id sid tid token lemma pos cana-1834 1 1 communications communication NOUN cana-1834 1 2 on on ADP cana-1834 1 3 applied apply VERB cana-1834 1 4 nonlinear nonlinear ADJ cana-1834 1 5 analysis analysis NOUN cana-1834 1 6 issn issn NOUN cana-1834 1 7 : : PUNCT cana-1834 1 8 1074 1074 NUM cana-1834 1 9 - - PUNCT cana-1834 1 10 133x 133x NUM cana-1834 1 11 vol vol NOUN cana-1834 1 12 32 32 NUM cana-1834 1 13 no no NOUN cana-1834 1 14 . . NOUN cana-1834 1 15 2 2 NUM cana-1834 1 16 ( ( PUNCT cana-1834 1 17 2025 2025 NUM cana-1834 1 18 ) ) PUNCT cana-1834 1 19 581 581 NUM cana-1834 1 20 https://internationalpubls.com https://internationalpubls.com X cana-1834 1 21 an an DET cana-1834 1 22 introduction introduction NOUN cana-1834 1 23 to to PART cana-1834 1 24 ternary ternary VERB cana-1834 1 25 𝛀 𝛀 PROPN cana-1834 1 26 βˆ’semirings βˆ’semiring NOUN cana-1834 1 27 g. g. PROPN cana-1834 1 28 chandrasekhar1,2 chandrasekhar1,2 PROPN cana-1834 1 29 , , PUNCT cana-1834 1 30 d.madhusudana d.madhusudana ADJ cana-1834 1 31 rao3,p.siva rao3,p.siva VERB cana-1834 1 32 prasad4 prasad4 NOUN cana-1834 1 33 1 1 NUM cana-1834 1 34 research research NOUN cana-1834 1 35 scholar scholar NOUN cana-1834 1 36 , , PUNCT cana-1834 1 37 department department NOUN cana-1834 1 38 of of ADP cana-1834 1 39 mathematics mathematics PROPN cana-1834 1 40 , , PUNCT cana-1834 1 41 acharya acharya NOUN cana-1834 1 42 nagarjunauniversity nagarjunauniversity PROPN cana-1834 1 43 , , PUNCT cana-1834 1 44 guntur guntur PROPN cana-1834 1 45 , , PUNCT cana-1834 1 46 a. a. NOUN cana-1834 1 47 p. p. NOUN cana-1834 1 48 ,india ,india PUNCT cana-1834 1 49 . . PUNCT cana-1834 2 1 2lecturer 2lecturer NUM cana-1834 2 2 in in ADP cana-1834 2 3 mathematics mathematic NOUN cana-1834 2 4 , , PUNCT cana-1834 2 5 government government NOUN cana-1834 2 6 college college NOUN cana-1834 2 7 ( ( PUNCT cana-1834 2 8 autonomous autonomous ADJ cana-1834 2 9 ) ) PUNCT cana-1834 2 10 , , PUNCT cana-1834 2 11 rajahmundry rajahmundry ADJ cana-1834 2 12 , , PUNCT cana-1834 2 13 a. a. NOUN cana-1834 2 14 p. p. PROPN cana-1834 2 15 , , PUNCT cana-1834 2 16 india india PROPN cana-1834 2 17 . . PUNCT cana-1834 3 1 3professor 3professor NUM cana-1834 3 2 of of ADP cana-1834 3 3 mathematics mathematic NOUN cana-1834 3 4 , , PUNCT cana-1834 3 5 government government NOUN cana-1834 3 6 college college NOUN cana-1834 3 7 for for ADP cana-1834 3 8 women(a women(a PROPN cana-1834 3 9 ) ) PUNCT cana-1834 3 10 , , PUNCT cana-1834 3 11 samba samba PROPN cana-1834 3 12 siva siva PROPN cana-1834 3 13 peta peta PROPN cana-1834 3 14 rd rd PROPN cana-1834 3 15 , , PUNCT cana-1834 3 16 opp opp PROPN cana-1834 3 17 : : PUNCT cana-1834 3 18 ac ac PROPN cana-1834 3 19 college college PROPN cana-1834 3 20 , , PUNCT cana-1834 3 21 samba samba PROPN cana-1834 3 22 siva siva PROPN cana-1834 3 23 pet pet PROPN cana-1834 3 24 , , PUNCT cana-1834 3 25 guntur guntur PROPN cana-1834 3 26 , , PUNCT cana-1834 3 27 andhra andhra PROPN cana-1834 3 28 pradesh pradesh PROPN cana-1834 3 29 , , PUNCT cana-1834 3 30 india,mailid:dmrmaths@gmail.com india,mailid:dmrmaths@gmail.com X cana-1834 3 31 4associate 4associate NUM cana-1834 3 32 professor professor NOUN cana-1834 3 33 , , PUNCT cana-1834 3 34 department department NOUN cana-1834 3 35 of of ADP cana-1834 3 36 computer computer NOUN cana-1834 3 37 science science NOUN cana-1834 3 38 engineering engineering NOUN cana-1834 3 39 school school NOUN cana-1834 3 40 of of ADP cana-1834 3 41 computing computing PROPN cana-1834 3 42 & & CCONJ cana-1834 3 43 informatics informatics PROPN cana-1834 3 44 , , PUNCT cana-1834 3 45 vfstr vfstr NOUN cana-1834 3 46 deemed deem VERB cana-1834 3 47 to to ADP cana-1834 3 48 university university NOUN cana-1834 3 49 , , PUNCT cana-1834 3 50 vadlamudi vadlamudi NOUN cana-1834 3 51 , , PUNCT cana-1834 3 52 guntur guntur PROPN cana-1834 3 53 , , PUNCT cana-1834 3 54 a.p a.p PROPN cana-1834 3 55 , , PUNCT cana-1834 3 56 india india PROPN cana-1834 3 57 , , PUNCT cana-1834 3 58 mailid:pusapatisivaprasad@gmail.com mailid:pusapatisivaprasad@gmail.com X cana-1834 3 59 article article NOUN cana-1834 3 60 history history NOUN cana-1834 3 61 : : PUNCT cana-1834 3 62 received receive VERB cana-1834 3 63 : : PUNCT cana-1834 3 64 07 07 NUM cana-1834 3 65 - - SYM cana-1834 3 66 08 08 NUM cana-1834 3 67 - - PUNCT cana-1834 3 68 2024 2024 NUM cana-1834 3 69 revised revise VERB cana-1834 3 70 : : PUNCT cana-1834 3 71 17 17 NUM cana-1834 3 72 - - SYM cana-1834 3 73 09 09 NUM cana-1834 3 74 - - PUNCT cana-1834 3 75 2024 2024 NUM cana-1834 3 76 accepted accept VERB cana-1834 3 77 : : PUNCT cana-1834 3 78 25 25 NUM cana-1834 3 79 - - PUNCT cana-1834 3 80 09 09 NUM cana-1834 3 81 - - PUNCT cana-1834 3 82 2024 2024 NUM cana-1834 3 83 abstract abstract NOUN cana-1834 3 84 : : PUNCT cana-1834 3 85 in in ADP cana-1834 3 86 this this DET cana-1834 3 87 research research NOUN cana-1834 3 88 article article NOUN cana-1834 3 89 has have VERB cana-1834 3 90 to to PART cana-1834 3 91 introduce introduce VERB cana-1834 3 92 the the DET cana-1834 3 93 concept concept NOUN cana-1834 3 94 of of ADP cana-1834 3 95 ternary ternary ADJ cana-1834 3 96 Ξ³-semirings.we Ξ³-semirings.we PROPN cana-1834 3 97 first first ADV cana-1834 3 98 consider consider VERB cana-1834 3 99 the the DET cana-1834 3 100 congruences congruence NOUN cana-1834 3 101 and and CCONJ cana-1834 3 102 ideals ideal NOUN cana-1834 3 103 of of ADP cana-1834 3 104 ternary ternary ADJ cana-1834 3 105 Ξ³ Ξ³ NOUN cana-1834 3 106 - - NOUN cana-1834 3 107 semirings semiring NOUN cana-1834 3 108 then then ADV cana-1834 3 109 we we PRON cana-1834 3 110 construct construct VERB cana-1834 3 111 a a DET cana-1834 3 112 new new ADJ cana-1834 3 113 ternary ternary ADJ cana-1834 3 114 Ξ³ Ξ³ X cana-1834 3 115 - - PUNCT cana-1834 3 116 semiring semiring NOUN cana-1834 3 117 and and CCONJ cana-1834 3 118 to to PART cana-1834 3 119 be be AUX cana-1834 3 120 discussed discuss VERB cana-1834 3 121 formation formation NOUN cana-1834 3 122 of of ADP cana-1834 3 123 ideals ideal NOUN cana-1834 3 124 on on ADP cana-1834 3 125 this this DET cana-1834 3 126 ternary ternary ADJ cana-1834 3 127 Ξ³ Ξ³ NOUN cana-1834 3 128 - - PUNCT cana-1834 3 129 semiring semiring NOUN cana-1834 3 130 . . PUNCT cana-1834 4 1 also also ADV cana-1834 4 2 with with ADP cana-1834 4 3 the the DET cana-1834 4 4 help help NOUN cana-1834 4 5 of of ADP cana-1834 4 6 congruences congruence NOUN cana-1834 4 7 induced induce VERB cana-1834 4 8 by by ADP cana-1834 4 9 homomorphism homomorphism NOUN cana-1834 4 10 of of ADP cana-1834 4 11 a a DET cana-1834 4 12 ternary ternary ADJ cana-1834 4 13 Ξ³-semiring.after Ξ³-semiring.after NOUN cana-1834 4 14 that that SCONJ cana-1834 4 15 we we PRON cana-1834 4 16 will will AUX cana-1834 4 17 establish establish VERB cana-1834 4 18 some some PRON cana-1834 4 19 of of ADP cana-1834 4 20 isomorphism isomorphism NOUN cana-1834 4 21 theorems theorem NOUN cana-1834 4 22 and and CCONJ cana-1834 4 23 identified identify VERB cana-1834 4 24 the the DET cana-1834 4 25 some some PRON cana-1834 4 26 of of ADP cana-1834 4 27 commutativity commutativity NOUN cana-1834 4 28 in in ADP cana-1834 4 29 the the DET cana-1834 4 30 diagrams diagram NOUN cana-1834 4 31 . . PUNCT cana-1834 5 1 particularly particularly ADV cana-1834 5 2 some some DET cana-1834 5 3 fundamental fundamental ADJ cana-1834 5 4 results result NOUN cana-1834 5 5 of of ADP cana-1834 5 6 ternary ternary ADJ cana-1834 5 7 Ξ³ Ξ³ X cana-1834 5 8 - - PUNCT cana-1834 5 9 semiring semiring NOUN cana-1834 5 10 were be AUX cana-1834 5 11 proved prove VERB cana-1834 5 12 and and CCONJ cana-1834 5 13 strengthened strengthen VERB cana-1834 5 14 . . PUNCT cana-1834 6 1 keywords keyword NOUN cana-1834 6 2 : : PUNCT cana-1834 6 3 ternaryΞ³ ternaryΞ³ NOUN cana-1834 6 4 - - PUNCT cana-1834 6 5 semiring semiring ADJ cana-1834 6 6 , , PUNCT cana-1834 6 7 ideal ideal ADJ cana-1834 6 8 , , PUNCT cana-1834 6 9 congruence congruence NOUN cana-1834 6 10 , , PUNCT cana-1834 6 11 quotient quotient VERB cana-1834 6 12 ternary ternary ADJ cana-1834 6 13 Ξ³ Ξ³ PROPN cana-1834 6 14 - - PUNCT cana-1834 6 15 semiring semiring NOUN cana-1834 6 16 , , PUNCT cana-1834 6 17 homomorphism homomorphism PROPN cana-1834 6 18 and and CCONJ cana-1834 6 19 isomorphism isomorphism PROPN cana-1834 6 20 mathematics mathematic NOUN cana-1834 6 21 subject subject ADJ cana-1834 6 22 classification classification NOUN cana-1834 6 23 : : PUNCT cana-1834 6 24 16y60,06b10 16y60,06b10 NUM cana-1834 6 25 . . PUNCT cana-1834 7 1 1.introduction 1.introduction NUM cana-1834 7 2 : : PUNCT cana-1834 7 3 the the DET cana-1834 7 4 notion notion NOUN cana-1834 7 5 of of ADP cana-1834 7 6 𝛀 𝛀 PROPN cana-1834 7 7 βˆ’semiring βˆ’semire VERB cana-1834 7 8 was be AUX cana-1834 7 9 studied study VERB cana-1834 7 10 by by ADP cana-1834 7 11 m.k.rao.as m.k.rao.as PROPN cana-1834 7 12 a a DET cana-1834 7 13 generalization generalization NOUN cana-1834 7 14 of of ADP cana-1834 7 15 𝛀 𝛀 PROPN cana-1834 7 16 βˆ’ring βˆ’re VERB cana-1834 7 17 as as ADV cana-1834 7 18 well well ADV cana-1834 7 19 as as ADP cana-1834 7 20 of of ADP cana-1834 7 21 semiring semiring NOUN cana-1834 7 22 . . PUNCT cana-1834 8 1 in in ADP cana-1834 8 2 the the DET cana-1834 8 3 year year NOUN cana-1834 8 4 of of ADP cana-1834 8 5 1964 1964 NUM cana-1834 8 6 𝛀 𝛀 PROPN cana-1834 8 7 βˆ’ring βˆ’re VERB cana-1834 8 8 was be AUX cana-1834 8 9 introduced introduce VERB cana-1834 8 10 by by ADP cana-1834 8 11 n.nobusawa n.nobusawa NOUN cana-1834 8 12 there there ADV cana-1834 8 13 have have AUX cana-1834 8 14 been be AUX cana-1834 8 15 a a DET cana-1834 8 16 few few ADJ cana-1834 8 17 definitions definition NOUN cana-1834 8 18 for for ADP cana-1834 8 19 a a DET cana-1834 8 20 𝛀 𝛀 PROPN cana-1834 8 21 βˆ’ring.the βˆ’ring.the DET cana-1834 8 22 concepts concept NOUN cana-1834 8 23 of of ADP cana-1834 8 24 ternary ternary ADJ cana-1834 8 25 𝛀 𝛀 PROPN cana-1834 8 26 βˆ’semirings βˆ’semiring NOUN cana-1834 8 27 and and CCONJ cana-1834 8 28 ternary ternary ADJ cana-1834 8 29 sub sub NOUN cana-1834 8 30 𝛀 𝛀 PROPN cana-1834 8 31 βˆ’semiring βˆ’semire VERB cana-1834 8 32 with with ADP cana-1834 8 33 left left ADJ cana-1834 8 34 , , PUNCT cana-1834 8 35 right right INTJ cana-1834 8 36 , , PUNCT cana-1834 8 37 lateral lateral ADJ cana-1834 8 38 was be AUX cana-1834 8 39 studied study VERB cana-1834 8 40 by by ADP cana-1834 8 41 d.madhusudana d.madhusudana PROPN cana-1834 8 42 rao rao PROPN cana-1834 8 43 and and CCONJ cana-1834 8 44 m.sajani m.sajani NOUN cana-1834 8 45 lavanya lavanya NOUN cana-1834 8 46 in in ADP cana-1834 8 47 the the DET cana-1834 8 48 year year NOUN cana-1834 8 49 of of ADP cana-1834 8 50 2007.t.k 2007.t.k ADJ cana-1834 8 51 datta datta NOUN cana-1834 8 52 and and CCONJ cana-1834 8 53 m.l m.l PROPN cana-1834 8 54 das das PROPN cana-1834 8 55 were be AUX cana-1834 8 56 introduced introduce VERB cana-1834 8 57 and and CCONJ cana-1834 8 58 studied study VERB cana-1834 8 59 the the DET cana-1834 8 60 ideals ideal NOUN cana-1834 8 61 , , PUNCT cana-1834 8 62 prime prime ADJ cana-1834 8 63 ideals ideal NOUN cana-1834 8 64 semiprime semiprime NOUN cana-1834 8 65 ideals ideal NOUN cana-1834 8 66 k k PROPN cana-1834 8 67 - - PUNCT cana-1834 8 68 idelas idela NOUN cana-1834 8 69 and and CCONJ cana-1834 8 70 h h NOUN cana-1834 8 71 - - PUNCT cana-1834 8 72 ideals ideal NOUN cana-1834 8 73 of of ADP cana-1834 8 74 a a DET cana-1834 8 75 ternary ternary ADJ cana-1834 8 76 𝛀 𝛀 PROPN cana-1834 8 77 βˆ’semiring βˆ’semire VERB cana-1834 8 78 , , PUNCT cana-1834 8 79 regular regular ADJ cana-1834 8 80 ternary ternary ADJ cana-1834 8 81 𝛀 𝛀 PROPN cana-1834 8 82 βˆ’semiring βˆ’semire VERB cana-1834 8 83 respectively respectively ADV cana-1834 8 84 2 2 NUM cana-1834 8 85 .. .. PUNCT cana-1834 8 86 priliminaries priliminarie NOUN cana-1834 8 87 : : PUNCT cana-1834 8 88 definition definition NOUN cana-1834 8 89 2.1 2.1 NUM cana-1834 8 90 : : PUNCT cana-1834 8 91 let(𝑇 let(𝑇 NUM cana-1834 8 92 , , PUNCT cana-1834 8 93 + + NOUN cana-1834 8 94 ) ) PUNCT cana-1834 8 95 and and CCONJ cana-1834 8 96 ( ( PUNCT cana-1834 8 97 Ξ³ Ξ³ X cana-1834 8 98 , , PUNCT cana-1834 8 99 + + NOUN cana-1834 8 100 ) ) PUNCT cana-1834 8 101 be be VERB cana-1834 8 102 two two NUM cana-1834 8 103 additive additive ADJ cana-1834 8 104 commutative commutative ADJ cana-1834 8 105 semigroups semigroup NOUN cana-1834 8 106 then then ADV cana-1834 8 107 𝑇 𝑇 PROPN cana-1834 8 108 is be AUX cana-1834 8 109 known know VERB cana-1834 8 110 as as ADP cana-1834 8 111 ternary ternary ADJ cana-1834 8 112 𝛀 𝛀 PROPN cana-1834 8 113 βˆ’semiring βˆ’semire VERB cana-1834 8 114 if if SCONJ cana-1834 8 115 there there PRON cana-1834 8 116 exist exist VERB cana-1834 8 117 a a DET cana-1834 8 118 mapping mapping NOUN cana-1834 8 119 from from ADP cana-1834 8 120 𝑇 𝑇 PROPN cana-1834 8 121 Γ— Γ— NOUN cana-1834 8 122 𝛀 𝛀 PROPN cana-1834 8 123 Γ— Γ— NOUN cana-1834 8 124 𝑇 𝑇 PROPN cana-1834 8 125 Γ— Γ— NOUN cana-1834 8 126 𝛀 𝛀 PROPN cana-1834 8 127 Γ— Γ— VERB cana-1834 8 128 𝑇 𝑇 NOUN cana-1834 8 129 to to ADP cana-1834 8 130 𝑇 𝑇 PROPN cana-1834 8 131 which which DET cana-1834 8 132 maps map VERB cana-1834 8 133 ( ( PUNCT cana-1834 8 134 π‘Ž π‘Ž X cana-1834 8 135 , , PUNCT cana-1834 8 136 𝛼 𝛼 NOUN cana-1834 8 137 , , PUNCT cana-1834 8 138 𝑏 𝑏 NOUN cana-1834 8 139 , , PUNCT cana-1834 8 140 𝛽 𝛽 NOUN cana-1834 8 141 , , PUNCT cana-1834 8 142 𝑐 𝑐 NOUN cana-1834 8 143 ) ) PUNCT cana-1834 8 144 β†’ β†’ PUNCT cana-1834 9 1 [ [ X cana-1834 9 2 π‘Žπ›Όπ‘π›½π‘ π‘Žπ›Όπ‘π›½π‘ X cana-1834 9 3 ] ] X cana-1834 9 4 satisfying satisfy VERB cana-1834 9 5 the the DET cana-1834 9 6 following follow VERB cana-1834 9 7 conditions condition NOUN cana-1834 9 8 𝑖 𝑖 NOUN cana-1834 9 9 ) ) PUNCT cana-1834 9 10 [ [ X cana-1834 9 11 [ [ PUNCT cana-1834 9 12 π‘Žπ›Όπ‘π›½π‘]𝛾𝑑𝛿𝑒 π‘Žπ›Όπ‘π›½π‘]𝛾𝑑𝛿𝑒 X cana-1834 9 13 ] ] X cana-1834 9 14 = = PUNCT cana-1834 10 1 [ [ X cana-1834 10 2 π‘Žπ›Ό[𝑏𝛽𝑐𝛾𝑑]𝛿𝑒 π‘Žπ›Ό[𝑏𝛽𝑐𝛾𝑑]𝛿𝑒 X cana-1834 10 3 ] ] X cana-1834 10 4 = = PUNCT cana-1834 11 1 [ [ X cana-1834 11 2 π‘Žπ›Όπ‘π›½[𝑐𝛾𝑑𝛿𝑒 π‘Žπ›Όπ‘π›½[𝑐𝛾𝑑𝛿𝑒 X cana-1834 11 3 ] ] X cana-1834 11 4 ] ] X cana-1834 11 5 𝑖𝑖)[(π‘Ž 𝑖𝑖)[(π‘Ž NOUN cana-1834 12 1 + + X cana-1834 12 2 𝑏)𝛼𝑐𝛽𝑑 𝑏)𝛼𝑐𝛽𝑑 NOUN cana-1834 12 3 ] ] X cana-1834 12 4 = = PUNCT cana-1834 13 1 [ [ X cana-1834 13 2 π‘Žπ›Όπ‘π›½π‘‘ π‘Žπ›Όπ‘π›½π‘‘ X cana-1834 13 3 ] ] PUNCT cana-1834 13 4 + + CCONJ cana-1834 13 5 [ [ X cana-1834 13 6 𝑏𝛼𝑐𝛽𝑑 𝑏𝛼𝑐𝛽𝑑 X cana-1834 13 7 ] ] PUNCT cana-1834 13 8 𝑖𝑖𝑖)[π‘Žπ›Ό(𝑏 𝑖𝑖𝑖)[π‘Žπ›Ό(𝑏 PUNCT cana-1834 13 9 + + PUNCT cana-1834 13 10 𝑐)𝛽𝑑 𝑐)𝛽𝑑 ADJ cana-1834 13 11 ] ] X cana-1834 13 12 = = PUNCT cana-1834 14 1 [ [ X cana-1834 14 2 π‘Žπ›Όπ‘π›½π‘‘ π‘Žπ›Όπ‘π›½π‘‘ NOUN cana-1834 14 3 ] ] PUNCT cana-1834 14 4 + + CCONJ cana-1834 14 5 [ [ PUNCT cana-1834 14 6 π‘Žπ›Όπ‘π›½π‘‘ π‘Žπ›Όπ‘π›½π‘‘ X cana-1834 14 7 ] ] PUNCT cana-1834 14 8 𝑖𝑣)[π‘Žπ›Όπ‘π›½(𝑐 𝑖𝑣)[π‘Žπ›Όπ‘π›½(𝑐 PUNCT cana-1834 14 9 + + CCONJ cana-1834 14 10 𝑑 𝑑 NOUN cana-1834 14 11 ) ) PUNCT cana-1834 14 12 ] ] PUNCT cana-1834 14 13 = = PUNCT cana-1834 15 1 [ [ X cana-1834 15 2 π‘Žπ›Όπ‘π›½π‘ π‘Žπ›Όπ‘π›½π‘ X cana-1834 15 3 ] ] X cana-1834 15 4 + + CCONJ cana-1834 15 5 [ [ X cana-1834 15 6 π‘Žπ›Όπ‘π›½π‘‘]for π‘Žπ›Όπ‘π›½π‘‘]for ADP cana-1834 15 7 all all DET cana-1834 15 8 π‘Ž π‘Ž ADJ cana-1834 15 9 , , PUNCT cana-1834 15 10 𝑏 𝑏 NOUN cana-1834 15 11 , , PUNCT cana-1834 15 12 𝑐 𝑐 PROPN cana-1834 15 13 , , PUNCT cana-1834 15 14 𝑑 𝑑 PROPN cana-1834 15 15 ∈ ∈ PROPN cana-1834 15 16 𝑇 𝑇 PROPN cana-1834 15 17 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ cana-1834 15 18 𝛼 𝛼 PROPN cana-1834 15 19 , , PUNCT cana-1834 