & & CCONJ american_scientific_journal-707 181 32 sons son NOUN american_scientific_journal-707 181 33 , , PUNCT american_scientific_journal-707 181 34 1989 1989 NUM american_scientific_journal-707 181 35 [ [ X american_scientific_journal-707 181 36 11 11 NUM american_scientific_journal-707 181 37 ] ] X american_scientific_journal-707 181 38 pierce pierce PROPN american_scientific_journal-707 181 39 , , PUNCT american_scientific_journal-707 181 40 r.s r.s PROPN american_scientific_journal-707 181 41 . . PROPN american_scientific_journal-707 181 42 , , PUNCT american_scientific_journal-707 181 43 introduction introduction NOUN american_scientific_journal-707 181 44 to to ADP american_scientific_journal-707 181 45 the the DET american_scientific_journal-707 181 46 theory theory NOUN american_scientific_journal-707 181 47 of of ADP american_scientific_journal-707 181 48 abstract abstract ADJ american_scientific_journal-707 181 49 algebra algebra NOUN american_scientific_journal-707 181 50 , , PUNCT american_scientific_journal-707 181 51 halt halt NOUN american_scientific_journal-707 181 52 , , PUNCT american_scientific_journal-707 181 53 rinehart rinehart PROPN american_scientific_journal-707 181 54 and and CCONJ american_scientific_journal-707 181 55 winston winston PROPN american_scientific_journal-707 181 56 , , PUNCT american_scientific_journal-707 181 57 new new PROPN american_scientific_journal-707 181 58 york york PROPN american_scientific_journal-707 181 59 , , PUNCT american_scientific_journal-707 181 60 1968 1968 NUM american_scientific_journal-707 181 61 73 73 NUM american_scientific_journal-707 181 62 https://www.google.com.et/search?tbo=p&tbm=bks&q=inauthor:%22clifford+bergman%22&source=gbs_metadata_r&cad=5 https://www.google.com.et/search?tbo=p&tbm=bks&q=inauthor:%22clifford+bergman%22&source=gbs_metadata_r&cad=5 ADJ american_scientific_journal-707 181 63 https://www.google.com.et/search?tbo=p&tbm=bks&q=inauthor:%22klaus+denecke%22&source=gbs_metadata_r&cad=5 https://www.google.com.et/search?tbo=p&tbm=bks&q=inauthor:%22klaus+denecke%22&source=gbs_metadata_r&cad=5 NOUN american_scientific_journal-707 181 64 https://www.google.com.et/search?tbo=p&tbm=bks&q=inauthor:%22shelly+l.+wismath%22&source=gbs_metadata_r&cad=5 https://www.google.com.et/search?tbo=p&tbm=bks&q=inauthor:%22shelly+l.+wismath%22&source=gbs_metadata_r&cad=5 ADJ american_scientific_journal-707 181 65 https://www.google.com.et/search?tbo=p&tbm=bks&q=inauthor:%22klaus+denecke%22&source=gbs_metadata_r&cad=7 https://www.google.com.et/search?tbo=p&tbm=bks&q=inauthor:%22klaus+denecke%22&source=gbs_metadata_r&cad=7 NOUN american_scientific_journal-707 181 66 https://www.google.com.et/search?tbo=p&tbm=bks&q=inauthor:%22shelly+l.+wismath%22&source=gbs_metadata_r&cad=7 https://www.google.com.et/search?tbo=p&tbm=bks&q=inauthor:%22shelly+l.+wismath%22&source=gbs_metadata_r&cad=7 ADJ american_scientific_journal-707 181 67 https://www.google.com.et/search?tbo=p&tbm=bks&q=inauthor:%22l.+szab%c3%b3%22&source=gbs_metadata_r&cad=5 https://www.google.com.et/search?tbo=p&tbm=bks&q=inauthor:%22l.+szab%c3%b3%22&source=gbs_metadata_r&cad=5 ADJ american_scientific_journal-707 181 68 https://www.google.com.et/search?tbo=p&tbm=bks&q=inauthor:%22a.+szendrei%22&source=gbs_metadata_r&cad=5 https://www.google.com.et/search?tbo=p&tbm=bks&q=inauthor:%22a.+szendrei%22&source=gbs_metadata_r&cad=5 ADJ 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https://www.google.com.au/search?tbo=p&tbm=bks&q=bibliogroup:%22wiley+classics+library%22&source=gbs_metadata_r&cad=7 PROPN american_scientific_journal-707 181 76 acknowledgements acknowledgement NOUN american_scientific_journal-707 181 77 references reference NOUN