15 20 𝛽 𝛽 NOUN cana-1834 15 21 , , PUNCT cana-1834 15 22 𝛾 𝛾 PROPN cana-1834 15 23 , , PUNCT cana-1834 15 24 π›Ώπœ–π›€ π›Ώπœ–π›€ NOUN cana-1834 15 25 communications communication NOUN cana-1834 15 26 on on ADP cana-1834 15 27 applied apply VERB cana-1834 15 28 nonlinear nonlinear ADJ cana-1834 15 29 analysis analysis NOUN cana-1834 15 30 issn issn NOUN cana-1834 15 31 : : PUNCT cana-1834 15 32 1074 1074 NUM cana-1834 15 33 - - PUNCT cana-1834 15 34 133x 133x NUM cana-1834 15 35 vol vol NOUN cana-1834 15 36 32 32 NUM cana-1834 15 37 no no NOUN cana-1834 15 38 . . NOUN cana-1834 15 39 2 2 NUM cana-1834 15 40 ( ( PUNCT cana-1834 15 41 2025 2025 NUM cana-1834 15 42 ) ) PUNCT cana-1834 15 43 582 582 NUM cana-1834 15 44 https://internationalpubls.com https://internationalpubls.com X cana-1834 15 45 definition definition NOUN cana-1834 15 46 2.2 2.2 NUM cana-1834 15 47 : : PUNCT cana-1834 15 48 a a DET cana-1834 15 49 ternary𝛀 ternary𝛀 PROPN cana-1834 15 50 βˆ’semiring βˆ’semire VERB cana-1834 15 51 𝑇 𝑇 PROPN cana-1834 15 52 is be AUX cana-1834 15 53 said say VERB cana-1834 15 54 to to PART cana-1834 15 55 have have VERB cana-1834 15 56 a a DET cana-1834 15 57 zero zero NUM cana-1834 15 58 element element NOUN cana-1834 15 59 provided provide VERB cana-1834 15 60 0 0 PUNCT cana-1834 16 1 + + CCONJ cana-1834 17 1 π‘₯ π‘₯ NOUN cana-1834 17 2 = = PUNCT cana-1834 17 3 π‘₯ π‘₯ NOUN cana-1834 17 4 = = PUNCT cana-1834 17 5 π‘₯ π‘₯ PROPN cana-1834 18 1 + + NOUN cana-1834 18 2 0 0 NUM cana-1834 18 3 and and CCONJ cana-1834 18 4 [ [ X cana-1834 18 5 0π›Όπ‘Žπ›½π‘ 0π›Όπ‘Žπ›½π‘ X cana-1834 18 6 ] ] X cana-1834 18 7 = = X cana-1834 19 1 [ [ X cana-1834 19 2 π‘Žπ›Ό0𝛽𝑏 π‘Žπ›Ό0𝛽𝑏 X cana-1834 19 3 ] ] X cana-1834 19 4 = = PUNCT cana-1834 20 1 [ [ X cana-1834 20 2 π‘Žπ›Όπ‘π›½0 π‘Žπ›Όπ‘π›½0 X cana-1834 20 3 ] ] X cana-1834 20 4 = = SYM cana-1834 20 5 0 0 NUM cana-1834 20 6 , , PUNCT cana-1834 20 7 βˆ€π‘Ž βˆ€π‘Ž PROPN cana-1834 20 8 , , PUNCT cana-1834 20 9 𝑏 𝑏 NOUN cana-1834 20 10 , , PUNCT cana-1834 20 11 π‘₯ π‘₯ PRON cana-1834 20 12 ∈ ∈ PROPN cana-1834 20 13 𝑇 𝑇 PROPN cana-1834 20 14 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ cana-1834 20 15 𝛼 𝛼 NOUN cana-1834 20 16 , , PUNCT cana-1834 20 17 π›½πœ–π›€ π›½πœ–π›€ NOUN cana-1834 20 18 definition definition NOUN cana-1834 20 19 2.3 2.3 NUM cana-1834 20 20 : : PUNCT cana-1834 20 21 a a DET cana-1834 20 22 ternary𝛀 ternary𝛀 PROPN cana-1834 20 23 βˆ’semiring βˆ’semire VERB cana-1834 20 24 𝑇 𝑇 PROPN cana-1834 20 25 is be AUX cana-1834 20 26 known know VERB cana-1834 20 27 as as ADP cana-1834 20 28 commutative commutative ADJ cana-1834 20 29 ternary𝛀 ternary𝛀 PROPN cana-1834 20 30 βˆ’semiring βˆ’semire VERB cana-1834 20 31 𝑇 𝑇 PROPN cana-1834 20 32 provided provide VERB cana-1834 20 33 [ [ PRON cana-1834 20 34 π‘Žπ›Όπ‘π›½π‘ π‘Žπ›Όπ‘π›½π‘ X cana-1834 20 35 ] ] X cana-1834 20 36 = = PUNCT cana-1834 21 1 [ [ X cana-1834 21 2 π‘π›Όπ‘π›½π‘Ž π‘π›Όπ‘π›½π‘Ž X cana-1834 21 3 ] ] X cana-1834 21 4 = = PUNCT cana-1834 22 1 [ [ X cana-1834 22 2 π‘π›Όπ‘Žπ›½π‘ π‘π›Όπ‘Žπ›½π‘ X cana-1834 22 3 ] ] X cana-1834 22 4 = = PUNCT cana-1834 23 1 [ [ X cana-1834 23 2 π‘π›Όπ‘Žπ›½π‘ π‘π›Όπ‘Žπ›½π‘ X cana-1834 23 3 ] ] X cana-1834 23 4 = = PUNCT cana-1834 24 1 [ [ X cana-1834 24 2 π‘π›Όπ‘π›½π‘Ž π‘π›Όπ‘π›½π‘Ž X cana-1834 24 3 ] ] X cana-1834 24 4 = = PUNCT cana-1834 25 1 [ [ X cana-1834 25 2 π‘Žπ›Όπ‘π›½π‘ π‘Žπ›Όπ‘π›½π‘ X cana-1834 25 3 ] ] PUNCT cana-1834 25 4 for for ADP cana-1834 25 5 allπ‘Ž allπ‘Ž PROPN cana-1834 25 6 , , PUNCT cana-1834 25 7 𝑏 𝑏 NOUN cana-1834 25 8 , , PUNCT cana-1834 25 9 𝑐 𝑐 PROPN cana-1834 25 10 ∈ ∈ PROPN cana-1834 25 11 𝑇. 𝑇. PROPN cana-1834 25 12 definition definition NOUN cana-1834 25 13 2.4 2.4 NUM cana-1834 25 14 : : PUNCT cana-1834 25 15 let let VERB cana-1834 25 16 𝑆 𝑆 PROPN cana-1834 25 17 be be AUX cana-1834 25 18 a a DET cana-1834 25 19 non non X cana-1834 25 20 empty empty ADJ cana-1834 25 21 subset subset NOUN cana-1834 25 22 of of ADP cana-1834 25 23 a a DET cana-1834 25 24 ternary ternary ADJ cana-1834 25 25 𝛀 𝛀 PROPN cana-1834 25 26 βˆ’semiring βˆ’semire VERB cana-1834 25 27 𝑇 𝑇 PROPN cana-1834 25 28 is be AUX cana-1834 25 29 said say VERB cana-1834 25 30 to to PART cana-1834 25 31 be be AUX cana-1834 25 32 a a DET cana-1834 25 33 ternary ternary ADJ cana-1834 25 34 sub sub NOUN cana-1834 25 35 𝛀 𝛀 PROPN cana-1834 25 36 βˆ’semiring βˆ’semire VERB cana-1834 25 37 of of ADP cana-1834 25 38 𝑇 𝑇 PROPN cana-1834 25 39 if if SCONJ cana-1834 25 40 and and CCONJ cana-1834 25 41 only only ADV cana-1834 25 42 if if SCONJ cana-1834 25 43 𝑆 𝑆 PROPN cana-1834 25 44 + + CCONJ cana-1834 25 45 𝑆 𝑆 PROPN cana-1834 25 46 βŠ† βŠ† NUM cana-1834 25 47 𝑆 𝑆 PROPN cana-1834 25 48 and and CCONJ cana-1834 25 49 [ [ X cana-1834 25 50 𝑆𝛼𝑆𝛽𝑆 𝑆𝛼𝑆𝛽𝑆 X cana-1834 25 51 ] ] X cana-1834 25 52 βŠ† βŠ† NUM cana-1834 25 53 𝑆 𝑆 PROPN cana-1834 25 54 for for ADP cana-1834 25 55 all all DET cana-1834 25 56 𝛼 𝛼 PROPN cana-1834 25 57 , , PUNCT cana-1834 25 58 π›½πœ–π›€. π›½πœ–π›€. PROPN cana-1834 25 59 definition definition NOUN cana-1834 25 60 2.5 2.5 NUM cana-1834 25 61 : : PUNCT cana-1834 25 62 let let VERB cana-1834 25 63 𝑇 𝑇 PROPN cana-1834 25 64 be be AUX cana-1834 25 65 a a DET cana-1834 25 66 ternary𝛀 ternary𝛀 PROPN cana-1834 25 67 βˆ’semiring βˆ’semiring NOUN cana-1834 25 68 and and CCONJ cana-1834 25 69 𝐴 𝐴 PROPN cana-1834 25 70 be be VERB cana-1834 25 71 a a DET cana-1834 25 72 non non X cana-1834 25 73 empty empty ADJ cana-1834 25 74 subset subset NOUN cana-1834 25 75 of of ADP cana-1834 25 76 𝑇 𝑇 PROPN cana-1834 25 77 is be AUX cana-1834 25 78 said say VERB cana-1834 25 79 to to PART cana-1834 25 80 be be AUX cana-1834 25 81 a a DET cana-1834 25 82 left left ADJ cana-1834 25 83 ternary ternary ADJ cana-1834 25 84 𝛀-ideal 𝛀-ideal PROPN cana-1834 25 85 of of ADP cana-1834 25 86 𝑇 𝑇 PROPN cana-1834 25 87 if if SCONJ cana-1834 25 88 𝑖)(π‘Ž 𝑖)(π‘Ž PRON cana-1834 25 89 + + NUM cana-1834 25 90 𝑏 𝑏 NOUN cana-1834 25 91 ) ) PUNCT cana-1834 25 92 ∈ ∈ PROPN cana-1834 25 93 𝐴 𝐴 PROPN cana-1834 25 94 and and CCONJ cana-1834 25 95 𝑖𝑖)𝑏 𝑖𝑖)𝑏 PROPN cana-1834 25 96 , , PUNCT cana-1834 25 97 𝑐 𝑐 PROPN cana-1834 25 98 ∈ ∈ PROPN cana-1834 25 99 𝑇 𝑇 PROPN cana-1834 25 100 , , PUNCT cana-1834 25 101 π‘Ž π‘Ž PROPN cana-1834 25 102 ∈ ∈ PROPN cana-1834 25 103 𝐴 𝐴 PROPN cana-1834 25 104 , , PUNCT cana-1834 25 105 𝛼 𝛼 PROPN cana-1834 25 106 , , PUNCT cana-1834 25 107 π›½πœ–π›€ π›½πœ–π›€ NOUN cana-1834 25 108 ⟹ ⟹ VERB cana-1834 26 1 [ [ X cana-1834 26 2 π‘π›Όπ‘π›½π‘Ž π‘π›Όπ‘π›½π‘Ž X cana-1834 26 3 ] ] X cana-1834 26 4 ∈ ∈ PROPN cana-1834 26 5 𝐴. 𝐴. PROPN cana-1834 26 6 definition definition NOUN cana-1834 26 7 2.6 2.6 NUM cana-1834 26 8 : : PUNCT cana-1834 26 9 let𝑇 let𝑇 NOUN cana-1834 26 10 be be AUX cana-1834 26 11 a a DET cana-1834 26 12 ternary𝛀 ternary𝛀 PROPN cana-1834 26 13 βˆ’semiring βˆ’semiring NOUN cana-1834 26 14 and and CCONJ cana-1834 26 15 𝐴 𝐴 PROPN cana-1834 26 16 be be VERB cana-1834 26 17 a a DET cana-1834 26 18 non non X cana-1834 26 19 empty empty ADJ cana-1834 26 20 subset subset NOUN cana-1834 26 21 of of ADP cana-1834 26 22 𝑇 𝑇 PROPN cana-1834 26 23 is be AUX cana-1834 26 24 said say VERB cana-1834 26 25 to to PART cana-1834 26 26 be be AUX cana-1834 26 27 a a DET cana-1834 26 28 lateral lateral ADJ cana-1834 26 29 ternary ternary ADJ cana-1834 26 30 𝛀-ideal 𝛀-ideal PROPN cana-1834 26 31 of of ADP cana-1834 26 32 𝑇 𝑇 PROPN cana-1834 26 33 if if SCONJ cana-1834 26 34 𝑖)(π‘Ž 𝑖)(π‘Ž PRON cana-1834 26 35 + + NUM cana-1834 26 36 𝑏 𝑏 NOUN cana-1834 26 37 ) ) PUNCT cana-1834 26 38 ∈ ∈ PROPN cana-1834 26 39 𝐴 𝐴 PROPN cana-1834 26 40 and and CCONJ cana-1834 26 41 𝑖𝑖)𝑏 𝑖𝑖)𝑏 PROPN cana-1834 26 42 , , PUNCT cana-1834 26 43 𝑐 𝑐 PROPN cana-1834 26 44 ∈ ∈ PROPN cana-1834 26 45 𝑇 𝑇 PROPN cana-1834 26 46 , , PUNCT cana-1834 26 47 π‘Ž π‘Ž PROPN cana-1834 26 48 ∈ ∈ PROPN cana-1834 26 49 𝐴 𝐴 PROPN cana-1834 26 50 , , PUNCT cana-1834 26 51 𝛼 𝛼 PROPN cana-1834 26 52 , , PUNCT cana-1834 26 53 π›½πœ–π›€ π›½πœ–π›€ NOUN cana-1834 26 54 ⟹ ⟹ PUNCT cana-1834 27 1 [ [ X cana-1834 27 2 π‘π›Όπ‘Žπ›½π‘ π‘π›Όπ‘Žπ›½π‘ X cana-1834 27 3 ] ] X cana-1834 27 4 ∈ ∈ PROPN cana-1834 27 5 𝐴. 𝐴. PROPN cana-1834 27 6 definition definition NOUN cana-1834 27 7 2.7 2.7 NUM cana-1834 27 8 : : PUNCT cana-1834 27 9 let𝑇 let𝑇 NOUN cana-1834 27 10 be be AUX cana-1834 27 11 a a DET cana-1834 27 12 ternary𝛀 ternary𝛀 PROPN cana-1834 27 13 βˆ’semiring βˆ’semiring NOUN cana-1834 27 14 and and CCONJ cana-1834 27 15 𝐴 𝐴 PROPN cana-1834 27 16 be be VERB cana-1834 27 17 a a DET cana-1834 27 18 non non X cana-1834 27 19 empty empty ADJ cana-1834 27 20 subset subset NOUN cana-1834 27 21 of of ADP cana-1834 27 22 𝑇 𝑇 PROPN cana-1834 27 23 is be AUX cana-1834 27 24 said say VERB cana-1834 27 25 to to PART cana-1834 27 26 be be AUX cana-1834 27 27 a a DET cana-1834 27 28 right right ADJ cana-1834 27 29 ternary ternary ADJ cana-1834 27 30 𝛀-ideal 𝛀-ideal PROPN cana-1834 27 31 of of ADP cana-1834 27 32 𝑇 𝑇 PROPN cana-1834 27 33 if if SCONJ cana-1834 27 34 𝑖)(π‘Ž 𝑖)(π‘Ž PRON cana-1834 27 35 + + NUM cana-1834 27 36 𝑏 𝑏 NOUN cana-1834 27 37 ) ) PUNCT cana-1834 27 38 ∈ ∈ PROPN cana-1834 27 39 𝐴 𝐴 PROPN cana-1834 27 40 and and CCONJ cana-1834 27 41 𝑖𝑖)𝑏 𝑖𝑖)𝑏 PROPN cana-1834 27 42 , , PUNCT cana-1834 27 43 𝑐 𝑐 PROPN cana-1834 27 44 ∈ ∈ PROPN cana-1834 27 45 𝑇 𝑇 PROPN cana-1834 27 46 , , PUNCT cana-1834 27 47 π‘Ž π‘Ž PROPN cana-1834 27 48 ∈ ∈ PROPN cana-1834 27 49 𝐴 𝐴 PROPN cana-1834 27 50 , , PUNCT cana-1834 27 51 𝛼 𝛼 PROPN cana-1834 27 52 , , PUNCT cana-1834 27 53 π›½πœ–π›€ π›½πœ–π›€ NOUN cana-1834 27 54 ⟹ ⟹ PUNCT cana-1834 28 1 [ [ X cana-1834 28 2 π‘Žπ›Όπ‘π›½π‘ π‘Žπ›Όπ‘π›½π‘ X cana-1834 28 3 ] ] X cana-1834 28 4 ∈ ∈ PROPN cana-1834 28 5 𝐴. 𝐴. PROPN cana-1834 28 6 definition definition NOUN cana-1834 28 7 2.8 2.8 NUM cana-1834 28 8 : : PUNCT cana-1834 28 9 let let VERB cana-1834 28 10 𝑇 𝑇 PROPN cana-1834 28 11 be be AUX cana-1834 28 12 a a DET cana-1834 28 13 ternary𝛀 ternary𝛀 PROPN cana-1834 28 14 βˆ’semiring βˆ’semiring NOUN cana-1834 28 15 and and CCONJ cana-1834 28 16 𝐴 𝐴 PROPN cana-1834 28 17 be be VERB cana-1834 28 18 a a DET cana-1834 28 19 non non X cana-1834 28 20 empty empty ADJ cana-1834 28 21 subset subset NOUN cana-1834 28 22 of of ADP cana-1834 28 23 𝑇 𝑇 PROPN cana-1834 28 24 is be AUX cana-1834 28 25 said say VERB cana-1834 28 26 to to PART cana-1834 28 27 be be AUX cana-1834 28 28 a a DET cana-1834 28 29 ternary ternary ADJ cana-1834 28 30 𝛀-ideal 𝛀-ideal PROPN cana-1834 28 31 of of ADP cana-1834 28 32 𝑇 𝑇 PROPN cana-1834 28 33 if if SCONJ cana-1834 28 34 and and CCONJ cana-1834 28 35 only only ADV cana-1834 28 36 if if SCONJ cana-1834 28 37 it it PRON cana-1834 28 38 is be AUX cana-1834 28 39 a a DET cana-1834 28 40 left left ADJ cana-1834 28 41 ternary ternary ADJ cana-1834 28 42 𝛀-ideal 𝛀-ideal PROPN cana-1834 28 43 , , PUNCT cana-1834 28 44 lateral lateral ADJ cana-1834 28 45 ternary ternary ADJ cana-1834 28 46 𝛀-ideal 𝛀-ideal PROPN cana-1834 28 47 , , PUNCT cana-1834 28 48 right right ADJ cana-1834 28 49 ternary ternary NOUN cana-1834 28 50 𝛀ideal 𝛀ideal PROPN cana-1834 28 51 of of ADP cana-1834 28 52 𝑇. 𝑇. PROPN cana-1834 28 53 3.ideals 3.ideals NUM cana-1834 28 54 of of ADP cana-1834 28 55 a a DET cana-1834 28 56 ternary ternary ADJ cana-1834 28 57 𝛀 𝛀 PROPN cana-1834 28 58 βˆ’semiring βˆ’semire VERB cana-1834 28 59 entire entire ADJ cana-1834 28 60 of of ADP cana-1834 28 61 this this DET cana-1834 28 62 research research NOUN cana-1834 28 63 article article NOUN cana-1834 28 64 t t PROPN cana-1834 28 65 be be AUX cana-1834 28 66 a a DET cana-1834 28 67 ternary𝛀 ternary𝛀 PROPN cana-1834 28 68 βˆ’semiringunless βˆ’semiringunless NOUN cana-1834 28 69 otherwise otherwise ADV cana-1834 28 70 specified specify VERB cana-1834 28 71 . . PUNCT cana-1834 29 1 the the DET cana-1834 29 2 below below ADJ cana-1834 29 3 theorems theorem NOUN cana-1834 29 4 are be AUX cana-1834 29 5 easily easily ADV cana-1834 29 6 to to PART cana-1834 29 7 prove prove VERB cana-1834 29 8 . . PUNCT cana-1834 30 1 lemma3.1 lemma3.1 NOUN cana-1834 30 2 : : PUNCT cana-1834 30 3 letbe letbe X cana-1834 30 4 a a DET cana-1834 30 5 non non X cana-1834 30 6 empty empty ADJ cana-1834 30 7 index index NOUN cana-1834 30 8 set set VERB cana-1834 30 9 and and CCONJ cana-1834 30 10 { { PUNCT cana-1834 30 11 } } PUNCT cana-1834 30 12 i i PROPN cana-1834 30 13 ο¬οƒŽο„ ο¬οƒŽο„ NUM cana-1834 30 14 be be AUX cana-1834 30 15 a a DET cana-1834 30 16 family family NOUN cana-1834 30 17 of of ADP cana-1834 30 18 ideals ideal NOUN cana-1834 30 19 of of ADP cana-1834 30 20 ( ( PUNCT cana-1834 30 21 t,𝛀).then t,𝛀).then X cana-1834 30 22 i i PROPN cana-1834 30 23 ο¬οƒŽο„ ο¬οƒŽο„ X cana-1834 30 24   PROPN cana-1834 30 25 is be AUX cana-1834 30 26 an an DET cana-1834 30 27 ideal ideal NOUN cana-1834 30 28 of of ADP cana-1834 30 29 ( ( PUNCT cana-1834 30 30 t,𝛀 t,𝛀 PROPN cana-1834 30 31 ) ) PUNCT cana-1834 30 32 . . PUNCT cana-1834 31 1 lemma3.2 lemma3.2 PROPN cana-1834 31 2 : : PUNCT cana-1834 31 3 let let VERB cana-1834 31 4 Β£ Β£ SYM cana-1834 31 5 ( ( PUNCT cana-1834 31 6 t,𝛀)be t,𝛀)be VERB cana-1834 31 7 the the DET cana-1834 31 8 set set NOUN cana-1834 31 9 of of ADP cana-1834 31 10 all all DET cana-1834 31 11 ideals ideal NOUN cana-1834 31 12 of of ADP cana-1834 31 13 ( ( PUNCT cana-1834 31 14 t,𝛀).then t,𝛀).then X cana-1834 31 15 Β£ Β£ PROPN cana-1834 31 16 , , PUNCT cana-1834 31 17 Ξ³ Ξ³ NOUN cana-1834 31 18 ) ) PUNCT cana-1834 31 19 , , PUNCT cana-1834 31 20 , , PUNCT cana-1834 31 21 ( ( PUNCT cana-1834 31 22 ( ( PUNCT cana-1834 31 23 , , PUNCT cana-1834 31 24 ) ) PUNCT cana-1834 31 25 t t PROPN cana-1834 31 26   PROPN cana-1834 31 27   PROPN cana-1834 31 28 οƒš οƒš NOUN cana-1834 31 29 is be AUX cana-1834 31 30 a a DET cana-1834 31 31 complete complete ADJ cana-1834 31 32 lattice lattice NOUN cana-1834 31 33 , , PUNCT cana-1834 31 34 where where SCONJ cana-1834 31 35 i i PRON cana-1834 31 36 j j VERB cana-1834 32 1 i i PRON cana-1834 32 2 j j NOUN cana-1834 32 3 = = PUNCT cana-1834 32 4   PUNCT cana-1834 32 5 and and CCONJ cana-1834 32 6 i i PRON cana-1834 32 7 j j PROPN cana-1834 33 1 i i PRON cana-1834 33 2 jοƒš jοƒš PROPN cana-1834 33 3 = = PUNCT cana-1834 33 4 οƒˆ οƒˆ NOUN cana-1834 33 5 is be AUX cana-1834 33 6 the the DET cana-1834 33 7 unique unique ADJ cana-1834 33 8 smallest small ADJ cana-1834 33 9 ideal ideal NOUN cana-1834 33 10 containing contain VERB cana-1834 33 11 i i PRON cana-1834 33 12 jοƒˆ jοƒˆ PROPN cana-1834 33 13 . . PUNCT cana-1834 34 1 theorem theorem VERB cana-1834 34 2 3.3 3.3 NUM cana-1834 34 3 : : PUNCT cana-1834 34 4 lett lett PROPN cana-1834 34 5 be be AUX cana-1834 34 6 a a DET cana-1834 34 7 ternary ternary ADJ cana-1834 34 8 𝛀 𝛀 PROPN cana-1834 34 9 βˆ’semiring βˆ’semire VERB cana-1834 34 10 with with ADP cana-1834 34 11 zero zero NUM cana-1834 34 12 and and CCONJ cana-1834 34 13 i i PRON cana-1834 34 14 be be VERB cana-1834 34 15 an an DET cana-1834 34 16 ideal ideal NOUN cana-1834 34 17 of of ADP cana-1834 34 18 ( ( PUNCT cana-1834 34 19 t,𝛀).then t,𝛀).then X cana-1834 34 20 { { PUNCT cana-1834 34 21 p p NOUN cana-1834 34 22 / / SYM cana-1834 34 23 p p NOUN cana-1834 34 24 } } PUNCT cana-1834 34 25 r r NOUN cana-1834 34 26 i i NOUN cana-1834 35 1 r r VERB cana-1834 35 2 i i NOUN cana-1834 35 3 = = PUNCT cana-1834 36 1 + + NUM cana-1834 36 2 οƒŽ οƒŽ NOUN cana-1834 36 3 is be AUX cana-1834 36 4 a a DET cana-1834 36 5 ternary ternary ADJ cana-1834 36 6 𝛀 𝛀 PROPN cana-1834 36 7 βˆ’semiring βˆ’semire VERB cana-1834 36 8 with with ADP cana-1834 36 9 the the DET cana-1834 36 10 mapping mapping NOUN cana-1834 36 11 Ξ³ Ξ³ X cana-1834 36 12 : : PUNCT cana-1834 36 13 Ξ³ Ξ³ X cana-1834 36 14 * * PUNCT cana-1834 36 15 r r NOUN cana-1834 37 1 r r NOUN cana-1834 38 1 r r NOUN cana-1834 39 1 r r NOUN cana-1834 40 1 i i PRON cana-1834 41 1 i i PRON cana-1834 42 1 i i PRON cana-1834 43 1 i i PRON cana-1834 43 2 ο‚΄ ο‚΄ VERB cana-1834 43 3 ο‚΄ ο‚΄ PROPN cana-1834 43 4 ο‚΄ ο‚΄ PROPN cana-1834 43 5 ο‚΄ ο‚΄ PROPN cana-1834 43 6 β†’ β†’ PUNCT cana-1834 43 7 defined define VERB cana-1834 43 8 by by ADP cana-1834 43 9 ( ( PUNCT cana-1834 43 10 p p NOUN cana-1834 43 11 ) ) PUNCT cana-1834 43 12 * * PUNCT cana-1834 44 1 * * PUNCT cana-1834 44 2 ( ( PUNCT cana-1834 44 3 q q NOUN cana-1834 44 4 ) ) PUNCT cana-1834 44 5 * * PUNCT cana-1834 45 1 * * PUNCT cana-1834 45 2 ( ( PUNCT cana-1834 45 3 r r NOUN cana-1834 45 4 ) ) PUNCT cana-1834 45 5 i i PRON cana-1834 45 6 i i PRON cana-1834 46 1 i i PRON cana-1834 46 2 p p VERB cana-1834 46 3 q q NOUN cana-1834 46 4 r r NOUN cana-1834 46 5 i i NOUN cana-1834 46 6   NUM cana-1834 46 7   NUM cana-1834 47 1 + + X cana-1834 48 1 + + PUNCT cana-1834 49 1 + + PUNCT cana-1834 49 2 = = X cana-1834 50 1 + + CCONJ cana-1834 50 2 for for ADP cana-1834 50 3 all all DET cana-1834 50 4 p p NOUN cana-1834 50 5 , , PUNCT cana-1834 50 6 q q NOUN cana-1834 50 7 , , PUNCT cana-1834 50 8 r r NOUN cana-1834 50 9 Ξ³r Ξ³r NOUN cana-1834 50 10 and and CCONJ cana-1834 50 11 ο§οƒŽ ο§οƒŽ PRON cana-1834 50 12 οƒŽ οƒŽ NOUN cana-1834 50 13 proof proof NOUN cana-1834 50 14 : : PUNCT cana-1834 50 15 first first ADV cana-1834 50 16 we we PRON cana-1834 50 17 define define VERB cana-1834 50 18 an an DET cana-1834 50 19 operation operation NOUN cana-1834 50 20 οƒ… οƒ… ADJ cana-1834 50 21 on on ADP cana-1834 50 22 r r PROPN cana-1834 50 23 i i PRON cana-1834 50 24 by by ADP cana-1834 50 25 ( ( PUNCT cana-1834 50 26 p p NOUN cana-1834 50 27 ) ) PUNCT cana-1834 50 28 ( ( PUNCT cana-1834 50 29 q q NOUN cana-1834 50 30 ) ) PUNCT cana-1834 50 31 ( ( PUNCT cana-1834 50 32 r r NOUN cana-1834 50 33 ) ) PUNCT cana-1834 50 34 i i PRON cana-1834 51 1 i i PRON cana-1834 51 2 irp irp VERB cana-1834 51 3 qi+ qi+ ADV cana-1834 52 1 + + X cana-1834 52 2 + + CCONJ cana-1834 52 3 = = ADJ cana-1834 52 4 οƒ… οƒ… ADJ cana-1834 52 5 οƒ… οƒ… ADJ cana-1834 52 6 οƒ… οƒ… PROPN cana-1834 53 1 + + NOUN cana-1834 53 2 οƒ… οƒ… ADJ cana-1834 53 3 for for ADP cana-1834 53 4 all all DET cana-1834 53 5 p p NOUN cana-1834 53 6 i+ i+ NOUN cana-1834 53 7 , , PUNCT cana-1834 53 8 q q X cana-1834 53 9 i+ i+ NOUN cana-1834 53 10 , , PUNCT cana-1834 53 11 & & CCONJ cana-1834 53 12 r r PROPN cana-1834 53 13 i+ i+ NUM cana-1834 53 14 οƒŽ οƒŽ NOUN cana-1834 53 15 r r NOUN cana-1834 53 16 i i PROPN cana-1834 53 17 .it .it PUNCT cana-1834 53 18 is be AUX cana-1834 53 19 easy easy ADJ cana-1834 53 20 to to PART cana-1834 53 21 see see VERB cana-1834 53 22 that that DET cana-1834 53 23 οƒ… οƒ… PROPN cana-1834 53 24 and and CCONJ cana-1834 53 25 * * NOUN cana-1834 53 26 are be AUX cana-1834 53 27 well well ADV cana-1834 53 28 defined define VERB cana-1834 53 29 . . PUNCT cana-1834 54 1 consequently consequently ADV cana-1834 54 2 we we PRON cana-1834 54 3 can can AUX cana-1834 54 4 verify verify VERB cana-1834 54 5 that that SCONJ cana-1834 54 6 ( ( PUNCT cana-1834 54 7 r r NOUN cana-1834 54 8 i i PRON cana-1834 54 9 , , PUNCT cana-1834 54 10 οƒ… οƒ… ADJ cana-1834 54 11 , , PUNCT cana-1834 54 12 [ [ PUNCT cana-1834 54 13 ] ] X cana-1834 54 14 ) ) PUNCT cana-1834 54 15 is be AUX cana-1834 54 16 a a DET cana-1834 54 17 commutative commutative ADJ cana-1834 54 18 semigroup semigroup NOUN cana-1834 54 19 and and CCONJ cana-1834 54 20 we we PRON cana-1834 54 21 have have VERB cana-1834 54 22 the the DET cana-1834 54 23 following follow VERB cana-1834 54 24 equalities equality NOUN cana-1834 54 25 . . PUNCT cana-1834 55 1 left leave VERB cana-1834 55 2 distributive distributive ADJ cana-1834 55 3 communications communication NOUN cana-1834 55 4 on on ADP cana-1834 55 5 applied apply VERB cana-1834 55 6 nonlinear nonlinear ADJ cana-1834 55 7 analysis analysis NOUN cana-1834 55 8 issn issn NOUN cana-1834 55 9 : : PUNCT cana-1834 55 10 1074 1074 NUM cana-1834 55 11 - - PUNCT cana-1834 55 12 133x 133x NUM cana-1834 55 13 vol vol NOUN cana-1834 55 14 32 32 NUM cana-1834 55 15 no no NOUN cana-1834 55 16 . . NOUN cana-1834 55 17 2 2 NUM cana-1834 55 18 ( ( PUNCT cana-1834 55 19 2025 2025 NUM cana-1834 55 20 ) ) PUNCT cana-1834 56 1 583 583 NUM cana-1834 56 2 https://internationalpubls.com https://internationalpubls.com X cana-1834 57 1 [ [ X cana-1834 57 2 ( ( PUNCT cana-1834 57 3 ) ) PUNCT cana-1834 57 4 ( ( PUNCT cana-1834 57 5 ) ) PUNCT cana-1834 57 6 ( ( PUNCT cana-1834 57 7 ( ( PUNCT cana-1834 57 8 ) ) PUNCT cana-1834 57 9 ( ( PUNCT cana-1834 57 10 ) ) PUNCT cana-1834 57 11 ) ) PUNCT cana-1834 58 1 [ [ X cana-1834 58 2 ( ( PUNCT cana-1834 58 3 ) ) PUNCT cana-1834 58 4 ( ( PUNCT cana-1834 58 5 ) ) PUNCT cana-1834 58 6 ( ( PUNCT cana-1834 58 7 ) ) PUNCT cana-1834 58 8 ] ] PUNCT cana-1834 59 1 [ [ X cana-1834 59 2 p p X cana-1834 59 3 q q X cana-1834 59 4 ( ( PUNCT cana-1834 59 5 r r NOUN cana-1834 59 6 s s PROPN cana-1834 59 7 ) ) PUNCT cana-1834 59 8 ] ] PUNCT cana-1834 60 1 i i PRON cana-1834 60 2 [ [ X cana-1834 60 3 ( ( PUNCT cana-1834 60 4 p p X cana-1834 60 5 q q X cana-1834 60 6 r r NOUN cana-1834 60 7 ) ) PUNCT cana-1834 60 8 ( ( PUNCT cana-1834 60 9 ( ( PUNCT cana-1834 60 10 p p X cana-1834 60 11 q q NOUN cana-1834 60 12 ) ) PUNCT cana-1834 60 13 ] ] PUNCT cana-1834 61 1 [ [ X cana-1834 61 2 [ [ X cana-1834 61 3 p p X cana-1834 61 4 q q X cana-1834 61 5 r r X cana-1834 61 6 ] ] X cana-1834 61 7 i i PRON cana-1834 61 8 ] ] PUNCT cana-1834 62 1 [ [ X cana-1834 62 2 [ [ X cana-1834 62 3 p p X cana-1834 62 4 q q X cana-1834 62 5 ] ] PUNCT cana-1834 62 6 ] ] PUNCT cana-1834 63 1 [ [ X cana-1834 63 2 ( ( PUNCT cana-1834 63 3 ) ) PUNCT cana-1834 63 4 ( ( PUNCT cana-1834 63 5 ) ) PUNCT cana-1834 63 6 ( ( PUNCT cana-1834 63 7 ) ) PUNCT cana-1834 63 8 ] ] PUNCT cana-1834 64 1 [ [ X cana-1834 64 2 ( ( PUNCT cana-1834 64 3 ) ) PUNCT cana-1834 64 4 ( ( PUNCT cana-1834 64 5 ) ) PUNCT cana-1834 64 6 ( ( PUNCT cana-1834 64 7 ) ) PUNCT cana-1834 64 8 ] ] PUNCT cana-1834 65 1 p p X cana-1834 66 1 i i PRON cana-1834 66 2 q q NOUN cana-1834 67 1 i i PRON cana-1834 67 2 r r VERB cana-1834 68 1 i i PRON cana-1834 68 2 s s VERB cana-1834 69 1 i i PRON cana-1834 69 2 p p NOUN cana-1834 70 1 i i PRON cana-1834 70 2 q q NOUN cana-1834 71 1 i i PRON cana-1834 71 2 r r NOUN cana-1834 71 3 s s VERB cana-1834 72 1 i i PRON cana-1834 72 2 s s VERB cana-1834 72 3 i i PRON cana-1834 72 4 s s VERB cana-1834 73 1 i i PRON cana-1834 73 2 p p NOUN cana-1834 74 1 i i PRON cana-1834 74 2 q q NOUN cana-1834 75 1 i i PRON cana-1834 75 2 r r VERB cana-1834 76 1 i i PRON cana-1834 76 2 p p X cana-1834 77 1 i i PRON cana-1834 77 2 q q NOUN cana-1834 78 1 i i PRON cana-1834 78 2 s s VERB cana-1834 78 3 i i PRON cana-1834 78 4   PRON cana-1834 78 5   PROPN cana-1834 78 6   NOUN cana-1834 78 7   PROPN cana-1834 78 8   NOUN cana-1834 78 9   PROPN cana-1834 78 10   NOUN cana-1834 78 11   PROPN cana-1834 78 12   NOUN cana-1834 78 13   PROPN cana-1834 78 14   NOUN cana-1834 78 15   PROPN cana-1834 78 16   NOUN cana-1834 78 17   PROPN cana-1834 78 18   NOUN cana-1834 78 19   PROPN cana-1834 78 20   NOUN cana-1834 78 21   PROPN cana-1834 78 22 + + CCONJ cana-1834 78 23 + + CCONJ cana-1834 78 24 + + CCONJ cana-1834 78 25 οƒ… οƒ… ADJ cana-1834 78 26 + + X cana-1834 78 27 = = PUNCT cana-1834 79 1 + + PUNCT cana-1834 79 2 + + PUNCT cana-1834 79 3 + + PUNCT cana-1834 79 4 + + PUNCT cana-1834 79 5 = = SYM cana-1834 80 1 + + PUNCT cana-1834 80 2 + + CCONJ cana-1834 80 3 = = NOUN cana-1834 80 4 οƒ… οƒ… ADJ cana-1834 80 5 + + CCONJ cana-1834 80 6 = = PUNCT cana-1834 81 1 + + NUM cana-1834 81 2 οƒ… οƒ… ADJ cana-1834 81 3 + + X cana-1834 81 4 = = PUNCT cana-1834 82 1 + + PUNCT cana-1834 82 2 + + CCONJ cana-1834 82 3 + + CCONJ cana-1834 82 4 οƒ… οƒ… ADJ cana-1834 82 5 + + CCONJ cana-1834 82 6 + + CCONJ cana-1834 82 7 + + CCONJ cana-1834 82 8 lateral lateral ADJ cana-1834 82 9 distributive distributive ADJ cana-1834 82 10 ( ( PUNCT cana-1834 82 11 ) ) PUNCT cana-1834 83 1 [ [ X cana-1834 83 2 ( ( PUNCT cana-1834 83 3 ) ) PUNCT cana-1834 83 4 ( ( PUNCT cana-1834 83 5 ) ) PUNCT cana-1834 83 6 ] ] PUNCT cana-1834 83 7 ( ( PUNCT cana-1834 83 8 q q NOUN cana-1834 83 9 ) ) PUNCT cana-1834 84 1 [ [ X cana-1834 84 2 ( ( PUNCT cana-1834 84 3 ) ) PUNCT cana-1834 84 4 ( ( PUNCT cana-1834 84 5 ) ) PUNCT cana-1834 84 6 ] ] PUNCT cana-1834 84 7 ( ( PUNCT cana-1834 84 8 ) ) PUNCT cana-1834 84 9 ] ] PUNCT cana-1834 85 1 [ [ X cana-1834 85 2 p p X cana-1834 85 3 ( ( PUNCT cana-1834 85 4 r r NOUN cana-1834 85 5 s s PROPN cana-1834 85 6 ) ) PUNCT cana-1834 85 7 i i NOUN cana-1834 85 8 ] ] X cana-1834 85 9 ( ( PUNCT cana-1834 85 10 q q NOUN cana-1834 85 11 i i NOUN cana-1834 85 12 ) ) PUNCT cana-1834 85 13 [ [ PUNCT cana-1834 85 14 ( ( PUNCT cana-1834 85 15 ) ) PUNCT cana-1834 85 16 ( ( PUNCT cana-1834 85 17 ) ) PUNCT cana-1834 85 18 ] ] PUNCT cana-1834 85 19 [ [ PUNCT cana-1834 85 20 ( ( PUNCT cana-1834 85 21 ) ) PUNCT cana-1834 85 22 q q NOUN cana-1834 85 23 ] ] PUNCT cana-1834 86 1 [ [ X cana-1834 86 2 [ [ X cana-1834 86 3 ] ] X cana-1834 86 4 [ [ X cana-1834 86 5 ] ] X cana-1834 86 6 ] ] PUNCT cana-1834 87 1 [ [ X cana-1834 87 2 [ [ X cana-1834 87 3 ] ] X cana-1834 87 4 i i PRON cana-1834 87 5 ] ] PUNCT cana-1834 88 1 [ [ X cana-1834 88 2 [ [ X cana-1834 88 3 ] ] X cana-1834 88 4 ] ] PUNCT cana-1834 89 1 [ [ X cana-1834 89 2 ( ( PUNCT cana-1834 89 3 ) ) PUNCT cana-1834 89 4 ( ( PUNCT cana-1834 89 5 ) ) PUNCT cana-1834 89 6 ( ( PUNCT cana-1834 89 7 ) ) PUNCT cana-1834 89 8 ] ] PUNCT cana-1834 90 1 [ [ X cana-1834 90 2 ( ( PUNCT cana-1834 90 3 ) ) PUNCT cana-1834 90 4 ( ( PUNCT cana-1834 90 5 s s NOUN cana-1834 90 6 ) ) PUNCT cana-1834 90 7 ( ( PUNCT cana-1834 90 8 ) ) PUNCT cana-1834 90 9 ] ] PUNCT cana-1834 91 1 p p X cana-1834 92 1 i i PRON cana-1834 92 2 r r VERB cana-1834 93 1 i i PRON cana-1834 93 2 s s VERB cana-1834 93 3 i i PRON cana-1834 94 1 i i PRON cana-1834 94 2 p p X cana-1834 95 1 i i PRON cana-1834 95 2 r r NOUN cana-1834 95 3 s s VERB cana-1834 96 1 i i INTJ cana-1834 96 2 q q NOUN cana-1834 97 1 i i PRON cana-1834 97 2 p p NOUN cana-1834 97 3 r r NOUN cana-1834 97 4 s s VERB cana-1834 97 5 q q NOUN cana-1834 98 1 i i PRON cana-1834 98 2 p p NOUN cana-1834 98 3 r r NOUN cana-1834 98 4 s s VERB cana-1834 99 1 i i NOUN cana-1834 99 2 p p NOUN cana-1834 99 3 r r NOUN cana-1834 99 4 q q X cana-1834 99 5 p p X cana-1834 99 6 s s X cana-1834 99 7 q q NOUN cana-1834 100 1 i i X cana-1834 100 2 p p NOUN cana-1834 100 3 r r NOUN cana-1834 100 4 q q X cana-1834 100 5 p p X cana-1834 100 6 s s X cana-1834 100 7 q q NOUN cana-1834 101 1 i i PRON cana-1834 101 2 p p X cana-1834 102 1 i i PRON cana-1834 102 2 r r VERB cana-1834 103 1 i i PRON cana-1834 103 2 q q NOUN cana-1834 104 1 i i PRON cana-1834 104 2 p p X cana-1834 105 1 i i PRON cana-1834 105 2 i i PRON cana-1834 105 3 q q VERB cana-1834 106 1 i i PRON cana-1834 106 2   NUM cana-1834 106 3   PROPN cana-1834 106 4   NOUN cana-1834 106 5   PROPN cana-1834 106 6   NOUN cana-1834 106 7   PROPN cana-1834 106 8   NOUN cana-1834 106 9   PROPN cana-1834 106 10   NOUN cana-1834 106 11   PROPN cana-1834 106 12   NOUN cana-1834 106 13   PROPN cana-1834 106 14   NOUN cana-1834 106 15   PROPN cana-1834 106 16   NOUN cana-1834 106 17   PROPN cana-1834 106 18   NOUN cana-1834 106 19   PROPN cana-1834 106 20   NOUN cana-1834 106 21   PROPN cana-1834 106 22   NOUN cana-1834 106 23   PROPN cana-1834 106 24 + + CCONJ cana-1834 106 25 + + CCONJ cana-1834 106 26 οƒ… οƒ… ADJ cana-1834 106 27 + + X cana-1834 106 28 + + PUNCT cana-1834 106 29 = = X cana-1834 107 1 + + PUNCT cana-1834 107 2 + + PUNCT cana-1834 107 3 + + PUNCT cana-1834 107 4 + + PUNCT cana-1834 107 5 = = X cana-1834 108 1 + + PUNCT cana-1834 108 2 + + PUNCT cana-1834 108 3 + + PUNCT cana-1834 108 4 = = SYM cana-1834 108 5 + + PUNCT cana-1834 108 6 + + CCONJ cana-1834 108 7 = = SYM cana-1834 109 1 + + PUNCT cana-1834 109 2 + + CCONJ cana-1834 109 3 = = NOUN cana-1834 109 4 οƒ… οƒ… ADJ cana-1834 109 5 + + CCONJ cana-1834 109 6 = = PUNCT cana-1834 110 1 + + NUM cana-1834 110 2 οƒ… οƒ… ADJ cana-1834 110 3 + + X cana-1834 110 4 = = PUNCT cana-1834 111 1 + + PUNCT cana-1834 111 2 + + CCONJ cana-1834 111 3 + + CCONJ cana-1834 111 4 οƒ… οƒ… ADJ cana-1834 111 5 + + X cana-1834 111 6 + + CCONJ cana-1834 112 1 + + CCONJ cana-1834 112 2 right right ADJ cana-1834 112 3 distributive distributive ADJ cana-1834 112 4 [ [ X cana-1834 112 5 ( ( PUNCT cana-1834 112 6 ) ) PUNCT cana-1834 112 7 ( ( PUNCT cana-1834 112 8 ) ) PUNCT cana-1834 112 9 ] ] PUNCT cana-1834 112 10 ( ( PUNCT cana-1834 112 11 ) ) PUNCT cana-1834 112 12 ( ( PUNCT cana-1834 112 13 ) ) PUNCT cana-1834 112 14 ) ) PUNCT cana-1834 113 1 [ [ X cana-1834 113 2 ( ( PUNCT cana-1834 113 3 ) ) PUNCT cana-1834 113 4 ] ] PUNCT cana-1834 113 5 ( ( PUNCT cana-1834 113 6 ) ) PUNCT cana-1834 113 7 ( ( PUNCT cana-1834 113 8 ) ) PUNCT cana-1834 114 1 [ [ X cana-1834 114 2 ( ( PUNCT cana-1834 114 3 ) ) PUNCT cana-1834 114 4 ] ] PUNCT cana-1834 115 1 [ [ X cana-1834 115 2 ( ( PUNCT cana-1834 115 3 ) ) PUNCT cana-1834 115 4 ( ( PUNCT cana-1834 115 5 ) ) PUNCT cana-1834 115 6 ] ] PUNCT cana-1834 116 1 [ [ X cana-1834 116 2 [ [ X cana-1834 116 3 ] ] X cana-1834 116 4 i i PRON cana-1834 116 5 ] ] X cana-1834 117 1 [ [ X cana-1834 117 2 [ [ X cana-1834 117 3 ] ] X cana-1834 117 4 i i PRON cana-1834 117 5 ] ] X cana-1834 118 1 [ [ X cana-1834 118 2 ( ( PUNCT cana-1834 118 3 ) ) PUNCT cana-1834 118 4 ( ( PUNCT cana-1834 118 5 ) ) PUNCT cana-1834 118 6 ( ( PUNCT cana-1834 118 7 ) ) PUNCT cana-1834 118 8 ] ] PUNCT cana-1834 119 1 [ [ X cana-1834 119 2 ( ( PUNCT cana-1834 119 3 ) ) PUNCT cana-1834 119 4 ( ( PUNCT cana-1834 119 5 ) ) PUNCT cana-1834 119 6 ( ( PUNCT cana-1834 119 7 ) ) PUNCT cana-1834 119 8 ] ] X cana-1834 120 1 r r NOUN cana-1834 121 1 i i PRON cana-1834 121 2 s s VERB cana-1834 122 1 i i PRON cana-1834 122 2 p p NOUN cana-1834 123 1 i i PRON cana-1834 123 2 q q NOUN cana-1834 124 1 i i PRON cana-1834 124 2 r r NOUN cana-1834 124 3 s s VERB cana-1834 125 1 i i PRON cana-1834 125 2 p p NOUN cana-1834 126 1 i i PRON cana-1834 126 2 q q NOUN cana-1834 127 1 i i PRON cana-1834 127 2 r r NOUN cana-1834 127 3 s s NOUN cana-1834 127 4 p p X cana-1834 127 5 q q X cana-1834 128 1 i i NOUN cana-1834 128 2 r r NOUN cana-1834 128 3 p p NOUN cana-1834 128 4 q q PROPN cana-1834 128 5 s s X cana-1834 128 6 p p X cana-1834 128 7 q q X cana-1834 129 1 i i NOUN cana-1834 129 2 r r NOUN cana-1834 129 3 p p NOUN cana-1834 129 4 q q PROPN cana-1834 129 5 s s X cana-1834 129 6 p p NOUN cana-1834 129 7 q q NOUN cana-1834 129 8 r r NOUN cana-1834 130 1 i i PRON cana-1834 130 2 p p X cana-1834 131 1 i i PRON cana-1834 131 2 q q NOUN cana-1834 132 1 i i PRON cana-1834 132 2 s s VERB cana-1834 133 1 i i PRON cana-1834 133 2 p p NOUN cana-1834 134 1 i i PRON cana-1834 134 2 q q PROPN cana-1834 134 3 i i PRON cana-1834 134 4   NUM cana-1834 134 5   PROPN cana-1834 134 6   NOUN cana-1834 134 7   PROPN cana-1834 134 8   NOUN cana-1834 134 9   PROPN cana-1834 134 10   NOUN cana-1834 134 11   PROPN cana-1834 134 12   NOUN cana-1834 134 13   PROPN cana-1834 134 14   NOUN cana-1834 134 15   PROPN cana-1834 134 16   NOUN cana-1834 134 17   PROPN cana-1834 134 18   NOUN cana-1834 134 19   PROPN cana-1834 134 20   NOUN cana-1834 134 21   PROPN cana-1834 134 22 + + NUM cana-1834 134 23 οƒ… οƒ… ADJ cana-1834 134 24 + + X cana-1834 134 25 + + PUNCT cana-1834 134 26 + + PUNCT cana-1834 134 27 = = X cana-1834 135 1 + + PUNCT cana-1834 135 2 + + PUNCT cana-1834 135 3 + + PUNCT cana-1834 135 4 + + PUNCT cana-1834 135 5 = = SYM cana-1834 136 1 + + PUNCT cana-1834 136 2 + + CCONJ cana-1834 136 3 = = NOUN cana-1834 136 4 οƒ… οƒ… ADJ cana-1834 136 5 + + CCONJ cana-1834 136 6 = = PUNCT cana-1834 137 1 + + NUM cana-1834 137 2 οƒ… οƒ… ADJ cana-1834 137 3 + + X cana-1834 137 4 = = PUNCT cana-1834 138 1 + + PUNCT cana-1834 138 2 + + CCONJ cana-1834 138 3 + + CCONJ cana-1834 138 4 οƒ… οƒ… ADJ cana-1834 138 5 + + X cana-1834 139 1 + + PUNCT cana-1834 139 2 + + CCONJ cana-1834 139 3 these these PRON cana-1834 139 4 shows show VERB cana-1834 139 5 that that PRON cana-1834 139 6 is be AUX cana-1834 139 7 a a DET cana-1834 139 8 r r NOUN cana-1834 139 9 i i PRON cana-1834 139 10 ternary ternary VERB cana-1834 139 11 𝛀 𝛀 PROPN cana-1834 139 12 βˆ’semiring βˆ’semire VERB cana-1834 139 13 . . PUNCT cana-1834 140 1 theorem theorem PROPN cana-1834 140 2 3.4(correspondance 3.4(correspondance NOUN cana-1834 140 3 theorem theorem NOUN cana-1834 140 4 ): ): PUNCT cana-1834 140 5 let let VERB cana-1834 140 6 r r PRON cana-1834 140 7 be be AUX cana-1834 140 8 a a DET cana-1834 140 9 ternary𝛀 ternary𝛀 PROPN cana-1834 140 10 βˆ’semiring βˆ’semire VERB cana-1834 140 11 with with ADP cana-1834 140 12 zero zero NUM cana-1834 140 13 and and CCONJ cana-1834 140 14 j j PROPN cana-1834 140 15 an an DET cana-1834 140 16 ideal ideal NOUN cana-1834 140 17 of of ADP cana-1834 140 18 r r NOUN cana-1834 140 19 such such ADJ cana-1834 140 20 that that SCONJ cana-1834 140 21 i i PRON cana-1834 140 22 j j VERB cana-1834 141 1 then then ADV cana-1834 141 2 j j PROPN cana-1834 141 3 i i PRON cana-1834 141 4 is be AUX cana-1834 141 5 an an DET cana-1834 141 6 ideal ideal NOUN cana-1834 141 7 of of ADP cana-1834 141 8   ADJ cana-1834 141 9   PROPN cana-1834 141 10 , , PUNCT cana-1834 141 11 , , PUNCT cana-1834 141 12 Β Β  Β Β  SPACE cana-1834 141 13 ( ( PUNCT cana-1834 141 14 ) ) PUNCT cana-1834 141 15 r r NOUN cana-1834 141 16 i i PRON cana-1834 141 17 οƒ… οƒ… VERB cana-1834 141 18 .conversely .conversely ADV cana-1834 141 19 , , PUNCT cana-1834 141 20 if if SCONJ cana-1834 141 21 k k PROPN cana-1834 141 22 is be AUX cana-1834 141 23 an an DET cana-1834 141 24 ideal ideal NOUN cana-1834 141 25 of of ADP cana-1834 141 26   ADJ cana-1834 141 27   PROPN cana-1834 141 28 , , PUNCT cana-1834 141 29 , , PUNCT cana-1834 141 30 Β Β  Β Β  SPACE cana-1834 141 31 ( ( PUNCT cana-1834 141 32 ) ) PUNCT cana-1834 142 1 r r NOUN cana-1834 142 2 i i PRON cana-1834 142 3 οƒ… οƒ… VERB cana-1834 142 4 then then ADV cana-1834 142 5 there there PRON cana-1834 142 6 exist exist VERB cana-1834 142 7 an an DET cana-1834 142 8 ideal ideal ADJ cana-1834 142 9 j j PROPN cana-1834 142 10 of of ADP cana-1834 142 11 ( ( PUNCT cana-1834 142 12 , , PUNCT cana-1834 142 13 , , PUNCT cana-1834 142 14 [ [ PUNCT cana-1834 142 15 ] ] X cana-1834 142 16 ) ) PUNCT cana-1834 142 17 r r NOUN cana-1834 142 18   VERB cana-1834 142 19 such such ADJ cana-1834 142 20 that that SCONJ cana-1834 142 21 i i PRON cana-1834 142 22 j j VERB cana-1834 143 1 and and CCONJ cana-1834 144 1 j j PROPN cana-1834 144 2 k k PROPN cana-1834 144 3 i i PROPN cana-1834 144 4 = = PUNCT cana-1834 144 5 . . PUNCT cana-1834 145 1 proof proof NOUN cana-1834 145 2 : : PUNCT cana-1834 145 3 the the DET cana-1834 145 4 proof proof NOUN cana-1834 145 5 of of ADP cana-1834 145 6 this this DET cana-1834 145 7 theorem theorem NOUN cana-1834 145 8 is be AUX cana-1834 145 9 similar similar ADJ cana-1834 145 10 to to ADP cana-1834 145 11 the the DET cana-1834 145 12 above above ADJ cana-1834 145 13 . . PUNCT cana-1834 146 1 4.commutative 4.commutative NUM cana-1834 146 2 ternary ternary ADJ cana-1834 146 3 𝛀 𝛀 PROPN cana-1834 146 4 βˆ’semirings βˆ’semiring NOUN cana-1834 146 5 and and CCONJ cana-1834 146 6 congruence congruence NOUN cana-1834 146 7 ’s ’s X cana-1834 146 8 on on ADP cana-1834 146 9 a a DET cana-1834 146 10 ternary ternary ADJ cana-1834 146 11 𝛀 𝛀 PROPN cana-1834 146 12 βˆ’semirings βˆ’semiring NOUN cana-1834 146 13 in in ADP cana-1834 146 14 this this DET cana-1834 146 15 entire entire ADJ cana-1834 146 16 section section NOUN cana-1834 146 17 r r NOUN cana-1834 146 18 is be AUX cana-1834 146 19 a a DET cana-1834 146 20 commutative commutative ADJ cana-1834 146 21 ternary ternary NOUN cana-1834 146 22 𝛀 𝛀 PROPN cana-1834 146 23 βˆ’semiring.the βˆ’semiring.the DET cana-1834 146 24 following follow VERB cana-1834 146 25 theorems theorem NOUN cana-1834 146 26 are be AUX cana-1834 146 27 well well ADV cana-1834 146 28 known know VERB cana-1834 146 29 . . PUNCT cana-1834 147 1 theorem theorem VERB cana-1834 147 2 4.1 4.1 NUM cana-1834 147 3 : : PUNCT cana-1834 147 4 the the DET cana-1834 147 5 following follow VERB cana-1834 147 6 conditions condition NOUN cana-1834 147 7 on on ADP cana-1834 147 8 an an DET cana-1834 147 9 ideal ideal ADJ cana-1834 147 10 i i PRON cana-1834 147 11 of of ADP cana-1834 147 12 a a DET cana-1834 147 13 commutative commutative ADJ cana-1834 147 14 ternary ternary NOUN cana-1834 147 15 𝛀 𝛀 PROPN cana-1834 147 16 βˆ’semiring βˆ’semire VERB cana-1834 147 17 r r NOUN cana-1834 147 18 with with ADP cana-1834 147 19 zero zero NUM cana-1834 147 20 are be AUX cana-1834 147 21 equivalent equivalent ADJ cana-1834 147 22 ( ( PUNCT cana-1834 147 23 1 1 NUM cana-1834 147 24 ) ) PUNCT cana-1834 147 25 ( ( PUNCT cana-1834 148 1 0 0 NUM cana-1834 148 2 : : PUNCT cana-1834 148 3 i i NOUN cana-1834 148 4 ) ) PUNCT cana-1834 148 5 ( ( PUNCT cana-1834 148 6 : : PUNCT cana-1834 148 7 i)h i)h NOUN cana-1834 148 8 h h NOUN cana-1834 148 9 iοƒ… iοƒ… ADJ cana-1834 148 10 = = PUNCT cana-1834 148 11   VERB cana-1834 148 12 for for ADP cana-1834 148 13 all all DET cana-1834 148 14 ideals ideal NOUN cana-1834 148 15 h h NOUN cana-1834 148 16 of of ADP cana-1834 148 17 ( ( PUNCT cana-1834 148 18 , , PUNCT cana-1834 148 19 , , PUNCT cana-1834 148 20 [ [ PUNCT cana-1834 148 21 ] ] X cana-1834 148 22 ) ) PUNCT cana-1834 148 23 r r NOUN cana-1834 148 24   NOUN cana-1834 148 25 communications communication NOUN cana-1834 148 26 on on ADP cana-1834 148 27 applied apply VERB cana-1834 148 28 nonlinear nonlinear ADJ cana-1834 148 29 analysis analysis NOUN cana-1834 148 30 issn issn NOUN cana-1834 148 31 : : PUNCT cana-1834 148 32 1074 1074 NUM cana-1834 148 33 - - PUNCT cana-1834 148 34 133x 133x NUM cana-1834 148 35 vol vol NOUN cana-1834 148 36 32 32 NUM cana-1834 148 37 no no NOUN cana-1834 148 38 . . NOUN cana-1834 148 39 2 2 NUM cana-1834 148 40 ( ( PUNCT cana-1834 148 41 2025 2025 NUM cana-1834 148 42 ) ) PUNCT cana-1834 148 43 584 584 NUM cana-1834 148 44 https://internationalpubls.com https://internationalpubls.com X cana-1834 148 45 ( ( PUNCT cana-1834 148 46 2 2 NUM cana-1834 148 47 ) ) PUNCT cana-1834 148 48 h h NOUN cana-1834 149 1 i i NOUN cana-1834 149 2 k k NOUN cana-1834 150 1 i i PROPN cana-1834 150 2 = = PUNCT cana-1834 150 3   PROPN cana-1834 150 4 implies imply VERB cana-1834 150 5 that that SCONJ cana-1834 150 6 ( ( PUNCT cana-1834 150 7 0 0 NUM cana-1834 150 8 : : PUNCT cana-1834 150 9 ) ) PUNCT cana-1834 150 10 ( ( PUNCT cana-1834 150 11 0 0 NUM cana-1834 150 12 : : PUNCT cana-1834 150 13 ) ) PUNCT cana-1834 151 1 i i PRON cana-1834 151 2 h h VERB cana-1834 152 1 i i PRON cana-1834 152 2 k+ k+ VERB cana-1834 152 3 = = PUNCT cana-1834 153 1 + + CCONJ cana-1834 153 2 for for ADP cana-1834 153 3 all all DET cana-1834 153 4 ideals ideal NOUN cana-1834 153 5 h h NOUN cana-1834 153 6 and and CCONJ cana-1834 153 7 k k PROPN cana-1834 153 8 of of ADP cana-1834 153 9 ( ( PUNCT cana-1834 153 10 , , PUNCT cana-1834 153 11 , , PUNCT cana-1834 153 12 [ [ PUNCT cana-1834 153 13 ] ] X cana-1834 153 14 ) ) PUNCT cana-1834 153 15 r r NOUN cana-1834 153 16   VERB cana-1834 153 17 where where SCONJ cana-1834 153 18 ( ( PUNCT cana-1834 153 19 : : PUNCT cana-1834 153 20 ) ) PUNCT cana-1834 153 21 { { PUNCT cana-1834 153 22 p p NOUN cana-1834 153 23 / / SYM cana-1834 153 24 , , PUNCT cana-1834 153 25 , , PUNCT cana-1834 153 26 } } PUNCT cana-1834 153 27 i i PRON cana-1834 153 28 a a DET cana-1834 153 29 r r NOUN cana-1834 153 30 p p NOUN cana-1834 153 31 a a DET cana-1834 153 32 b b NOUN cana-1834 153 33 i i PRON cana-1834 153 34 a a PRON cana-1834 153 35 b b X cana-1834 153 36 aand aand X cana-1834 153 37   NUM cana-1834 153 38 = = PROPN cana-1834 153 39 οƒŽ οƒŽ PROPN cana-1834 153 40 οƒŽ οƒŽ PROPN cana-1834 153 41 ο€’ ο€’ VERB cana-1834 153 42 οƒŽ οƒŽ NOUN cana-1834 153 43 οƒŽο‡ οƒŽο‡ NOUN cana-1834 153 44 for for SCONJ cana-1834 153 45 any any DET cana-1834 153 46 .a .a ADJ cana-1834 153 47 r r VERB cana-1834 153 48   PROPN cana-1834 153 49 theorem theorem VERB cana-1834 153 50 4.2 4.2 NUM cana-1834 153 51 : : PUNCT cana-1834 153 52 assume assume VERB cana-1834 153 53 that that SCONJ cana-1834 153 54 r r NOUN cana-1834 153 55 be be AUX cana-1834 153 56 a a DET cana-1834 153 57 commutative commutative ADJ cana-1834 153 58 ternary ternary NOUN cana-1834 153 59 𝛀 𝛀 PROPN cana-1834 154 1 βˆ’semiring.if βˆ’semiring.if ADV cana-1834 154 2 i i PRON cana-1834 154 3 is be AUX cana-1834 154 4 an an DET cana-1834 154 5 ideal ideal NOUN cana-1834 154 6 of of ADP cana-1834 154 7 ( ( PUNCT cana-1834 154 8 , , PUNCT cana-1834 154 9 , , PUNCT cana-1834 154 10 [ [ PUNCT cana-1834 154 11 ] ] X cana-1834 154 12 ) ) PUNCT cana-1834 154 13 r r NOUN cana-1834 154 14   VERB cana-1834 154 15 , , PUNCT cana-1834 154 16 .a .a ADJ cana-1834 154 17 r r VERB cana-1834 154 18   PROPN cana-1834 154 19 and and CCONJ cana-1834 154 20   NUM cana-1834 154 21 οƒŽο‡ οƒŽο‡ NOUN cana-1834 154 22 then then ADV cana-1834 154 23 the the DET cana-1834 154 24 following following ADJ cana-1834 154 25 statements statement NOUN cana-1834 154 26 are be AUX cana-1834 154 27 holds hold VERB cana-1834 154 28 good good ADJ cana-1834 154 29 : : PUNCT cana-1834 154 30 ( ( PUNCT cana-1834 154 31 1 1 X cana-1834 154 32 ) ) PUNCT cana-1834 154 33 ( ( PUNCT cana-1834 154 34 : : PUNCT cana-1834 154 35 ) ) PUNCT cana-1834 154 36 ( ( PUNCT cana-1834 154 37 : : PUNCT cana-1834 154 38 ) ) PUNCT cana-1834 154 39 ( ( PUNCT cana-1834 154 40 : : PUNCT cana-1834 154 41 ) ) PUNCT cana-1834 154 42 , , PUNCT cana-1834 154 43 i i PRON cana-1834 154 44 i i PRON cana-1834 154 45 a a VERB cana-1834 154 46 i i PRON cana-1834 154 47 a a PRON cana-1834 154 48 a a DET cana-1834 154 49 a a DET cana-1834 154 50 i i PRON cana-1834 154 51 a a DET cana-1834 154 52 a a DET cana-1834 154 53 a a NOUN cana-1834 154 54   NUM cana-1834 154 55   NUM cana-1834 154 56   NUM cana-1834 154 57 ο§οƒŒ ο§οƒŒ PROPN cana-1834 154 58   PROPN cana-1834 154 59   PROPN cana-1834 154 60 ο€’ ο€’ NOUN cana-1834 154 61 οƒŽο‡ οƒŽο‡ NOUN cana-1834 154 62 ( ( PUNCT cana-1834 154 63 2 2 NUM cana-1834 154 64 ) ) PUNCT cana-1834 154 65 , , PUNCT cana-1834 154 66 ( ( PUNCT cana-1834 154 67 : : PUNCT cana-1834 154 68 ) ) PUNCT cana-1834 154 69 .if .if PUNCT cana-1834 155 1 a a DET cana-1834 155 2 i i PRON cana-1834 155 3 then then ADV cana-1834 155 4 i i PRON cana-1834 155 5 a a DET cana-1834 155 6 r r NOUN cana-1834 155 7 = = PUNCT cana-1834 155 8 theorem theorem VERB cana-1834 155 9 4.3 4.3 NUM cana-1834 155 10 : : PUNCT cana-1834 155 11 assume assume VERB cana-1834 155 12 that that SCONJ cana-1834 155 13 r r NOUN cana-1834 155 14 be be AUX cana-1834 155 15 a a DET cana-1834 155 16 commutative commutative ADJ cana-1834 155 17 ternary ternary NOUN cana-1834 155 18 𝛀 𝛀 PROPN cana-1834 156 1 βˆ’semiring.if βˆ’semiring.if ADV cana-1834 156 2 i i PRON cana-1834 156 3 is be AUX cana-1834 156 4 an an DET cana-1834 156 5 ideal ideal NOUN cana-1834 156 6 of of ADP cana-1834 156 7 ( ( PUNCT cana-1834 156 8 , , PUNCT cana-1834 156 9 , , PUNCT cana-1834 156 10 [ [ PUNCT cana-1834 156 11 ] ] X cana-1834 156 12 ) ) PUNCT cana-1834 156 13 r r NOUN cana-1834 156 14   VERB cana-1834 156 15 , , PUNCT cana-1834 156 16 .a .a ADJ cana-1834 156 17 r r NOUN cana-1834 156 18   PROPN cana-1834 156 19 then then ADV cana-1834 156 20 ( ( PUNCT cana-1834 156 21 : : PUNCT cana-1834 156 22 ) ) PUNCT cana-1834 156 23 ( ( PUNCT cana-1834 156 24 : : PUNCT cana-1834 156 25 \ \ NOUN cana-1834 156 26 ) ) PUNCT cana-1834 157 1 a a DET cana-1834 157 2 if if SCONJ cana-1834 157 3 a a PRON cana-1834 157 4 i i PRON cana-1834 158 1 i i PRON cana-1834 158 2 a a X cana-1834 159 1 i i PRON cana-1834 159 2 a a PRON cana-1834 159 3 i i NOUN cana-1834 159 4 ο‘οƒŽ ο‘οƒŽ ADV cana-1834 159 5   PROPN cana-1834 160 1 = = SYM cana-1834 160 2 = = PUNCT cana-1834 160 3 an an DET cana-1834 160 4 equivalence equivalence NOUN cana-1834 160 5 relation relation NOUN cana-1834 160 6   PROPN cana-1834 160 7 on on ADP cana-1834 160 8 ( ( PUNCT cana-1834 160 9 , , PUNCT cana-1834 160 10 , , PUNCT cana-1834 160 11 [ [ PUNCT cana-1834 160 12 ] ] X cana-1834 160 13 ) ) PUNCT cana-1834 160 14 r r NOUN cana-1834 160 15   NOUN cana-1834 160 16 is be AUX cana-1834 160 17 said say VERB cana-1834 160 18 to to PART cana-1834 160 19 be be AUX cana-1834 160 20 a a DET cana-1834 160 21 congruence congruence NOUN cana-1834 160 22 if if SCONJ cana-1834 160 23 for for ADP cana-1834 160 24 all all PRON cana-1834 160 25 , , PUNCT cana-1834 160 26 , , PUNCT cana-1834 160 27 , , PUNCT cana-1834 160 28 , , PUNCT cana-1834 160 29 p p NOUN cana-1834 160 30 q q NOUN cana-1834 160 31 r r NOUN cana-1834 160 32 r r NOUN cana-1834 160 33 ο§οƒŽ ο§οƒŽ NOUN cana-1834 160 34 οƒŽο‡ οƒŽο‡ NOUN cana-1834 160 35 we we PRON cana-1834 160 36 have have VERB cana-1834 160 37 ( ( PUNCT cana-1834 160 38 ) ) PUNCT cana-1834 160 39 ( ( PUNCT cana-1834 160 40 ) ) PUNCT cana-1834 160 41 ( ( PUNCT cana-1834 160 42 ) ) PUNCT cana-1834 160 43 ( ( PUNCT cana-1834 160 44 ) ) PUNCT cana-1834 160 45 ( ( PUNCT cana-1834 160 46 ) ) PUNCT cana-1834 160 47 ( ( PUNCT cana-1834 160 48 ) ) PUNCT cana-1834 160 49 ( ( PUNCT cana-1834 160 50 ) ) PUNCT cana-1834 160 51 ( ( PUNCT cana-1834 160 52 ) ) PUNCT cana-1834 160 53 ( ( PUNCT cana-1834 160 54 ) ) PUNCT cana-1834 161 1 p p X cana-1834 161 2 q q NOUN cana-1834 161 3 r r NOUN cana-1834 161 4 p p X cana-1834 161 5 s s X cana-1834 161 6 q q X cana-1834 161 7 s s X cana-1834 161 8 r r NOUN cana-1834 161 9 s s X cana-1834 161 10 p p NOUN cana-1834 161 11 q q PROPN cana-1834 161 12 p p X cana-1834 161 13 q q X cana-1834 161 14 s s X cana-1834 161 15 q q NOUN cana-1834 161 16 r r NOUN cana-1834 161 17 s s NOUN cana-1834 161 18 r r NOUN cana-1834 161 19 p p X cana-1834 161 20 s s NOUN cana-1834 161 21 and and CCONJ cana-1834 161 22 q q ADJ cana-1834 161 23 p p X cana-1834 161 24 s s NOUN cana-1834 161 25 r r NOUN cana-1834 161 26 q q X cana-1834 161 27 s s X cana-1834 161 28 p p NOUN cana-1834 161 29 r r NOUN cana-1834 161 30 s s NOUN cana-1834 161 31   X cana-1834 161 32   X cana-1834 161 33   PROPN cana-1834 161 34   PROPN cana-1834 161 35   PROPN cana-1834 161 36   NUM cana-1834 161 37   NUM cana-1834 161 38   X cana-1834 162 1   NUM cana-1834 162 2   NUM cana-1834 162 3   X cana-1834 163 1   NUM cana-1834 163 2   NUM cana-1834 163 3   NUM cana-1834 163 4   NUM cana-1834 163 5   X cana-1834 164 1   NUM cana-1834 164 2   NUM cana-1834 164 3   X cana-1834 165 1   NUM cana-1834 165 2   NUM cana-1834 165 3 οƒž οƒž NOUN cana-1834 166 1 + + X cana-1834 166 2 + + CCONJ cana-1834 166 3 + + CCONJ cana-1834 166 4 οƒž οƒž NOUN cana-1834 166 5 by by ADP cana-1834 166 6 r r NOUN cana-1834 166 7 : : PUNCT cana-1834 166 8   NOUN cana-1834 166 9 , , PUNCT cana-1834 166 10 we we PRON cana-1834 166 11 mean mean VERB cana-1834 166 12 the the DET cana-1834 166 13 set set NOUN cana-1834 166 14 of of ADP cana-1834 166 15 all all DET cana-1834 166 16 equivalence equivalence NOUN cana-1834 166 17 classes class NOUN cana-1834 166 18 of of ADP cana-1834 166 19 the the DET cana-1834 166 20 elements element NOUN cana-1834 166 21 of of ADP cana-1834 166 22 r r NOUN cana-1834 166 23 with with ADP cana-1834 166 24 respect respect NOUN cana-1834 166 25 to to ADP cana-1834 166 26 the the DET cana-1834 166 27 mapping mapping NOUN cana-1834 166 28   NOUN cana-1834 166 29 that that ADV cana-1834 166 30 is be AUX cana-1834 166 31 , , PUNCT cana-1834 166 32 : : PUNCT cana-1834 166 33 { { PUNCT cana-1834 166 34 ( ( PUNCT cana-1834 166 35 ) ) PUNCT cana-1834 166 36 / / SYM cana-1834 166 37 } } PUNCT cana-1834 166 38 .r .r NOUN cana-1834 166 39 x x X cana-1834 166 40 x x X cana-1834 166 41 r r PROPN cana-1834 166 42 = = PROPN cana-1834 166 43 οƒŽ οƒŽ PROPN cana-1834 166 44 lemma lemma PROPN cana-1834 166 45 4.4 4.4 NUM cana-1834 166 46 : : PUNCT cana-1834 166 47 let let PROPN cana-1834 166 48 be be AUX cana-1834 166 49 a a DET cana-1834 166 50 congruence congruence NOUN cana-1834 166 51 relation relation NOUN cana-1834 166 52 on on ADP cana-1834 166 53 ( ( PUNCT cana-1834 166 54 , , PUNCT cana-1834 166 55 , , PUNCT cana-1834 166 56 [ [ PUNCT cana-1834 166 57 ] ] X cana-1834 166 58 ) ) PUNCT cana-1834 166 59 r r NOUN cana-1834 166 60   NOUN cana-1834 166 61 .then .then ADP cana-1834 166 62 ( ( PUNCT cana-1834 166 63 ) ) PUNCT cana-1834 166 64 ( ( PUNCT cana-1834 166 65 ( ( PUNCT cana-1834 166 66 ) ) PUNCT cana-1834 166 67 ( ( PUNCT cana-1834 166 68 ) ) PUNCT cana-1834 166 69 ) ) PUNCT cana-1834 167 1 ( ( PUNCT cana-1834 167 2 ) ) PUNCT cana-1834 167 3 ( ( PUNCT cana-1834 167 4 ( ( PUNCT cana-1834 167 5 ) ) PUNCT cana-1834 167 6 ( ( PUNCT cana-1834 167 7 ) ) PUNCT cana-1834 167 8 ( ( PUNCT cana-1834 167 9 ) ) PUNCT cana-1834 167 10 ) ) PUNCT cana-1834 167 11 , , PUNCT cana-1834 167 12 , , PUNCT cana-1834 167 13 , , PUNCT cana-1834 167 14 .x .x PROPN cana-1834 168 1 y y PROPN cana-1834 168 2 x x PUNCT cana-1834 168 3 y y PROPN cana-1834 168 4 and and CCONJ cana-1834 168 5 x x PROPN cana-1834 168 6 y y PROPN cana-1834 168 7 z z NOUN cana-1834 168 8 x x VERB cana-1834 168 9 y y PROPN cana-1834 168 10 z z PROPN cana-1834 168 11 for for ADP cana-1834 168 12 all all PRON cana-1834 168 13 x x SYM cana-1834 168 14 y y PROPN cana-1834 168 15 z z NOUN cana-1834 168 16 r r NOUN cana-1834 168 17 and and PROPN cana-1834 168 18   X cana-1834 168 19   X cana-1834 168 20   PROPN cana-1834 168 21   PROPN cana-1834 169 1   NUM cana-1834 169 2   NUM cana-1834 169 3   X cana-1834 170 1   X cana-1834 170 2   PROPN cana-1834 170 3   VERB cana-1834 170 4 + + NOUN cana-1834 171 1 = = PUNCT cana-1834 172 1 + + PUNCT cana-1834 173 1 = = NOUN cana-1834 174 1 οƒŽ οƒŽ NOUN cana-1834 174 2 οƒŽο‡ οƒŽο‡ NOUN cana-1834 174 3 proof proof NOUN cana-1834 174 4 : : PUNCT cana-1834 174 5 first first ADV cana-1834 174 6 we we PRON cana-1834 174 7 observe observe VERB cana-1834 174 8 that that SCONJ cana-1834 174 9 ( ( PUNCT cana-1834 174 10 ) ) PUNCT cana-1834 174 11 ( ( PUNCT cana-1834 174 12 ( ( PUNCT cana-1834 174 13 ) ) PUNCT cana-1834 174 14 ( ( PUNCT cana-1834 174 15 ) ) PUNCT cana-1834 174 16 ) ) PUNCT cana-1834 175 1 x x SYM cana-1834 175 2 y y NOUN cana-1834 175 3 x x PUNCT cana-1834 175 4 y y X cana-1834 175 5   PROPN cana-1834 176 1   PROPN cana-1834 176 2 + + PROPN cana-1834 176 3   PROPN cana-1834 176 4 + + CCONJ cana-1834 176 5 and and CCONJ cana-1834 176 6 ( ( PUNCT cana-1834 176 7 ) ) PUNCT cana-1834 176 8 ( ( PUNCT cana-1834 176 9 ( ( PUNCT cana-1834 176 10 ) ) PUNCT cana-1834 176 11 ( ( PUNCT cana-1834 176 12 ) ) PUNCT cana-1834 176 13 ( ( PUNCT cana-1834 176 14 ) ) PUNCT cana-1834 176 15 ) ) PUNCT cana-1834 177 1 x x PUNCT cana-1834 177 2 y y PROPN cana-1834 177 3 z z NOUN cana-1834 177 4 x x SYM cana-1834 177 5 y y PROPN cana-1834 177 6 z z PROPN cana-1834 177 7   NUM cana-1834 177 8   NUM cana-1834 177 9   X cana-1834 178 1   X cana-1834 178 2   PROPN cana-1834 178 3   X cana-1834 178 4 .by .by PROPN cana-1834 178 5 routine routine ADJ cana-1834 178 6 checking checking NOUN cana-1834 178 7 , , PUNCT cana-1834 178 8 we we PRON cana-1834 178 9 can can AUX cana-1834 178 10 easily easily ADV cana-1834 178 11 verify verify VERB cana-1834 178 12 that that SCONJ cana-1834 178 13 the the DET cana-1834 178 14 above above ADJ cana-1834 178 15 equalities equality NOUN cana-1834 178 16 holds hold VERB cana-1834 178 17 . . PUNCT cana-1834 179 1 in in ADP cana-1834 179 2 the the DET cana-1834 179 3 next next ADJ cana-1834 179 4 theorem theorem NOUN cana-1834 179 5 , , PUNCT cana-1834 179 6 we we PRON cana-1834 179 7 demonstrate demonstrate VERB cana-1834 179 8 how how SCONJ cana-1834 179 9 to to PART cana-1834 179 10 construct construct VERB cana-1834 179 11 a a DET cana-1834 179 12 new new ADJ cana-1834 179 13 ternary ternary ADJ cana-1834 179 14 𝛀 𝛀 PROPN cana-1834 179 15 βˆ’semiring βˆ’semire VERB cana-1834 179 16 by by ADP cana-1834 179 17 using use VERB cana-1834 179 18 the the DET cana-1834 179 19 congruence congruence PROPN cana-1834 179 20 relations relation NOUN cana-1834 179 21 . . PUNCT cana-1834 180 1 theorem theorem VERB cana-1834 180 2 4.5 4.5 NUM cana-1834 180 3 : : PUNCT cana-1834 180 4 let let VERB cana-1834 180 5   PRON cana-1834 180 6 be be AUX cana-1834 180 7 a a DET cana-1834 180 8 congruence congruence NOUN cana-1834 180 9 relation relation NOUN cana-1834 180 10 on on ADP cana-1834 180 11 ( ( PUNCT cana-1834 180 12 , , PUNCT cana-1834 180 13 , , PUNCT cana-1834 180 14 [ [ PUNCT cana-1834 180 15 ] ] X cana-1834 180 16 ) ) PUNCT cana-1834 180 17 r r NOUN cana-1834 180 18   VERB cana-1834 180 19 . . PUNCT cana-1834 181 1 define define VERB cana-1834 181 2 οƒ… οƒ… ADJ cana-1834 181 3 on on ADP cana-1834 181 4 r r NOUN cana-1834 181 5 : : PUNCT cana-1834 181 6   X cana-1834 181 7 by by ADP cana-1834 181 8 ( ( PUNCT cana-1834 181 9 ) ) PUNCT cana-1834 181 10 ( ( PUNCT cana-1834 181 11 ) ) PUNCT cana-1834 181 12 ( ( PUNCT cana-1834 181 13 ) ) PUNCT cana-1834 181 14 , , PUNCT cana-1834 181 15 .x .x PROPN cana-1834 182 1 y y PROPN cana-1834 182 2 x x PUNCT cana-1834 182 3 y y PROPN cana-1834 182 4 for for ADP cana-1834 182 5 all all PRON cana-1834 182 6 x x PUNCT cana-1834 182 7 y y PROPN cana-1834 182 8 r r NOUN cana-1834 182 9   X cana-1834 182 10   PROPN cana-1834 182 11 = = PUNCT cana-1834 183 1 + + NUM cana-1834 183 2 οƒŽ οƒŽ NOUN cana-1834 183 3 then then ADV cana-1834 183 4 ( ( PUNCT cana-1834 183 5 : : PUNCT cana-1834 183 6 , , PUNCT cana-1834 183 7 , , PUNCT cana-1834 183 8 [ [ PUNCT cana-1834 183 9 ] ] X cana-1834 183 10 ) ) PUNCT cana-1834 183 11 r r NOUN cana-1834 183 12   NOUN cana-1834 183 13 οƒ… οƒ… PROPN cana-1834 183 14 is be AUX cana-1834 183 15 a a DET cana-1834 183 16 ternary ternary ADJ cana-1834 183 17 𝛀 𝛀 PROPN cana-1834 183 18 βˆ’semiring βˆ’semire VERB cana-1834 183 19 with with ADP cana-1834 183 20 the the DET cana-1834 183 21 following follow VERB cana-1834 183 22 mapping mapping NOUN cana-1834 183 23 : : PUNCT cana-1834 183 24 ( ( PUNCT cana-1834 183 25 : : PUNCT cana-1834 183 26 ) ) PUNCT cana-1834 183 27 ( ( PUNCT cana-1834 183 28 : : PUNCT cana-1834 183 29 ) ) PUNCT cana-1834 183 30 ( ( PUNCT cana-1834 183 31 : : PUNCT cana-1834 183 32 ) ) PUNCT cana-1834 183 33 ( ( PUNCT cana-1834 183 34 : : PUNCT cana-1834 183 35 ) ) PUNCT cana-1834 183 36 , , PUNCT cana-1834 183 37 r r NOUN cana-1834 183 38 r r NOUN cana-1834 183 39 r r NOUN cana-1834 183 40 r r NOUN cana-1834 183 41   PROPN cana-1834 183 42   PROPN cana-1834 183 43   PROPN cana-1834 183 44   PROPN cana-1834 183 45 β†’ β†’ PUNCT cana-1834 183 46 defined define VERB cana-1834 183 47 by by ADP cana-1834 183 48 ( ( PUNCT cana-1834 183 49 ) ) PUNCT cana-1834 183 50 ( ( PUNCT cana-1834 183 51 y y NOUN cana-1834 183 52 ) ) PUNCT cana-1834 183 53 ( ( PUNCT cana-1834 183 54 z z NOUN cana-1834 183 55 ) ) PUNCT cana-1834 183 56 ( ( PUNCT cana-1834 183 57 ) ) PUNCT cana-1834 183 58 , , PUNCT cana-1834 183 59 for for ADP cana-1834 183 60 all all DET cana-1834 183 61 x x NOUN cana-1834 183 62 , , PUNCT cana-1834 183 63 y y PROPN cana-1834 183 64 , , PUNCT cana-1834 183 65 z z NOUN cana-1834 183 66 r r NOUN cana-1834 183 67 , , PUNCT cana-1834 183 68 .x .x NOUN cana-1834 184 1 x x PUNCT cana-1834 184 2 y y PROPN cana-1834 184 3 z z PROPN cana-1834 184 4   NUM cana-1834 184 5   PROPN cana-1834 184 6   NUM cana-1834 184 7   X cana-1834 185 1   X cana-1834 185 2   NUM cana-1834 185 3   NUM cana-1834 185 4 = = NOUN cana-1834 185 5 οƒŽ οƒŽ PROPN cana-1834 185 6 οƒŽο‡ οƒŽο‡ NOUN cana-1834 185 7 proof proof NOUN cana-1834 185 8 : : PUNCT cana-1834 185 9 let let VERB cana-1834 185 10 ' ' PUNCT cana-1834 185 11 ' ' PUNCT cana-1834 185 12 ' ' PUNCT cana-1834 185 13 ( ( PUNCT cana-1834 185 14 ) ) PUNCT cana-1834 185 15 ( ( PUNCT cana-1834 185 16 x x X cana-1834 185 17 ) ) PUNCT cana-1834 185 18 ( ( PUNCT cana-1834 185 19 y y NOUN cana-1834 185 20 ) ) PUNCT cana-1834 185 21 ( ( PUNCT cana-1834 185 22 y y PROPN cana-1834 185 23 ) ) PUNCT cana-1834 185 24 ( ( PUNCT cana-1834 185 25 z z NOUN cana-1834 185 26 ) ) PUNCT cana-1834 185 27 ( ( PUNCT cana-1834 185 28 z z NOUN cana-1834 185 29 ) ) PUNCT cana-1834 185 30 x x PROPN cana-1834 185 31 and and CCONJ cana-1834 185 32 and and PROPN cana-1834 185 33   PROPN cana-1834 185 34   X cana-1834 185 35   X cana-1834 185 36   X cana-1834 186 1 = = X cana-1834 187 1 = = PUNCT cana-1834 187 2 = = PUNCT cana-1834 187 3 then then ADV cana-1834 187 4 by by ADP cana-1834 187 5 lemma lemma PROPN cana-1834 187 6 4.4 4.4 NUM cana-1834 187 7 we we PRON cana-1834 187 8 have have VERB cana-1834 187 9 the the DET cana-1834 187 10 following follow VERB cana-1834 187 11 equality equality NOUN cana-1834 187 12 ' ' PUNCT cana-1834 187 13 ' ' PUNCT cana-1834 187 14 ' ' PUNCT cana-1834 187 15 ' ' PUNCT cana-1834 187 16 ( ( PUNCT cana-1834 187 17 ) ) PUNCT cana-1834 187 18 ( ( PUNCT cana-1834 187 19 y y NOUN cana-1834 187 20 ) ) PUNCT cana-1834 187 21 ( ( PUNCT cana-1834 187 22 x x NOUN cana-1834 187 23 y y NOUN cana-1834 187 24 ) ) PUNCT cana-1834 187 25 ( ( PUNCT cana-1834 187 26 ( ( PUNCT cana-1834 187 27 x x X cana-1834 187 28 ) ) PUNCT cana-1834 187 29 ( ( PUNCT cana-1834 187 30 y y NOUN cana-1834 187 31 ) ) PUNCT cana-1834 187 32 ) ) PUNCT cana-1834 187 33 ( ( PUNCT cana-1834 187 34 ( ( PUNCT cana-1834 187 35 x x X cana-1834 187 36 ) ) PUNCT cana-1834 187 37 ( ( PUNCT cana-1834 187 38 y y PROPN cana-1834 187 39 ) ) PUNCT cana-1834 187 40 ) ) PUNCT cana-1834 188 1 ( ( PUNCT cana-1834 188 2 x x X cana-1834 188 3 ) ) PUNCT cana-1834 188 4 ( ( PUNCT cana-1834 188 5 y y PROPN cana-1834 188 6 ) ) PUNCT cana-1834 188 7 x x PUNCT cana-1834 188 8   X cana-1834 189 1   X cana-1834 189 2   X cana-1834 189 3   X cana-1834 189 4   X cana-1834 189 5   X cana-1834 189 6   PROPN cana-1834 189 7   PROPN cana-1834 189 8   PROPN cana-1834 189 9   PROPN cana-1834 189 10 οƒ… οƒ… PROPN cana-1834 189 11 = = PUNCT cana-1834 190 1 + + PUNCT cana-1834 190 2 = = SYM cana-1834 191 1 + + PUNCT cana-1834 191 2 = = SYM cana-1834 192 1 + + CCONJ cana-1834 192 2 = = NOUN cana-1834 192 3 οƒ… οƒ… ADJ cana-1834 192 4 communications communication NOUN cana-1834 192 5 on on ADP cana-1834 192 6 applied apply VERB cana-1834 192 7 nonlinear nonlinear ADJ cana-1834 192 8 analysis analysis NOUN cana-1834 192 9 issn issn NOUN cana-1834 192 10 : : PUNCT cana-1834 192 11 1074 1074 NUM cana-1834 192 12 - - PUNCT cana-1834 192 13 133x 133x NUM cana-1834 192 14 vol vol NOUN cana-1834 192 15 32 32 NUM cana-1834 192 16 no no NOUN cana-1834 192 17 . . NOUN cana-1834 192 18 2 2 NUM cana-1834 192 19 ( ( PUNCT cana-1834 192 20 2025 2025 NUM cana-1834 192 21 ) ) PUNCT cana-1834 192 22 585 585 NUM cana-1834 192 23 https://internationalpubls.com https://internationalpubls.com X cana-1834 193 1 also also ADV cana-1834 193 2 we we PRON cana-1834 193 3 have have VERB cana-1834 193 4 an an DET cana-1834 193 5 additional additional ADJ cana-1834 193 6 euality euality NOUN cana-1834 193 7 ' ' PUNCT cana-1834 193 8 ' ' PUNCT cana-1834 193 9 ' ' PUNCT cana-1834 193 10 ' ' PUNCT cana-1834 193 11 ' ' PUNCT cana-1834 193 12 ' ' PUNCT cana-1834 193 13 ( ( PUNCT cana-1834 193 14 ) ) PUNCT cana-1834 193 15 ( ( PUNCT cana-1834 193 16 ) ) PUNCT cana-1834 193 17 ( ( PUNCT cana-1834 193 18 z z NOUN cana-1834 193 19 ) ) PUNCT cana-1834 193 20 ( ( PUNCT cana-1834 193 21 x x PUNCT cana-1834 193 22 y y PROPN cana-1834 193 23 z z PROPN cana-1834 193 24 ) ) PUNCT cana-1834 193 25 ( ( PUNCT cana-1834 193 26 ( ( PUNCT cana-1834 193 27 ) ) PUNCT cana-1834 193 28 ( ( PUNCT cana-1834 193 29 y y NOUN cana-1834 193 30 ) ) PUNCT cana-1834 193 31 ( ( PUNCT cana-1834 193 32 z z NOUN cana-1834 193 33 ) ) PUNCT cana-1834 193 34 ) ) PUNCT cana-1834 194 1 ( ( PUNCT cana-1834 194 2 ( ( PUNCT cana-1834 194 3 x x X cana-1834 194 4 ) ) PUNCT cana-1834 194 5 ( ( PUNCT cana-1834 194 6 ) ) PUNCT cana-1834 194 7 ( ( PUNCT cana-1834 194 8 ) ) PUNCT cana-1834 194 9 ( ( PUNCT cana-1834 194 10 x x X cana-1834 194 11 ) ) PUNCT cana-1834 194 12 ( ( PUNCT cana-1834 194 13 ) ) PUNCT cana-1834 194 14 ( ( PUNCT cana-1834 194 15 ) ) PUNCT cana-1834 194 16 x x PUNCT cana-1834 194 17 y y NOUN cana-1834 194 18 x x PUNCT cana-1834 194 19 y y PROPN cana-1834 194 20 z z PROPN cana-1834 194 21 y y PROPN cana-1834 194 22 z z PROPN cana-1834 194 23   PROPN cana-1834 194 24   NUM cana-1834 194 25   PROPN cana-1834 194 26   NUM cana-1834 194 27   X cana-1834 194 28   X cana-1834 194 29   NUM cana-1834 194 30   NUM cana-1834 194 31   X cana-1834 195 1   X cana-1834 195 2   PROPN cana-1834 195 3   ADV cana-1834 195 4   PROPN cana-1834 196 1   PROPN cana-1834 196 2   VERB cana-1834 196 3   ADV cana-1834 196 4   PROPN cana-1834 196 5   NUM cana-1834 196 6   PROPN cana-1834 196 7   NUM cana-1834 196 8   X cana-1834 197 1 = = PUNCT cana-1834 197 2 = = PUNCT cana-1834 198 1 = = PUNCT cana-1834 198 2 = = PUNCT cana-1834 198 3 thus thus ADV cana-1834 198 4 andοƒ… andοƒ… INTJ cana-1834 198 5 are be AUX cana-1834 198 6 well well ADV cana-1834 198 7 defined define VERB cana-1834 198 8 . . PUNCT cana-1834 199 1 hence hence ADV cana-1834 199 2 , , PUNCT cana-1834 199 3 the the DET cana-1834 199 4 author author NOUN cana-1834 199 5 cam cam NOUN cana-1834 199 6 verify verify VERB cana-1834 199 7 that that SCONJ cana-1834 199 8 ( ( PUNCT cana-1834 199 9 : : PUNCT cana-1834 199 10 , , PUNCT cana-1834 199 11 , , PUNCT cana-1834 199 12 [ [ PUNCT cana-1834 199 13 ] ] X cana-1834 199 14 ) ) PUNCT cana-1834 199 15 r r NOUN cana-1834 199 16   NOUN cana-1834 199 17 οƒ… οƒ… PROPN cana-1834 199 18 is be AUX cana-1834 199 19 a a DET cana-1834 199 20 commutative commutative ADJ cana-1834 199 21 ternary ternary ADJ cana-1834 199 22 semigroup semigroup NOUN cana-1834 199 23 . . PUNCT cana-1834 200 1 now now ADV cana-1834 200 2 we we PRON cana-1834 200 3 deduce deduce VERB cana-1834 200 4 that that PRON cana-1834 200 5 ( ( PUNCT cana-1834 200 6 ) ) PUNCT cana-1834 200 7 ( ( PUNCT cana-1834 200 8 ) ) PUNCT cana-1834 201 1 [ [ X cana-1834 201 2 ( ( PUNCT cana-1834 201 3 ( ( PUNCT cana-1834 201 4 ) ) PUNCT cana-1834 201 5 ( ( PUNCT cana-1834 201 6 ) ) PUNCT cana-1834 201 7 ) ) PUNCT cana-1834 201 8 ] ] PUNCT cana-1834 201 9 ( ( PUNCT cana-1834 201 10 ) ) PUNCT cana-1834 201 11 ( ( PUNCT cana-1834 201 12 ) ) PUNCT cana-1834 201 13 ( ( PUNCT cana-1834 201 14 ) ) PUNCT cana-1834 201 15 ( ( PUNCT cana-1834 201 16 ( ( PUNCT cana-1834 201 17 ) ) PUNCT cana-1834 201 18 ) ) PUNCT cana-1834 202 1 ( ( PUNCT cana-1834 202 2 [ [ PUNCT cana-1834 202 3 ] ] X cana-1834 202 4 [ [ PUNCT cana-1834 202 5 ] ] X cana-1834 202 6 ) ) PUNCT cana-1834 202 7 [ [ PUNCT cana-1834 202 8 ] ] X cana-1834 202 9 [ [ PUNCT cana-1834 202 10 ] ] X cana-1834 202 11 [ [ PUNCT cana-1834 202 12 ( ( PUNCT cana-1834 202 13 ) ) PUNCT cana-1834 202 14 ( ( PUNCT cana-1834 202 15 y y NOUN cana-1834 202 16 ) ) PUNCT cana-1834 202 17 ( ( PUNCT cana-1834 202 18 z z NOUN cana-1834 202 19 ) ) PUNCT cana-1834 202 20 ] ] PUNCT cana-1834 202 21 [ [ PUNCT cana-1834 202 22 ( ( PUNCT cana-1834 202 23 ) ) PUNCT cana-1834 202 24 ( ( PUNCT cana-1834 202 25 y y NOUN cana-1834 202 26 ) ) PUNCT cana-1834 202 27 ( ( PUNCT cana-1834 202 28 t t PROPN cana-1834 202 29 ) ) PUNCT cana-1834 202 30 ] ] PUNCT cana-1834 202 31 x x PUNCT cana-1834 203 1 y y PROPN cana-1834 203 2 z z PROPN cana-1834 203 3 t t PROPN cana-1834 203 4 x x PUNCT cana-1834 203 5 y y PROPN cana-1834 203 6 z z PROPN cana-1834 203 7 t t PROPN cana-1834 203 8 x x PUNCT cana-1834 203 9 y y PROPN cana-1834 203 10 z z PROPN cana-1834 203 11 t t PROPN cana-1834 203 12 x x PUNCT cana-1834 203 13 y y PROPN cana-1834 203 14 z z NOUN cana-1834 203 15 x x VERB cana-1834 203 16 y y NOUN cana-1834 203 17 t t NOUN cana-1834 203 18 x x PUNCT cana-1834 203 19 y y PROPN cana-1834 203 20 z z NOUN cana-1834 203 21 x x VERB cana-1834 203 22 y y NOUN cana-1834 203 23 t t NOUN cana-1834 203 24 x x PUNCT cana-1834 203 25 x x PUNCT cana-1834 203 26   X cana-1834 203 27   NUM cana-1834 203 28   X cana-1834 203 29   X cana-1834 203 30   PROPN cana-1834 203 31   PROPN cana-1834 203 32   NUM cana-1834 203 33   X cana-1834 203 34   X cana-1834 203 35   PROPN cana-1834 203 36   NUM cana-1834 203 37   NUM cana-1834 203 38   X cana-1834 204 1   NUM cana-1834 204 2   NUM cana-1834 204 3   NUM cana-1834 204 4   NUM cana-1834 204 5   X cana-1834 205 1   NUM cana-1834 205 2   NUM cana-1834 205 3   X cana-1834 206 1   NUM cana-1834 206 2   NUM cana-1834 206 3   X cana-1834 207 1   NUM cana-1834 207 2   PROPN cana-1834 207 3   NUM cana-1834 207 4   X cana-1834 208 1   X cana-1834 208 2   NUM cana-1834 208 3   PROPN cana-1834 208 4   NUM cana-1834 208 5   X cana-1834 208 6 οƒ… οƒ… PROPN cana-1834 208 7 = = PUNCT cana-1834 209 1 + + PUNCT cana-1834 209 2 = = SYM cana-1834 210 1 + + PUNCT cana-1834 210 2 = = SYM cana-1834 211 1 + + CCONJ cana-1834 211 2 = = SYM cana-1834 211 3 οƒ… οƒ… ADJ cana-1834 211 4 = = SYM cana-1834 211 5 οƒ… οƒ… NOUN cana-1834 211 6 in in ADP cana-1834 211 7 that that DET cana-1834 211 8 similar similar ADJ cana-1834 211 9 way way NOUN cana-1834 211 10 prove prove VERB cana-1834 211 11 that that SCONJ cana-1834 211 12 [ [ PUNCT cana-1834 211 13 ( ( PUNCT cana-1834 211 14 ) ) PUNCT cana-1834 211 15 [ [ PUNCT cana-1834 211 16 ( ( PUNCT cana-1834 211 17 ) ) PUNCT cana-1834 211 18 ( ( PUNCT cana-1834 211 19 ) ) PUNCT cana-1834 211 20 ] ] PUNCT cana-1834 211 21 ( ( PUNCT cana-1834 211 22 ) ) PUNCT cana-1834 211 23 [ [ PUNCT cana-1834 211 24 ( ( PUNCT cana-1834 211 25 ) ) PUNCT cana-1834 211 26 ( ( PUNCT cana-1834 211 27 ) ) PUNCT cana-1834 211 28 ( ( PUNCT cana-1834 211 29 ) ) PUNCT cana-1834 211 30 ] ] PUNCT cana-1834 212 1 [ [ PUNCT cana-1834 212 2 ( ( PUNCT cana-1834 212 3 ) ) PUNCT cana-1834 212 4 ( ( PUNCT cana-1834 212 5 z z NOUN cana-1834 212 6 ) ) PUNCT cana-1834 212 7 ( ( PUNCT cana-1834 212 8 ) ) PUNCT cana-1834 212 9 ] ] X cana-1834 212 10 x x X cana-1834 212 11 y y PROPN cana-1834 212 12 z z PROPN cana-1834 212 13 t t PROPN cana-1834 212 14 x x X cana-1834 212 15 y y PROPN cana-1834 212 16 t t PROPN cana-1834 212 17 x x PUNCT cana-1834 212 18 t t NOUN cana-1834 212 19   NUM cana-1834 212 20   X cana-1834 213 1   X cana-1834 213 2   NUM cana-1834 213 3   X cana-1834 214 1   X cana-1834 214 2   NUM cana-1834 214 3   X cana-1834 215 1   X cana-1834 215 2   PROPN cana-1834 215 3   NUM cana-1834 215 4   PROPN cana-1834 216 1   PROPN cana-1834 216 2 = = PUNCT cana-1834 216 3 οƒ… οƒ… PROPN cana-1834 216 4 [ [ PUNCT cana-1834 216 5 ( ( PUNCT cana-1834 216 6 ) ) PUNCT cana-1834 216 7 ( ( PUNCT cana-1834 216 8 ) ) PUNCT cana-1834 216 9 ] ] PUNCT cana-1834 216 10 ( ( PUNCT cana-1834 216 11 ) ) PUNCT cana-1834 216 12 ( ( PUNCT cana-1834 216 13 ) ) PUNCT cana-1834 216 14 [ [ PUNCT cana-1834 216 15 ( ( PUNCT cana-1834 216 16 x x X cana-1834 216 17 ) ) PUNCT cana-1834 216 18 ( ( PUNCT cana-1834 216 19 ) ) PUNCT cana-1834 216 20 ( ( PUNCT cana-1834 216 21 ) ) PUNCT cana-1834 216 22 ] ] PUNCT cana-1834 217 1 [ [ PUNCT cana-1834 217 2 ( ( PUNCT cana-1834 217 3 y y NOUN cana-1834 217 4 ) ) PUNCT cana-1834 217 5 ( ( PUNCT cana-1834 217 6 ) ) PUNCT cana-1834 217 7 ( ( PUNCT cana-1834 217 8 ) ) PUNCT cana-1834 217 9 ] ] X cana-1834 217 10 x x X cana-1834 217 11 y y PROPN cana-1834 217 12 z z PROPN cana-1834 217 13 t t PROPN cana-1834 217 14 z z PROPN cana-1834 217 15 t t PROPN cana-1834 217 16 z z PROPN cana-1834 217 17 t t NOUN cana-1834 217 18   PROPN cana-1834 217 19   NUM cana-1834 217 20   PROPN cana-1834 217 21   NUM cana-1834 217 22   X cana-1834 218 1   X cana-1834 218 2   NUM cana-1834 218 3   PROPN cana-1834 218 4   NUM cana-1834 218 5   X cana-1834 219 1   X cana-1834 219 2   NUM cana-1834 219 3   PROPN cana-1834 219 4   NUM cana-1834 219 5   PROPN cana-1834 219 6 = = PUNCT cana-1834 219 7 οƒ… οƒ… ADJ cana-1834 219 8 also also ADV cana-1834 220 1 [ [ X cana-1834 220 2 [ [ PUNCT cana-1834 220 3 ( ( PUNCT cana-1834 220 4 ) ) PUNCT cana-1834 220 5 ( ( PUNCT cana-1834 220 6 y y NOUN cana-1834 220 7 ) ) PUNCT cana-1834 220 8 ( ( PUNCT cana-1834 220 9 ) ) PUNCT cana-1834 220 10 ] ] PUNCT cana-1834 220 11 ( ( PUNCT cana-1834 220 12 s s X cana-1834 220 13 ) ) PUNCT cana-1834 220 14 ( ( PUNCT cana-1834 220 15 t t PROPN cana-1834 220 16 ) ) PUNCT cana-1834 220 17 ] ] PUNCT cana-1834 221 1 [ [ PUNCT cana-1834 221 2 ( ( PUNCT cana-1834 221 3 ) ) PUNCT cana-1834 221 4 [ [ PUNCT cana-1834 221 5 ( ( PUNCT cana-1834 221 6 y y NOUN cana-1834 221 7 ) ) PUNCT cana-1834 221 8 ( ( PUNCT cana-1834 221 9 ) ) PUNCT cana-1834 221 10 ( ( PUNCT cana-1834 221 11 s s NOUN cana-1834 221 12 ) ) PUNCT cana-1834 221 13 ] ] PUNCT cana-1834 221 14 ( ( PUNCT cana-1834 221 15 t t NOUN cana-1834 221 16 ) ) PUNCT cana-1834 221 17 ] ] PUNCT cana-1834 222 1 [ [ PUNCT cana-1834 222 2 ( ( PUNCT cana-1834 222 3 ) ) PUNCT cana-1834 222 4 ( ( PUNCT cana-1834 222 5 y y NOUN cana-1834 222 6 ) ) PUNCT cana-1834 222 7 [ [ PUNCT cana-1834 222 8 ( ( PUNCT cana-1834 222 9 ) ) PUNCT cana-1834 222 10 ( ( PUNCT cana-1834 222 11 s s X cana-1834 222 12 ) ) PUNCT cana-1834 222 13 ( ( PUNCT cana-1834 222 14 t t PROPN cana-1834 222 15 ) ) PUNCT cana-1834 222 16 ] ] PUNCT cana-1834 222 17 ] ] X cana-1834 223 1 x x PUNCT cana-1834 223 2 z z NOUN cana-1834 223 3 x x X cana-1834 223 4 z z NOUN cana-1834 223 5 x x X cana-1834 223 6 z z NOUN cana-1834 223 7   NOUN cana-1834 223 8   X cana-1834 223 9   X cana-1834 223 10   NOUN cana-1834 223 11   PROPN cana-1834 223 12   NUM cana-1834 223 13   PROPN cana-1834 224 1   NUM cana-1834 224 2   X cana-1834 224 3   X cana-1834 224 4   X cana-1834 224 5   PROPN cana-1834 224 6   NOUN cana-1834 224 7   PROPN cana-1834 224 8   NUM cana-1834 224 9   PROPN cana-1834 225 1   NUM cana-1834 225 2   X cana-1834 225 3   X cana-1834 225 4   X cana-1834 225 5   PROPN cana-1834 225 6   NOUN cana-1834 225 7   PROPN cana-1834 225 8   NUM cana-1834 225 9   PROPN cana-1834 226 1   NUM cana-1834 226 2   X cana-1834 226 3 = = SYM cana-1834 226 4 = = PRON cana-1834 226 5 therefore therefore ADV cana-1834 226 6 r: r: PROPN cana-1834 226 7 is be AUX cana-1834 226 8 a a DET cana-1834 226 9 ternary ternary ADJ cana-1834 226 10 𝛀 𝛀 PROPN cana-1834 226 11 βˆ’semiring βˆ’semire VERB cana-1834 226 12 . . PUNCT cana-1834 227 1 lemma lemma PROPN cana-1834 227 2 4.6 4.6 NUM cana-1834 227 3 : : PUNCT cana-1834 227 4 if if SCONJ cana-1834 227 5 : : PUNCT cana-1834 227 6 : : PUNCT cana-1834 227 7 r r NOUN cana-1834 227 8 r r NOUN cana-1834 227 9 r r NOUN cana-1834 227 10   PUNCT cana-1834 227 11 β†’ β†’ PUNCT cana-1834 227 12 is be AUX cana-1834 227 13 defined define VERB cana-1834 227 14 by by ADP cana-1834 227 15 ( ( PUNCT cana-1834 227 16 ) ) PUNCT cana-1834 227 17 ( ( PUNCT cana-1834 227 18 ) ) PUNCT cana-1834 227 19 r r NOUN cana-1834 227 20 x x PUNCT cana-1834 228 1 x x PROPN cana-1834 228 2 = = PRON cana-1834 228 3 and and CCONJ cana-1834 228 4 ri ri PROPN cana-1834 228 5 is be AUX cana-1834 228 6 the the DET cana-1834 228 7 identity identity NOUN cana-1834 228 8 function function NOUN cana-1834 228 9 on on ADP cana-1834 228 10   PROPN cana-1834 228 11 , , PUNCT cana-1834 228 12 then then ADV cana-1834 228 13 ( ( PUNCT cana-1834 228 14 , , PUNCT cana-1834 228 15 ) ) PUNCT cana-1834 228 16 : : PUNCT cana-1834 228 17 ( ( PUNCT cana-1834 228 18 , , PUNCT cana-1834 228 19 ) ) PUNCT cana-1834 228 20 ( ( PUNCT cana-1834 228 21 : : PUNCT cana-1834 228 22 , , PUNCT cana-1834 228 23 ) ) PUNCT cana-1834 228 24 r r NOUN cana-1834 228 25 ri ri NOUN cana-1834 228 26 r r NOUN cana-1834 228 27 r r NOUN cana-1834 228 28   PUNCT cana-1834 228 29   VERB cana-1834 228 30 β†’ β†’ PUNCT cana-1834 228 31   VERB cana-1834 228 32 is be AUX cana-1834 228 33 an an DET cana-1834 228 34 epi epi NOUN cana-1834 228 35 - - NOUN cana-1834 228 36 morphism morphism NOUN cana-1834 228 37 . . PUNCT cana-1834 229 1 proof proof NOUN cana-1834 229 2 : : PUNCT cana-1834 229 3 let let VERB cana-1834 229 4 , , PUNCT cana-1834 229 5 x x PUNCT cana-1834 229 6 y y NOUN cana-1834 229 7 r r NOUN cana-1834 229 8 and and CCONJ cana-1834 229 9 ο§οƒŽ ο§οƒŽ NOUN cana-1834 229 10 οƒŽο‡ οƒŽο‡ NOUN cana-1834 229 11 .then .then PUNCT cana-1834 229 12 it it PRON cana-1834 229 13 is be AUX cana-1834 229 14 easy easy ADJ cana-1834 229 15 to to PART cana-1834 229 16 see see VERB cana-1834 229 17 that that PRON cana-1834 229 18 ( ( PUNCT cana-1834 229 19 ) ) PUNCT cana-1834 229 20 ( ( PUNCT cana-1834 229 21 ) ) PUNCT cana-1834 229 22 ( ( PUNCT cana-1834 229 23 ) ) PUNCT cana-1834 229 24 ( ( PUNCT cana-1834 229 25 y y NOUN cana-1834 229 26 ) ) PUNCT cana-1834 229 27 ( ( PUNCT cana-1834 229 28 ) ) PUNCT cana-1834 229 29 ( ( PUNCT cana-1834 229 30 y y X cana-1834 229 31 ) ) PUNCT cana-1834 230 1 r r NOUN cana-1834 230 2 r r NOUN cana-1834 230 3 r r NOUN cana-1834 230 4 x x PUNCT cana-1834 230 5 y y NOUN cana-1834 230 6 x x PUNCT cana-1834 230 7 y y NOUN cana-1834 230 8 x x X cana-1834 230 9 x x PUNCT cana-1834 230 10   X cana-1834 230 11   PROPN cana-1834 230 12   X cana-1834 231 1 + + PUNCT cana-1834 231 2 = = SYM cana-1834 232 1 + + CCONJ cana-1834 232 2 = = NOUN cana-1834 232 3 οƒ… οƒ… ADJ cana-1834 232 4 = = SYM cana-1834 232 5   PROPN cana-1834 232 6   PROPN cana-1834 232 7 also also ADV cana-1834 232 8 ( ( PUNCT cana-1834 232 9 ) ) PUNCT cana-1834 232 10 ( ( PUNCT cana-1834 232 11 ) ) PUNCT cana-1834 232 12 ( ( PUNCT cana-1834 232 13 ) ) PUNCT cana-1834 232 14 ( ( PUNCT cana-1834 232 15 y y NOUN cana-1834 232 16 ) ) PUNCT cana-1834 232 17 ( ( PUNCT cana-1834 232 18 z z NOUN cana-1834 232 19 ) ) PUNCT cana-1834 232 20 ( ( PUNCT cana-1834 232 21 ) ) PUNCT cana-1834 232 22 ( ( PUNCT cana-1834 232 23 y y NOUN cana-1834 232 24 ) ) PUNCT cana-1834 232 25 ( ( PUNCT cana-1834 232 26 z z X cana-1834 232 27 ) ) PUNCT cana-1834 232 28 r r NOUN cana-1834 232 29 r r NOUN cana-1834 232 30 r r NOUN cana-1834 232 31 r r NOUN cana-1834 232 32 x x PUNCT cana-1834 232 33 y y NOUN cana-1834 232 34 z z NOUN cana-1834 232 35 x x PROPN cana-1834 232 36 y y PROPN cana-1834 232 37 z z NOUN cana-1834 232 38 x x PUNCT cana-1834 232 39 x x SYM cana-1834 232 40   NUM cana-1834 232 41   NUM cana-1834 232 42   X cana-1834 233 1   NUM cana-1834 233 2   NUM cana-1834 233 3   X cana-1834 234 1   NUM cana-1834 234 2   X cana-1834 235 1   NUM cana-1834 235 2   PUNCT cana-1834 236 1 = = PUNCT cana-1834 237 1 = = PUNCT cana-1834 238 1 = = NUM cana-1834 238 2   PROPN cana-1834 238 3   NUM cana-1834 238 4   NUM cana-1834 238 5 clearly clearly ADV cana-1834 238 6 r r PROPN cana-1834 238 7 is be AUX cana-1834 238 8 surjective surjective ADJ cana-1834 238 9 and and CCONJ cana-1834 238 10 ( ( PUNCT cana-1834 238 11 , , PUNCT cana-1834 238 12 ) ) PUNCT cana-1834 238 13 r r NOUN cana-1834 238 14 ri ri NOUN cana-1834 238 15 is be AUX cana-1834 238 16 an an DET cana-1834 238 17 epi epi NOUN cana-1834 238 18 - - NOUN cana-1834 238 19 morphism morphism NOUN cana-1834 238 20 . . PUNCT cana-1834 239 1 communications communication NOUN cana-1834 239 2 on on ADP cana-1834 239 3 applied apply VERB cana-1834 239 4 nonlinear nonlinear ADJ cana-1834 239 5 analysis analysis NOUN cana-1834 239 6 issn issn NOUN cana-1834 239 7 : : PUNCT cana-1834 239 8 1074 1074 NUM cana-1834 239 9 - - PUNCT cana-1834 239 10 133x 133x NUM cana-1834 239 11 vol vol NOUN cana-1834 239 12 32 32 NUM cana-1834 239 13 no no NOUN cana-1834 239 14 . . NOUN cana-1834 239 15 2 2 NUM cana-1834 239 16 ( ( PUNCT cana-1834 239 17 2025 2025 NUM cana-1834 239 18 ) ) PUNCT cana-1834 239 19 586 586 NUM cana-1834 239 20 https://internationalpubls.com https://internationalpubls.com X cana-1834 239 21 5 5 NUM cana-1834 239 22 congruences congruence NOUN cana-1834 239 23 and and CCONJ cana-1834 239 24 products product NOUN cana-1834 239 25 of of ADP cana-1834 239 26 ternary ternary ADJ cana-1834 239 27 Ξ³ Ξ³ NOUN cana-1834 239 28 - - NOUN cana-1834 239 29 semirings semiring NOUN cana-1834 239 30 in in ADP cana-1834 239 31 this this DET cana-1834 239 32 section section NOUN cana-1834 239 33 , , PUNCT cana-1834 239 34 we we PRON cana-1834 239 35 show show VERB cana-1834 239 36 how how SCONJ cana-1834 239 37 to to PART cana-1834 239 38 use use VERB cana-1834 239 39 a a DET cana-1834 239 40 ternary ternary ADJ cana-1834 239 41 Ξ³ Ξ³ NOUN cana-1834 239 42 - - NOUN cana-1834 239 43 ideal ideal NOUN cana-1834 239 44 and and CCONJ cana-1834 239 45 a a DET cana-1834 239 46 congruence congruence NOUN cana-1834 239 47 on on ADP cana-1834 239 48 a a DET cana-1834 239 49 ternary ternary ADJ cana-1834 239 50 Ξ³ Ξ³ NOUN cana-1834 239 51 - - PUNCT cana-1834 239 52 semiringrto semiringrto NOUN cana-1834 239 53 construct construct VERB cana-1834 239 54 a a DET cana-1834 239 55 new new ADJ cana-1834 239 56 ternary ternary ADJ cana-1834 239 57 Ξ³ Ξ³ NOUN cana-1834 239 58 - - NOUN cana-1834 239 59 ideal ideal NOUN cana-1834 239 60 of of ADP cana-1834 239 61 rand rand NOUN cana-1834 239 62 to to PART cana-1834 239 63 find find VERB cana-1834 239 64 the the DET cana-1834 239 65 relationship relationship NOUN cana-1834 239 66 between between ADP cana-1834 239 67 them they PRON cana-1834 239 68 . . PUNCT cana-1834 240 1 theorem5.1 theorem5.1 PUNCT cana-1834 240 2 . . PUNCT cana-1834 241 1 let let VERB cana-1834 241 2 ΞΈ ΞΈ NOUN cana-1834 241 3 be be AUX cana-1834 241 4 a a DET cana-1834 241 5 congruence congruence NOUN cana-1834 241 6 on on ADP cana-1834 241 7 ( ( PUNCT cana-1834 241 8 r r NOUN cana-1834 241 9 , , PUNCT cana-1834 241 10 Ξ³ Ξ³ X cana-1834 241 11 , , PUNCT cana-1834 241 12 [ [ PUNCT cana-1834 241 13 ] ] X cana-1834 241 14 ) ) PUNCT cana-1834 241 15 .if .if PUNCT cana-1834 242 1 i i PRON cana-1834 242 2 is be AUX cana-1834 242 3 a a DET cana-1834 242 4 ternary ternary ADJ cana-1834 242 5 Ξ³ Ξ³ X cana-1834 242 6 – – PUNCT cana-1834 242 7 ideal ideal NOUN cana-1834 242 8 of of ADP cana-1834 242 9 ( ( PUNCT cana-1834 242 10 r r NOUN cana-1834 242 11 , , PUNCT cana-1834 242 12 Ξ³ Ξ³ X cana-1834 242 13 , , PUNCT cana-1834 242 14 [ [ X cana-1834 242 15 ] ] X cana-1834 242 16 ) ) PUNCT cana-1834 242 17 , , PUNCT cana-1834 242 18 then then ADV cana-1834 242 19 ci={x ci={x VERB cana-1834 242 20 ∈t| ∈t| PROPN cana-1834 243 1 [ [ X cana-1834 243 2 xΞ³ΞΈΞ³a]βˆƒa xΞ³ΞΈΞ³a]βˆƒa PROPN cana-1834 243 3 ∈i}is ∈i}is PROPN cana-1834 243 4 a a DET cana-1834 243 5 ternary ternary ADJ cana-1834 243 6 Ξ³ Ξ³ NOUN cana-1834 243 7 - - NOUN cana-1834 243 8 ideal ideal NOUN cana-1834 243 9 of of ADP cana-1834 243 10 ( ( PUNCT cana-1834 243 11 r r NOUN cana-1834 243 12 , , PUNCT cana-1834 243 13 Ξ³ Ξ³ X cana-1834 243 14 , , PUNCT cana-1834 243 15 [ [ PUNCT cana-1834 243 16 ] ] X cana-1834 243 17 ) ) PUNCT cana-1834 243 18 and and CCONJ cana-1834 243 19 iβŠ†ci iβŠ†ci PROPN cana-1834 243 20 . . PUNCT cana-1834 244 1 proof proof NOUN cana-1834 244 2 : : PUNCT cana-1834 244 3 clearly clearly ADV cana-1834 244 4 iβŠ†ci iβŠ†ci PROPN cana-1834 244 5 . . PUNCT cana-1834 245 1 let let VERB cana-1834 245 2 x x PRON cana-1834 245 3 , , PUNCT cana-1834 245 4 y∈ci y∈ci ADV cana-1834 245 5 . . PUNCT cana-1834 246 1 then[xΞ³ΞΈΞ³a]and then[xΞ³ΞΈΞ³a]and NOUN cana-1834 247 1 [ [ X cana-1834 247 2 yΞ³ΞΈΞ³b]for yΞ³ΞΈΞ³b]for ADP cana-1834 247 3 some some DET cana-1834 247 4 a a DET cana-1834 247 5 , , PUNCT cana-1834 247 6 b∈i b∈i NOUN cana-1834 247 7 . . PUNCT cana-1834 248 1 on on ADP cana-1834 248 2 the the DET cana-1834 248 3 other other ADJ cana-1834 248 4 hand hand NOUN cana-1834 248 5 , , PUNCT cana-1834 248 6 ΞΈis ΞΈis VERB cana-1834 248 7 a a DET cana-1834 248 8 congruence congruence NOUN cana-1834 248 9 on on ADP cana-1834 248 10 rwhich rwhich PROPN cana-1834 248 11 implies imply VERB cana-1834 248 12 that that SCONJ cana-1834 248 13 [ [ X cana-1834 248 14 ( ( PUNCT cana-1834 248 15 x x X cana-1834 248 16 + + NOUN cana-1834 248 17 y)Ξ³ΞΈΞ³(a+b y)Ξ³ΞΈΞ³(a+b PROPN cana-1834 248 18 ) ) PUNCT cana-1834 248 19 ] ] PUNCT cana-1834 248 20 and and CCONJ cana-1834 248 21 x x X cana-1834 249 1 + + NOUN cana-1834 249 2 y y PROPN cana-1834 249 3 ∈ci ∈ci NOUN cana-1834 249 4 . . PUNCT cana-1834 250 1 now now ADV cana-1834 250 2 , , PUNCT cana-1834 250 3 let let VERB cana-1834 250 4 x∈ci x∈ci PROPN cana-1834 250 5 , , PUNCT cana-1834 250 6 r∈rand r∈rand NOUN cana-1834 250 7 π›ΎβˆˆΞ³ π›ΎβˆˆΞ³ NOUN cana-1834 250 8 . . PUNCT cana-1834 251 1 then then ADV cana-1834 251 2 , , PUNCT cana-1834 251 3 [ [ X cana-1834 251 4 xΞ³ΞΈΞ³a]for xΞ³ΞΈΞ³a]for X cana-1834 251 5 some some DET cana-1834 251 6 a∈i.in a∈i.in PROPN cana-1834 251 7 other other ADJ cana-1834 251 8 words word NOUN cana-1834 251 9 , , PUNCT cana-1834 251 10 ΞΈ ΞΈ PROPN cana-1834 251 11 is be AUX cana-1834 251 12 a a DET cana-1834 251 13 congruence congruence NOUN cana-1834 251 14 on on ADP cana-1834 251 15 t t PROPN cana-1834 251 16 which which PRON cana-1834 251 17 implies imply VERB cana-1834 251 18 that that SCONJ cana-1834 251 19 [ [ X cana-1834 251 20 ( ( PUNCT cana-1834 251 21 xγ𝛾γr)Ξ³ΞΈΞ³(aγ𝛾γr xγ𝛾γr)Ξ³ΞΈΞ³(aγ𝛾γr NOUN cana-1834 251 22 ) ) PUNCT cana-1834 251 23 ] ] PUNCT cana-1834 251 24 . . PUNCT cana-1834 252 1 thus thus ADV cana-1834 252 2 , , PUNCT cana-1834 252 3 [ [ X cana-1834 252 4 x x X cana-1834 252 5 Ξ³ Ξ³ X cana-1834 252 6 𝛾γ 𝛾γ X cana-1834 252 7 r]∈ci r]∈ci PROPN cana-1834 252 8 . . PUNCT cana-1834 253 1 similarly similarly ADV cana-1834 253 2 , , PUNCT cana-1834 253 3 we we PRON cana-1834 253 4 can can AUX cana-1834 253 5 prove prove VERB cana-1834 253 6 that that SCONJ cana-1834 253 7 [ [ X cana-1834 253 8 r r NOUN cana-1834 253 9 Ξ³ Ξ³ X cana-1834 253 10 𝛾γ 𝛾γ NOUN cana-1834 253 11 x x X cana-1834 253 12 ] ] X cana-1834 253 13 ∈ci ∈ci VERB cana-1834 253 14 , , PUNCT cana-1834 253 15 also also ADV cana-1834 253 16 we we PRON cana-1834 253 17 will will AUX cana-1834 253 18 prove prove VERB cana-1834 253 19 that that SCONJ cana-1834 254 1 [ [ X cana-1834 254 2 𝛾 𝛾 X cana-1834 254 3 Ξ³ Ξ³ X cana-1834 254 4 x x X cana-1834 254 5 Ξ³ Ξ³ X cana-1834 254 6 r r X cana-1834 254 7 ] ] X cana-1834 254 8 ci ci NOUN cana-1834 254 9 , , PUNCT cana-1834 254 10 . . PUNCT cana-1834 255 1 therefore therefore ADV cana-1834 255 2 , , PUNCT cana-1834 255 3 ci ci PROPN cana-1834 255 4 is be AUX cana-1834 255 5 an an DET cana-1834 255 6 ternary ternary ADJ cana-1834 255 7 Ξ³ Ξ³ NOUN cana-1834 255 8 -ideal -ideal NOUN cana-1834 255 9 of of ADP cana-1834 255 10 ( ( PUNCT cana-1834 255 11 r r NOUN cana-1834 255 12 , , PUNCT cana-1834 255 13 Ξ³ Ξ³ X cana-1834 255 14 , , PUNCT cana-1834 255 15 [ [ X cana-1834 255 16 ] ] X cana-1834 255 17 ) ) PUNCT cana-1834 255 18 . . PUNCT cana-1834 256 1 by by ADP cana-1834 256 2 using use VERB cana-1834 256 3 the the DET cana-1834 256 4 standard standard ADJ cana-1834 256 5 arguments argument NOUN cana-1834 256 6 , , PUNCT cana-1834 256 7 we we PRON cana-1834 256 8 can can AUX cana-1834 256 9 prove prove VERB cana-1834 256 10 the the DET cana-1834 256 11 following follow VERB cana-1834 256 12 theorem theorem VERB cana-1834 256 13 . . PUNCT cana-1834 257 1 theorem5.2 theorem5.2 NOUN cana-1834 257 2 : : PUNCT cana-1834 257 3 let let VERB cana-1834 257 4 rbe rbe PROPN cana-1834 257 5 a a DET cana-1834 257 6 ternary ternary ADJ cana-1834 257 7 Ξ³ Ξ³ NOUN cana-1834 257 8 -semiring -semiring NOUN cana-1834 257 9 with with ADP cana-1834 257 10 zero zero NUM cana-1834 257 11 and and CCONJ cana-1834 257 12 ΞΈ ΞΈ PROPN cana-1834 257 13 a a DET cana-1834 257 14 congruence congruence NOUN cana-1834 257 15 on on ADP cana-1834 257 16 ( ( PUNCT cana-1834 257 17 r r NOUN cana-1834 257 18 , , PUNCT cana-1834 257 19 Ξ³ Ξ³ X cana-1834 257 20 , , PUNCT cana-1834 257 21 [ [ X cana-1834 257 22 ] ] X cana-1834 257 23 ) ) PUNCT cana-1834 257 24 . . PUNCT cana-1834 258 1 then then ADV cana-1834 258 2 , , PUNCT cana-1834 258 3 ΞΈ ΞΈ PROPN cana-1834 258 4 ( ( PUNCT cana-1834 258 5 0 0 NUM cana-1834 258 6 ) ) PUNCT cana-1834 258 7 is be AUX cana-1834 258 8 an an DET cana-1834 258 9 ternary ternary ADJ cana-1834 258 10 𝛾-ideal 𝛾-ideal NOUN cana-1834 258 11 of of ADP cana-1834 258 12 ( ( PUNCT cana-1834 258 13 r r NOUN cana-1834 258 14 , , PUNCT cana-1834 258 15 Ξ³ Ξ³ X cana-1834 258 16 , , PUNCT cana-1834 258 17 [ [ PUNCT cana-1834 258 18 ] ] X cana-1834 258 19 ) ) PUNCT cana-1834 258 20 .in .in PUNCT cana-1834 258 21 the the DET cana-1834 258 22 next next ADJ cana-1834 258 23 two two NUM cana-1834 258 24 theorems theorem NOUN cana-1834 258 25 , , PUNCT cana-1834 258 26 we we PRON cana-1834 258 27 state state VERB cana-1834 258 28 the the DET cana-1834 258 29 connections connection NOUN cana-1834 258 30 between between ADP cana-1834 258 31 the the DET cana-1834 258 32 ideals ideal NOUN cana-1834 258 33 of of ADP cana-1834 258 34 ( ( PUNCT cana-1834 258 35 r r NOUN cana-1834 258 36 , , PUNCT cana-1834 258 37 Ξ³ Ξ³ X cana-1834 258 38 , , PUNCT cana-1834 258 39 [ [ PUNCT cana-1834 258 40 ] ] X cana-1834 258 41 ) ) PUNCT cana-1834 258 42 and and CCONJ cana-1834 258 43 ( ( PUNCT cana-1834 258 44 r r NOUN cana-1834 258 45 : : PUNCT cana-1834 258 46 ΞΈ ΞΈ NOUN cana-1834 258 47 , , PUNCT cana-1834 258 48 Ξ³ Ξ³ X cana-1834 258 49 , , PUNCT cana-1834 258 50 [ [ X cana-1834 258 51 ] ] X cana-1834 258 52 ) ) PUNCT cana-1834 258 53 . . PUNCT cana-1834 259 1 theorem5.3.if theorem5.3.if NOUN cana-1834 260 1 i i PRON cana-1834 260 2 is be AUX cana-1834 260 3 a a DET cana-1834 260 4 ternary ternary ADJ cana-1834 260 5 Ξ³ Ξ³ NOUN cana-1834 260 6 -ideal -ideal NOUN cana-1834 260 7 of of ADP cana-1834 260 8 ( ( PUNCT cana-1834 260 9 r r NOUN cana-1834 260 10 , , PUNCT cana-1834 260 11 Ξ³ Ξ³ X cana-1834 260 12 , , PUNCT cana-1834 260 13 [ [ PUNCT cana-1834 260 14 ] ] X cana-1834 260 15 ) ) PUNCT cana-1834 260 16 , , PUNCT cana-1834 260 17 then then ADV cana-1834 260 18 i i PRON cana-1834 260 19 : : PUNCT cana-1834 260 20 ΞΈ ΞΈ PROPN cana-1834 260 21 is be AUX cana-1834 260 22 an an DET cana-1834 260 23 ideal ideal NOUN cana-1834 260 24 of of ADP cana-1834 260 25 ( ( PUNCT cana-1834 260 26 r r NOUN cana-1834 260 27 : : PUNCT cana-1834 260 28 ΞΈ ΞΈ NOUN cana-1834 260 29 , , PUNCT cana-1834 260 30 Ξ³ Ξ³ X cana-1834 260 31 , , PUNCT cana-1834 260 32 [ [ X cana-1834 260 33 ] ] X cana-1834 260 34 ) ) PUNCT cana-1834 260 35 . . PUNCT cana-1834 261 1 theorem5.4 theorem5.4 PROPN cana-1834 261 2 : : PUNCT cana-1834 261 3 if if SCONJ cana-1834 261 4 j j PROPN cana-1834 261 5 is be AUX cana-1834 261 6 an an DET cana-1834 261 7 ternary ternary ADJ cana-1834 261 8 Ξ³ Ξ³ NOUN cana-1834 261 9 - - NOUN cana-1834 261 10 ideal ideal NOUN cana-1834 261 11 of of ADP cana-1834 261 12 ( ( PUNCT cana-1834 261 13 r r NOUN cana-1834 261 14 : : PUNCT cana-1834 261 15 ΞΈ ΞΈ NOUN cana-1834 261 16 , , PUNCT cana-1834 261 17 Ξ³ Ξ³ X cana-1834 261 18 , , PUNCT cana-1834 261 19 [ [ X cana-1834 261 20 ] ] X cana-1834 261 21 ) ) PUNCT cana-1834 261 22 , , PUNCT cana-1834 261 23 then then ADV cana-1834 261 24 there there PRON cana-1834 261 25 exists exist VERB cana-1834 261 26 a a DET cana-1834 261 27 ternary ternary ADJ cana-1834 261 28 Ξ³ Ξ³ NOUN cana-1834 261 29 -ideal -ideal NOUN cana-1834 261 30 i i PRON cana-1834 261 31 of of ADP cana-1834 261 32 ( ( PUNCT cana-1834 261 33 r r NOUN cana-1834 261 34 , , PUNCT cana-1834 261 35 Ξ³ Ξ³ X cana-1834 261 36 , , PUNCT cana-1834 261 37 [ [ PUNCT cana-1834 261 38 ] ] X cana-1834 261 39 ) ) PUNCT cana-1834 261 40 such such ADJ cana-1834 261 41 that that SCONJ cana-1834 261 42 j j PROPN cana-1834 261 43 = = VERB cana-1834 261 44 i i NOUN cana-1834 261 45 : : PUNCT cana-1834 261 46 ΞΈproof ΞΈproof NOUN cana-1834 261 47 : : PUNCT cana-1834 261 48 define define VERB cana-1834 261 49 i={x∈r i={x∈r NOUN cana-1834 261 50 / / SYM cana-1834 261 51 ΞΈ(x)∈j ΞΈ(x)∈j ADJ cana-1834 261 52 } } PUNCT cana-1834 261 53 . . PUNCT cana-1834 262 1 then then ADV cana-1834 262 2 wehave wehave PROPN cana-1834 262 3 ΞΈ(x ΞΈ(x PROPN cana-1834 262 4 ) ) PUNCT cana-1834 262 5 ∈j ∈j NUM cana-1834 262 6 β‡’x β‡’x VERB cana-1834 262 7 ∈iβ‡’ΞΈ(x ∈iβ‡’ΞΈ(x NOUN cana-1834 262 8 ) ) PUNCT cana-1834 262 9 ∈i ∈i ADP cana-1834 262 10 : : PUNCT cana-1834 262 11 ΞΈ ΞΈ NOUN cana-1834 262 12 , , PUNCT cana-1834 262 13 andΞΈ(x andΞΈ(x NOUN cana-1834 262 14 ) ) PUNCT cana-1834 262 15 ∈i ∈i ADP cana-1834 262 16 : : PUNCT cana-1834 262 17 ΞΈβ‡’βˆƒa ΞΈβ‡’βˆƒa NOUN cana-1834 262 18 ∈i ∈i ADP cana-1834 262 19 , , PUNCT cana-1834 262 20 ΞΈ(x ΞΈ(x PROPN cana-1834 262 21 ) ) PUNCT cana-1834 262 22 = = SYM cana-1834 262 23 ΞΈ(a ΞΈ(a PROPN cana-1834 262 24 ) ) PUNCT cana-1834 262 25 β‡’ΞΈ(x β‡’ΞΈ(x PROPN cana-1834 262 26 ) ) PUNCT cana-1834 262 27 = = PUNCT cana-1834 263 1 ΞΈ(a)∈j.thus ΞΈ(a)∈j.thus X cana-1834 263 2 , , PUNCT cana-1834 263 3 j j PROPN cana-1834 263 4 = = VERB cana-1834 263 5 i i PROPN cana-1834 263 6 : : PUNCT cana-1834 263 7 ΞΈnow ΞΈnow ADV cana-1834 263 8 , , PUNCT cana-1834 263 9 suppose suppose VERB cana-1834 263 10 that that SCONJ cana-1834 263 11 x x NOUN cana-1834 263 12 , , PUNCT cana-1834 263 13 y∈i y∈i NUM cana-1834 263 14 . . PUNCT cana-1834 264 1 then then ADV cana-1834 264 2 ΞΈ(x),ΞΈ(y)∈j ΞΈ(x),ΞΈ(y)∈j X cana-1834 264 3 and and CCONJ cana-1834 264 4 by by ADP cana-1834 264 5 theorem4.5 theorem4.5 PROPN cana-1834 264 6 , , PUNCT cana-1834 264 7 we we PRON cana-1834 264 8 have have VERB cana-1834 264 9 ΞΈ(x+y)=ΞΈ(x ΞΈ(x+y)=ΞΈ(x PROPN cana-1834 264 10 ) ) PUNCT cana-1834 264 11 βŠ•ΞΈ(y)∈j βŠ•ΞΈ(y)∈j NOUN cana-1834 264 12 . . PUNCT cana-1834 265 1 hence hence ADV cana-1834 265 2 , , PUNCT cana-1834 265 3 x+y∈i x+y∈i PROPN cana-1834 265 4 . . PUNCT cana-1834 266 1 also also ADV cana-1834 266 2 , , PUNCT cana-1834 266 3 assume assume VERB cana-1834 266 4 that that SCONJ cana-1834 266 5 x∈i x∈i PROPN cana-1834 266 6 , , PUNCT cana-1834 266 7 r∈rand r∈rand NOUN cana-1834 266 8 π›ΎβˆˆΞ³ π›ΎβˆˆΞ³ NOUN cana-1834 266 9 .then .then ADV cana-1834 266 10 , , PUNCT cana-1834 266 11 we we PRON cana-1834 266 12 have have VERB cana-1834 266 13 ΞΈ(x)∈j ΞΈ(x)∈j NOUN cana-1834 266 14 and and CCONJ cana-1834 266 15 by by ADP cana-1834 266 16 theorem theorem NOUN cana-1834 266 17 4.5 4.5 NUM cana-1834 266 18 , , PUNCT cana-1834 266 19 we we PRON cana-1834 266 20 have have VERB cana-1834 266 21 ΞΈ([x ΞΈ([x ADJ cana-1834 266 22 γ𝛾γr γ𝛾γr NOUN cana-1834 266 23 ] ] PUNCT cana-1834 266 24 ) ) PUNCT cana-1834 266 25 = = PUNCT cana-1834 267 1 [ [ X cana-1834 267 2 ΞΈ(x)βŠ™ ΞΈ(x)βŠ™ NUM cana-1834 267 3 Ξ³βŠ™π›ΎβŠ™ Ξ³βŠ™π›ΎβŠ™ ADP cana-1834 267 4 Ξ³βŠ™ΞΈ(r Ξ³βŠ™ΞΈ(r NOUN cana-1834 267 5 ) ) PUNCT cana-1834 267 6 ] ] PUNCT cana-1834 267 7 ∈j ∈j NUM cana-1834 267 8 . . PUNCT cana-1834 268 1 hence hence ADV cana-1834 268 2 , , PUNCT cana-1834 268 3 [ [ X cana-1834 268 4 x x X cana-1834 268 5 γ𝛾 γ𝛾 PROPN cana-1834 268 6 Ξ³ Ξ³ X cana-1834 268 7 r r X cana-1834 268 8 ] ] PUNCT cana-1834 268 9 ∈i ∈i NOUN cana-1834 268 10 . . PUNCT cana-1834 269 1 similarly similarly ADV cana-1834 269 2 , , PUNCT cana-1834 269 3 we we PRON cana-1834 269 4 can can AUX cana-1834 269 5 prove prove VERB cana-1834 269 6 that that SCONJ cana-1834 270 1 [ [ X cana-1834 270 2 r r X cana-1834 270 3 γ𝛾 γ𝛾 PROPN cana-1834 270 4 Ξ³ Ξ³ X cana-1834 270 5 x x X cana-1834 270 6 ] ] X cana-1834 270 7 ∈i ∈i NOUN cana-1834 270 8 and and CCONJ cana-1834 270 9 [ [ X cana-1834 270 10 𝛾 𝛾 X cana-1834 270 11 Ξ³ Ξ³ X cana-1834 270 12 x x X cana-1834 270 13 Ξ³ Ξ³ X cana-1834 270 14 r r X cana-1834 270 15 ] ] PUNCT cana-1834 270 16 ∈i ∈i NOUN cana-1834 270 17 .therefore .therefore PUNCT cana-1834 270 18 , , PUNCT cana-1834 270 19 iis iis PROPN cana-1834 270 20 an an DET cana-1834 270 21 ideal ideal NOUN cana-1834 270 22 of of ADP cana-1834 270 23 ( ( PUNCT cana-1834 270 24 r r NOUN cana-1834 270 25 , , PUNCT cana-1834 270 26 Ξ³ Ξ³ X cana-1834 270 27 , , PUNCT cana-1834 270 28 [ [ X cana-1834 270 29 ] ] X cana-1834 270 30 ) ) PUNCT cana-1834 270 31 . . PUNCT cana-1834 271 1 lemma5.5 lemma5.5 NOUN cana-1834 271 2 : : PUNCT cana-1834 271 3 let let VERB cana-1834 271 4 ribe ribe VERB cana-1834 271 5 a a DET cana-1834 271 6 ternary ternary ADJ cana-1834 271 7 Ξ³isemiring(1≀i≀n Ξ³isemiring(1≀i≀n NOUN cana-1834 271 8 ) ) PUNCT cana-1834 271 9 . . PUNCT cana-1834 272 1 then then ADV cana-1834 272 2 , , PUNCT cana-1834 272 3 r1Γ—Β·Β·Β·Γ—rn r1Γ—Β·Β·Β·Γ—rn PROPN cana-1834 272 4 is be AUX cana-1834 272 5 a a DET cana-1834 272 6 ternary ternary ADJ cana-1834 272 7 Ξ³ Ξ³ NOUN cana-1834 272 8 1Γ—Β·Β·Β·Γ—Ξ³n 1Γ—Β·Β·Β·Γ—Ξ³n NUM cana-1834 272 9 semiring.the semiring.the DET cana-1834 272 10 proof proof NOUN cana-1834 272 11 is be AUX cana-1834 272 12 standard standard ADJ cana-1834 272 13 and and CCONJ cana-1834 272 14 we we PRON cana-1834 272 15 hence hence ADV cana-1834 272 16 omit omit VERB cana-1834 272 17 the the DET cana-1834 272 18 details detail NOUN cana-1834 272 19 . . PUNCT cana-1834 273 1 it it PRON cana-1834 273 2 suffices suffice VERB cana-1834 273 3 we we PRON cana-1834 273 4 define(x1,Β·Β·Β·,xn)+(y1,Β·Β·Β·,yn)= define(x1,Β·Β·Β·,xn)+(y1,Β·Β·Β·,yn)= PROPN cana-1834 273 5 ( ( PUNCT cana-1834 273 6 x1+y1,,xn+yn),and x1+y1,,xn+yn),and NOUN cana-1834 273 7 o o X cana-1834 273 8 : : PUNCT cana-1834 273 9 ( ( PUNCT cana-1834 273 10 r1Γ—Β·Β·Β·Γ—rn)Γ— r1Γ—Β·Β·Β·Γ—rn)Γ— X cana-1834 273 11 ( ( PUNCT cana-1834 273 12 Ξ³ Ξ³ PROPN cana-1834 273 13 1Γ—Β·Β·Β·Γ— 1Γ—Β·Β·Β·Γ— NUM cana-1834 273 14 Ξ³ Ξ³ X cana-1834 273 15 n)Γ—(r1Γ—Β·Β·Β·Γ—rn)βˆ’β†’r1Γ—Γ—rn n)Γ—(r1Γ—Β·Β·Β·Γ—rn)βˆ’β†’r1Γ—Γ—rn NOUN cana-1834 273 16 by by ADV cana-1834 273 17 ( ( PUNCT cana-1834 273 18 x1,Β·Β·Β·,xn x1,Β·Β·Β·,xn PROPN cana-1834 273 19 ) ) PUNCT cana-1834 273 20 Ξ³ Ξ³ X cana-1834 273 21 o o X cana-1834 273 22 Ξ³ Ξ³ X cana-1834 273 23 ( ( PUNCT cana-1834 273 24 𝛾1 𝛾1 PROPN cana-1834 273 25 , , PUNCT cana-1834 273 26 Β· Β· PUNCT cana-1834 273 27 Β· Β· PUNCT cana-1834 273 28 Β· Β· PUNCT cana-1834 273 29 , , PUNCT cana-1834 273 30 𝛾 𝛾 NOUN cana-1834 273 31 n n CCONJ cana-1834 273 32 ) ) PUNCT cana-1834 273 33 Ξ³ Ξ³ NOUN cana-1834 273 34 o o X cana-1834 273 35 Ξ³ Ξ³ X cana-1834 273 36 ( ( PUNCT cana-1834 273 37 y1,Β·Β·Β·,yn)=[x1 y1,Β·Β·Β·,yn)=[x1 NOUN cana-1834 273 38 γ𝛾1 γ𝛾1 PROPN cana-1834 273 39 Ξ³ Ξ³ PROPN cana-1834 273 40 y1 y1 PROPN cana-1834 273 41 ] ] PUNCT cana-1834 273 42 , , PUNCT cana-1834 273 43 … … PUNCT cana-1834 273 44 .. .. PUNCT cana-1834 273 45 , , PUNCT cana-1834 273 46 [ [ X cana-1834 273 47 xn xn X cana-1834 273 48 γ𝛾n γ𝛾n VERB cana-1834 273 49 Ξ³yn],for Ξ³yn],for PROPN cana-1834 274 1 all all PRON cana-1834 274 2 ( ( PUNCT cana-1834 274 3 x1,Β·Β·Β·,xn),(y1,Β·Β·Β·,yn)∈t1Γ—Β·Β·Β·Γ—tnand x1,Β·Β·Β·,xn),(y1,Β·Β·Β·,yn)∈t1Γ—Β·Β·Β·Γ—tnand PROPN cana-1834 274 4 ( ( PUNCT cana-1834 274 5 𝛾1,Β·Β·Β·,𝛾n)∈ 𝛾1,Β·Β·Β·,𝛾n)∈ PROPN cana-1834 274 6 Ξ³ Ξ³ PROPN cana-1834 274 7 1Γ— 1Γ— PROPN cana-1834 274 8 … … SYM cana-1834 274 9 .Γ— .Γ— NOUN cana-1834 274 10 Ξ³ Ξ³ X cana-1834 274 11 n. n. PROPN cana-1834 274 12 in in ADP cana-1834 274 13 the the DET cana-1834 274 14 next next ADJ cana-1834 274 15 lemma lemma PROPN cana-1834 274 16 , , PUNCT cana-1834 274 17 we we PRON cana-1834 274 18 investigate investigate VERB cana-1834 274 19 the the DET cana-1834 274 20 behavior behavior NOUN cana-1834 274 21 of of ADP cana-1834 274 22 congruence congruence NOUN cana-1834 274 23 on on ADP cana-1834 274 24 the the DET cana-1834 274 25 products product NOUN cana-1834 274 26 of of ADP cana-1834 274 27 ternary ternary ADJ cana-1834 274 28 Ξ³ Ξ³ NOUN cana-1834 274 29 -semirings -semiring NOUN cana-1834 274 30 . . PUNCT cana-1834 275 1 lemma lemma PROPN cana-1834 275 2 5.6 5.6 NUM cana-1834 275 3 : : PUNCT cana-1834 275 4 let let VERB cana-1834 275 5 ΞΈi ΞΈi PART cana-1834 275 6 be be AUX cana-1834 275 7 a a DET cana-1834 275 8 congruence congruence NOUN cana-1834 275 9 on on ADP cana-1834 275 10 ( ( PUNCT cana-1834 275 11 ri ri NOUN cana-1834 275 12 , , PUNCT cana-1834 275 13 Ξ³i Ξ³i NOUN cana-1834 275 14 , , PUNCT cana-1834 275 15 [ [ PUNCT cana-1834 275 16 ] ] X cana-1834 275 17 ) ) PUNCT cana-1834 275 18 for for ADP cana-1834 275 19 1≀i≀n 1≀i≀n NUM cana-1834 275 20 .t .t PROPN cana-1834 275 21 hen hen NOUN cana-1834 275 22 ΞΈ ΞΈ PROPN cana-1834 275 23 is be AUX cana-1834 275 24 a a DET cana-1834 275 25 congruence congruence NOUN cana-1834 275 26 on on ADP cana-1834 275 27 ( ( PUNCT cana-1834 275 28 t1Γ—Β·Β·Β·Γ—tn t1Γ—Β·Β·Β·Γ—tn ADV cana-1834 275 29 , , PUNCT cana-1834 275 30 Ξ³ Ξ³ PROPN cana-1834 275 31 1Γ—Β·Β·Β·Γ— 1Γ—Β·Β·Β·Γ— NUM cana-1834 275 32 Ξ³ Ξ³ NOUN cana-1834 275 33 n n CCONJ cana-1834 275 34 , , PUNCT cana-1834 275 35 , , PUNCT cana-1834 275 36 [ [ PUNCT cana-1834 275 37 ] ] X cana-1834 275 38 ) ) PUNCT cana-1834 275 39 where where SCONJ cana-1834 275 40 ( ( PUNCT cana-1834 275 41 a1,Β·Β·Β·,an a1,Β·Β·Β·,an NUM cana-1834 275 42 ) ) PUNCT cana-1834 275 43 Ξ³βŠ™ Ξ³βŠ™ NOUN cana-1834 275 44 Ξ³ΞΈ Ξ³ΞΈ PROPN cana-1834 275 45 Ξ³βŠ™Ξ³ Ξ³βŠ™Ξ³ NOUN cana-1834 275 46 ( ( PUNCT cana-1834 275 47 b1…….bn b1…….bn NOUN cana-1834 275 48 ) ) PUNCT cana-1834 276 1 if if SCONJ cana-1834 276 2 and and CCONJ cana-1834 276 3 only only ADV cana-1834 276 4 if if SCONJ cana-1834 276 5 [ [ X cana-1834 276 6 aiΞ³iΞΈiΞ³i aiΞ³iΞΈiΞ³i NOUN cana-1834 276 7 bi]for bi]for ADP cana-1834 276 8 all all PRON cana-1834 276 9 ai ai VERB cana-1834 276 10 , , PUNCT cana-1834 276 11 bi∈tiand bi∈tiand PROPN cana-1834 276 12 1≀i≀n 1≀i≀n NUM cana-1834 276 13 . . PUNCT cana-1834 277 1 proof proof NOUN cana-1834 277 2 : : PUNCT cana-1834 277 3 if if SCONJ cana-1834 277 4 ( ( PUNCT cana-1834 277 5 x1,Β·Β·Β·,xn x1,Β·Β·Β·,xn PROPN cana-1834 277 6 ) ) PUNCT cana-1834 277 7 oΞ³io oΞ³io PROPN cana-1834 277 8 ΞΈΞ³io(y1,Β·Β·Β·,yn ΞΈΞ³io(y1,Β·Β·Β·,yn NOUN cana-1834 277 9 ) ) PUNCT cana-1834 277 10 , , PUNCT cana-1834 277 11 then then ADV cana-1834 277 12 [ [ X cana-1834 277 13 xi xi X cana-1834 277 14 oΞ³ioΞΈioΞ³ioyi]for oΞ³ioΞΈioΞ³ioyi]for ADP cana-1834 277 15 all all DET cana-1834 277 16 1≀i≀n 1≀i≀n NUM cana-1834 277 17 . . PUNCT cana-1834 278 1 hence hence ADV cana-1834 278 2 ( ( PUNCT cana-1834 278 3 xi+zi xi+zi NOUN cana-1834 278 4 ) ) PUNCT cana-1834 278 5 oΞ³ioΞΈi oΞ³ioΞΈi ADJ cana-1834 278 6 oΞ³io oΞ³io NOUN cana-1834 278 7 ( ( PUNCT cana-1834 278 8 yi+zi yi+zi NOUN cana-1834 278 9 ) ) PUNCT cana-1834 278 10 , , PUNCT cana-1834 278 11 for for ADP cana-1834 278 12 all all DET cana-1834 278 13 zi∈tiand zi∈tiand PROPN cana-1834 278 14 1≀i≀n 1≀i≀n NUM cana-1834 278 15 . . PUNCT cana-1834 279 1 this this PRON cana-1834 279 2 implies imply VERB cana-1834 279 3 that(x1,Β·Β·Β·,xn)βˆ“(z1,Β·Β·Β·,zn that(x1,Β·Β·Β·,xn)βˆ“(z1,Β·Β·Β·,zn NOUN cana-1834 279 4 ) ) PUNCT cana-1834 279 5 oΞ³ioΞΈioΞ³i(y1,Β·Β·Β·,yn)βˆ“(z1,Β·Β·Β·,zn).also oΞ³ioΞΈioΞ³i(y1,Β·Β·Β·,yn)βˆ“(z1,Β·Β·Β·,zn).also VERB cana-1834 279 6 , , PUNCT cana-1834 279 7 [ [ X cana-1834 279 8 xi xi X cana-1834 279 9 Ξ³iΞΈiΞ³iyi Ξ³iΞΈiΞ³iyi PROPN cana-1834 279 10 ] ] PUNCT cana-1834 279 11 for for ADP cana-1834 279 12 all all DET cana-1834 279 13 1≀i≀n 1≀i≀n NUM cana-1834 279 14 implies imply VERB cana-1834 279 15 that that SCONJ cana-1834 279 16 [ [ X cana-1834 279 17 xi xi X cana-1834 279 18 oΞ³io𝛾ioΞ³iozi]ΞΈ[yioΞ³io𝛾ioΞ³iozi oΞ³io𝛾ioΞ³iozi]ΞΈ[yioΞ³io𝛾ioΞ³iozi X cana-1834 279 19 ] ] X cana-1834 279 20 for for ADP cana-1834 279 21 all all DET cana-1834 279 22 zi∈ti zi∈ti NUM cana-1834 279 23 , , PUNCT cana-1834 279 24 𝛾iβˆˆπ›Ύiand 𝛾iβˆˆπ›Ύiand NOUN cana-1834 279 25 1 1 NUM cana-1834 279 26 ≀ ≀ NOUN cana-1834 279 27 i≀ i≀ NUM cana-1834 279 28 n. n. NOUN cana-1834 280 1 hence hence ADV cana-1834 280 2 [ [ X cana-1834 280 3 ( ( PUNCT cana-1834 280 4 x1,Β·Β·Β·,xn)oΞ³io(𝛾1,Β·Β·Β·,𝛾n)oΞ³io(z1,Β·Β·Β·,zn x1,Β·Β·Β·,xn)oΞ³io(𝛾1,Β·Β·Β·,𝛾n)oΞ³io(z1,Β·Β·Β·,zn NOUN cana-1834 280 5 ) ) PUNCT cana-1834 280 6 ] ] PUNCT cana-1834 280 7 oΞ³ioΞΈ oΞ³ioΞΈ ADJ cana-1834 280 8 oΞ³io oΞ³io NOUN cana-1834 281 1 [ [ X cana-1834 281 2 ( ( PUNCT cana-1834 281 3 y1,Β·Β·Β·,yn y1,Β·Β·Β·,yn NOUN cana-1834 281 4 ) ) PUNCT cana-1834 281 5 oΞ³io(𝛾1,Β·Β·Β·,𝛾n oΞ³io(𝛾1,Β·Β·Β·,𝛾n ADJ cana-1834 281 6 ) ) PUNCT cana-1834 281 7 oΞ³io(z1,Β·Β·Β·,zn)].similarly oΞ³io(z1,Β·Β·Β·,zn)].similarly ADV cana-1834 281 8 , , PUNCT cana-1834 281 9 we we PRON cana-1834 281 10 can can AUX cana-1834 281 11 prove prove VERB cana-1834 281 12 that that SCONJ cana-1834 281 13 [ [ X cana-1834 281 14 ( ( PUNCT cana-1834 281 15 z1,Β·Β·Β·,zn z1,Β·Β·Β·,zn NUM cana-1834 281 16 ) ) PUNCT cana-1834 281 17 oΞ³io(𝛾1,Β·Β·Β·,𝛾n oΞ³io(𝛾1,Β·Β·Β·,𝛾n ADJ cana-1834 281 18 ) ) PUNCT cana-1834 281 19 oΞ³io(x1,Β·Β·Β·,xn oΞ³io(x1,Β·Β·Β·,xn PROPN cana-1834 281 20 ) ) PUNCT cana-1834 281 21 ] ] PUNCT cana-1834 282 1 ] ] X cana-1834 282 2 oΞ³ioΞΈ oΞ³ioΞΈ ADJ cana-1834 282 3 oΞ³io oΞ³io NOUN cana-1834 282 4 [ [ X cana-1834 282 5 ( ( PUNCT cana-1834 282 6 z1,Β·Β·Β·,zn z1,Β·Β·Β·,zn NUM cana-1834 282 7 ) ) PUNCT cana-1834 282 8 oΞ³io(𝛾1,Β·Β·Β·,𝛾n oΞ³io(𝛾1,Β·Β·Β·,𝛾n ADJ cana-1834 282 9 ) ) PUNCT cana-1834 282 10 communications communication NOUN cana-1834 282 11 on on ADP cana-1834 282 12 applied apply VERB cana-1834 282 13 nonlinear nonlinear ADJ cana-1834 282 14 analysis analysis NOUN cana-1834 282 15 issn issn NOUN cana-1834 282 16 : : PUNCT cana-1834 282 17 1074 1074 NUM cana-1834 282 18 - - PUNCT cana-1834 282 19 133x 133x NUM cana-1834 282 20 vol vol NOUN cana-1834 282 21 32 32 NUM cana-1834 282 22 no no NOUN cana-1834 282 23 . . NOUN cana-1834 282 24 2 2 NUM cana-1834 282 25 ( ( PUNCT cana-1834 282 26 2025 2025 NUM cana-1834 282 27 ) ) PUNCT cana-1834 282 28 587 587 NUM cana-1834 282 29 https://internationalpubls.com https://internationalpubls.com X cana-1834 282 30 oΞ³io(y1,Β·Β·Β·,yn)]also oΞ³io(y1,Β·Β·Β·,yn)]also X cana-1834 283 1 [ [ X cana-1834 283 2 ( ( PUNCT cana-1834 283 3 y1,Β·Β·Β·,yn)oΞ³io(𝛾1,Β·Β·Β·,𝛾n y1,Β·Β·Β·,yn)oΞ³io(𝛾1,Β·Β·Β·,𝛾n NOUN cana-1834 283 4 ) ) PUNCT cana-1834 283 5 oΞ³io(x1,Β·Β·Β·,xn oΞ³io(x1,Β·Β·Β·,xn PROPN cana-1834 283 6 ) ) PUNCT cana-1834 283 7 ] ] PUNCT cana-1834 284 1 ] ] X cana-1834 284 2 oΞ³ioΞΈ oΞ³ioΞΈ ADJ cana-1834 284 3 oΞ³io oΞ³io NOUN cana-1834 284 4 [ [ X cana-1834 284 5 ( ( PUNCT cana-1834 284 6 z1,Β·Β·Β·,zn z1,Β·Β·Β·,zn NUM cana-1834 284 7 ) ) PUNCT cana-1834 284 8 oΞ³io(𝛾1,Β·Β·Β·,𝛾n oΞ³io(𝛾1,Β·Β·Β·,𝛾n ADJ cana-1834 284 9 ) ) PUNCT cana-1834 284 10 oΞ³io(y1,Β·Β·Β·,yn)]therefore oΞ³io(y1,Β·Β·Β·,yn)]therefore ADV cana-1834 284 11 , , PUNCT cana-1834 284 12 ΞΈ ΞΈ PROPN cana-1834 284 13 is be AUX cana-1834 284 14 a a DET cana-1834 284 15 congruence congruence NOUN cana-1834 284 16 on on ADP cana-1834 284 17 ( ( PUNCT cana-1834 284 18 r1Γ—Β·Β·Β·Γ—rn r1Γ—Β·Β·Β·Γ—rn PROPN cana-1834 284 19 , , PUNCT cana-1834 284 20 Ξ³1Γ—Β·Β·Β·Γ—Ξ³n Ξ³1Γ—Β·Β·Β·Γ—Ξ³n NOUN cana-1834 284 21 , , PUNCT cana-1834 284 22 [ [ PUNCT cana-1834 284 23 ] ] X cana-1834 284 24 ) ) PUNCT cana-1834 284 25 . . PUNCT cana-1834 285 1 6 6 NUM cana-1834 285 2 homomorphism homomorphism NOUN cana-1834 285 3 theorems theorem NOUN cana-1834 285 4 and and CCONJ cana-1834 285 5 isomorphism isomorphism NOUN cana-1834 285 6 theorems theorem NOUN cana-1834 285 7 of of ADP cana-1834 285 8 a a DET cana-1834 285 9 ternary ternary ADJ cana-1834 285 10 Ξ³ Ξ³ X cana-1834 285 11 – – PUNCT cana-1834 285 12 semiring semire VERB cana-1834 285 13 in in ADP cana-1834 285 14 the the DET cana-1834 285 15 following following NOUN cana-1834 285 16 theorem theorem NOUN cana-1834 285 17 , , PUNCT cana-1834 285 18 we we PRON cana-1834 285 19 prove prove VERB cana-1834 285 20 an an DET cana-1834 285 21 isomorphism isomorphism NOUN cana-1834 285 22 theorem theorem NOUN cana-1834 285 23 of of ADP cana-1834 285 24 products product NOUN cana-1834 285 25 of of ADP cana-1834 285 26 ternary ternary ADJ cana-1834 285 27 Ξ³semirings Ξ³semiring NOUN cana-1834 285 28 . . PUNCT cana-1834 286 1 theorem6.1 theorem6.1 NUM cana-1834 286 2 : : PUNCT cana-1834 286 3 let let AUX cana-1834 286 4 ΞΈibe ΞΈibe VERB cana-1834 286 5 a a DET cana-1834 286 6 congruence congruence NOUN cana-1834 286 7 on on ADP cana-1834 286 8 ( ( PUNCT cana-1834 286 9 ri ri NOUN cana-1834 286 10 , , PUNCT cana-1834 286 11 Ξ³i Ξ³i NOUN cana-1834 286 12 , , PUNCT cana-1834 286 13 [ [ PUNCT cana-1834 286 14 ] ] X cana-1834 286 15 ) ) PUNCT cana-1834 286 16 for for ADP cana-1834 286 17 1≀i≀n 1≀i≀n NUM cana-1834 286 18 and and CCONJ cana-1834 286 19 ΞΈ ΞΈ PRON cana-1834 286 20 the the DET cana-1834 286 21 congruence congruence NOUN cana-1834 286 22 on on ADP cana-1834 286 23 ( ( PUNCT cana-1834 286 24 r1Γ— r1Γ— X cana-1834 286 25 Β· Β· PUNCT cana-1834 286 26 Β· Β· PUNCT cana-1834 286 27 Β· Β· SYM cana-1834 286 28 Γ— Γ— PROPN cana-1834 286 29 rn rn PROPN cana-1834 286 30 , , PUNCT cana-1834 286 31 Ξ³1Γ— Ξ³1Γ— X cana-1834 286 32 Β· Β· PUNCT cana-1834 286 33 Β· Β· PUNCT cana-1834 286 34 Β· Β· SYM cana-1834 286 35 Γ— Γ— PROPN cana-1834 286 36 Ξ³n Ξ³n NUM cana-1834 286 37 , , PUNCT cana-1834 286 38 [ [ PUNCT cana-1834 286 39 ] ] X cana-1834 286 40 ) ) PUNCT cana-1834 286 41 defined define VERB cana-1834 286 42 in in ADP cana-1834 286 43 lemma lemma PROPN cana-1834 286 44 5.6 5.6 NUM cana-1834 286 45 . . PUNCT cana-1834 287 1 then then ADV cana-1834 287 2 ( ( PUNCT cana-1834 287 3 r1 r1 PROPN cana-1834 287 4 : : PUNCT cana-1834 287 5 ΞΈ1)Γ—Β·Β·Β·Γ—(rn ΞΈ1)Γ—Β·Β·Β·Γ—(rn NUM cana-1834 287 6 : : PUNCT cana-1834 287 7 ΞΈn),Ξ³1Γ—Β·Β·Β·Γ—Ξ³n ΞΈn),Ξ³1Γ—Β·Β·Β·Γ—Ξ³n VERB cana-1834 287 8 ∼ ∼ NOUN cana-1834 287 9 =( =( NOUN cana-1834 287 10 r1Γ—Β·Β·Β·Γ—rn r1Γ—Β·Β·Β·Γ—rn PROPN cana-1834 287 11 : : PUNCT cana-1834 287 12 ΞΈ ΞΈ NOUN cana-1834 287 13 , , PUNCT cana-1834 287 14 Ξ³1Γ— Ξ³1Γ— X cana-1834 287 15 … … PUNCT cana-1834 287 16 .Γ—Ξ³n .Γ—Ξ³n PUNCT cana-1834 287 17 ) ) PUNCT cana-1834 287 18 . . PUNCT cana-1834 288 1 proof proof NOUN cana-1834 288 2 : : PUNCT cana-1834 288 3 by by ADP cana-1834 288 4 theorem theorem VERB cana-1834 288 5 4.5 4.5 NUM cana-1834 288 6 and and CCONJ cana-1834 288 7 lemmas lemma VERB cana-1834 288 8 5.5 5.5 NUM cana-1834 288 9 and and CCONJ cana-1834 288 10 5.6 5.6 NUM cana-1834 288 11 , , PUNCT cana-1834 288 12 ( ( PUNCT cana-1834 288 13 r1 r1 PROPN cana-1834 288 14 : : PUNCT cana-1834 288 15 ΞΈ1)Γ—Β·Β·Β·Γ—(rn ΞΈ1)Γ—Β·Β·Β·Γ—(rn NUM cana-1834 288 16 : : PUNCT cana-1834 288 17 ΞΈn ΞΈn NOUN cana-1834 288 18 ) ) PUNCT cana-1834 288 19 and and CCONJ cana-1834 288 20 r1Γ—Β·Β·Β·Γ—rn r1Γ—Β·Β·Β·Γ—rn PROPN cana-1834 288 21 : : PUNCT cana-1834 288 22 ΞΈ ΞΈ NOUN cana-1834 288 23 are be AUX cana-1834 288 24 Ξ³1Γ—Β·Β·Β·Γ—Ξ³n Ξ³1Γ—Β·Β·Β·Γ—Ξ³n NOUN cana-1834 288 25 - - NOUN cana-1834 288 26 semirings semiring NOUN cana-1834 288 27 . . PUNCT cana-1834 289 1 define define VERB cana-1834 289 2 ψ:(r1 ψ:(r1 NOUN cana-1834 289 3 : : PUNCT cana-1834 289 4 ΞΈ1)Γ—Β·Β·Β·Γ—(rn ΞΈ1)Γ—Β·Β·Β·Γ—(rn NUM cana-1834 289 5 : : PUNCT cana-1834 289 6 ΞΈn)β†’r1Γ— ΞΈn)β†’r1Γ— ADJ cana-1834 289 7 … … PUNCT cana-1834 289 8 .. .. PUNCT cana-1834 289 9 Γ—rn Γ—rn NOUN cana-1834 289 10 : : PUNCT cana-1834 289 11 ΞΈ ΞΈ NOUN cana-1834 289 12 by by ADP cana-1834 289 13 : : PUNCT cana-1834 289 14 ψ ψ X cana-1834 289 15 ( ( PUNCT cana-1834 289 16 ΞΈ1(x1),Β·Β·Β·,ΞΈn(xn)=ΞΈ(x1, ΞΈ1(x1),Β·Β·Β·,ΞΈn(xn)=ΞΈ(x1, NOUN cana-1834 289 17 ... ... PUNCT cana-1834 289 18 ,xn)),for ,xn)),for PUNCT cana-1834 289 19 all all DET cana-1834 289 20 xi∈ti(1≀i≀n xi∈ti(1≀i≀n PROPN cana-1834 289 21 ) ) PUNCT cana-1834 289 22 . . PUNCT cana-1834 290 1 we we PRON cana-1834 290 2 can can AUX cana-1834 290 3 show show VERB cana-1834 290 4 that that SCONJ cana-1834 290 5 ( ( PUNCT cana-1834 290 6 ψ,1Ξ³1Γ—Β·Β·Β·Γ—Ξ³n ψ,1Ξ³1Γ—Β·Β·Β·Γ—Ξ³n NOUN cana-1834 290 7 ) ) PUNCT cana-1834 290 8 is be AUX cana-1834 290 9 an an DET cana-1834 290 10 isomorphism isomorphism NOUN cana-1834 290 11 between between ADP cana-1834 290 12 ( ( PUNCT cana-1834 290 13 ( ( PUNCT cana-1834 290 14 r1 r1 PROPN cana-1834 290 15 : : PUNCT cana-1834 290 16 ΞΈ1)Γ—Β·Β·Β·Γ—(rn ΞΈ1)Γ—Β·Β·Β·Γ—(rn ADP cana-1834 290 17 : : PUNCT cana-1834 290 18 ΞΈn ΞΈn ADJ cana-1834 290 19 ) ) PUNCT cana-1834 290 20 , , PUNCT cana-1834 290 21 Ξ³1Γ—Β·Β·Β·Γ—rn Ξ³1Γ—Β·Β·Β·Γ—rn PROPN cana-1834 290 22 , , PUNCT cana-1834 290 23 [ [ PUNCT cana-1834 290 24 ] ] X cana-1834 290 25 ) ) PUNCT cana-1834 290 26 and and CCONJ cana-1834 290 27 ( ( PUNCT cana-1834 290 28 r1Γ—Β·Β·Β·Γ—rn r1Γ—Β·Β·Β·Γ—rn ADJ cana-1834 290 29 : : SYM cana-1834 290 30 ΞΈ ΞΈ NOUN cana-1834 290 31 , , PUNCT cana-1834 290 32 Ξ³1Γ—Β·Β·Β·Γ—Ξ³n Ξ³1Γ—Β·Β·Β·Γ—Ξ³n NOUN cana-1834 290 33 , , PUNCT cana-1834 290 34 [ [ PUNCT cana-1834 290 35 ] ] X cana-1834 290 36 ) ) PUNCT cana-1834 290 37 . . PUNCT cana-1834 291 1 we we PRON cana-1834 291 2 have have VERB cana-1834 291 3 ΞΈ1(x1),Β·Β·Β·,ΞΈn(xn ΞΈ1(x1),Β·Β·Β·,ΞΈn(xn NOUN cana-1834 291 4 ) ) PUNCT cana-1834 291 5 ⇐ ⇐ ADJ cana-1834 291 6 β‡’ β‡’ PROPN cana-1834 291 7 ΞΈ1(y1 ΞΈ1(y1 NUM cana-1834 291 8 ) ) PUNCT cana-1834 291 9 , , PUNCT cana-1834 291 10 Β· Β· PUNCT cana-1834 291 11 Β· Β· PUNCT cana-1834 291 12 Β· Β· PUNCT cana-1834 291 13 , , PUNCT cana-1834 291 14 ΞΈn(yn ΞΈn(yn NOUN cana-1834 291 15 ) ) PUNCT cana-1834 291 16 ⇐ ⇐ ADJ cana-1834 291 17 β‡’ β‡’ NOUN cana-1834 291 18 ΞΈi(xi)=ΞΈi(yi ΞΈi(xi)=ΞΈi(yi ADV cana-1834 291 19 ) ) PUNCT cana-1834 291 20 , , PUNCT cana-1834 291 21 βˆ€1≀i≀n βˆ€1≀i≀n X cana-1834 291 22 ⇐ ⇐ ADJ cana-1834 291 23 β‡’ β‡’ PROPN cana-1834 291 24 xiΞ³iΞΈiΞ³iyi xiΞ³iΞΈiΞ³iyi PROPN cana-1834 291 25 , , PUNCT cana-1834 291 26 βˆ€1≀i≀n βˆ€1≀i≀n X cana-1834 291 27 ⇐ ⇐ ADJ cana-1834 291 28 β‡’ β‡’ NOUN cana-1834 291 29 ( ( PUNCT cana-1834 291 30 x1Β·Β·Β·xn)Ξ³iΞΈΞ³i(y1Β·Β·Β·yn x1Β·Β·Β·xn)Ξ³iΞΈΞ³i(y1Β·Β·Β·yn NUM cana-1834 291 31 ) ) PUNCT cana-1834 291 32 ⇐ ⇐ ADJ cana-1834 291 33 β‡’ β‡’ PROPN cana-1834 291 34 ΞΈ(x1Β·Β·Β·xn)=ΞΈ(y1Β·Β·Β·yn ΞΈ(x1Β·Β·Β·xn)=ΞΈ(y1Β·Β·Β·yn NOUN cana-1834 291 35 ) ) PUNCT cana-1834 291 36 ⇐ ⇐ ADJ cana-1834 291 37 β‡’ β‡’ NOUN cana-1834 291 38 ψθ1(x1),Β·Β·Β·,ΞΈn(xn)=ψθ1(y1),Β·Β·Β·,ΞΈn(yn ψθ1(x1),Β·Β·Β·,ΞΈn(xn)=ψθ1(y1),Β·Β·Β·,ΞΈn(yn PROPN cana-1834 291 39 ) ) PUNCT cana-1834 291 40 . . PUNCT cana-1834 292 1 hence hence ADV cana-1834 292 2 , , PUNCT cana-1834 292 3 ( ( PUNCT cana-1834 292 4 ψ,1Ξ³1Γ—Β·Β·Β·Γ—Ξ³n ψ,1Ξ³1Γ—Β·Β·Β·Γ—Ξ³n NOUN cana-1834 292 5 ) ) PUNCT cana-1834 292 6 is be AUX cana-1834 292 7 well well ADV cana-1834 292 8 - - PUNCT cana-1834 292 9 defined define VERB cana-1834 292 10 and and CCONJ cana-1834 292 11 one one NUM cana-1834 292 12 to to ADP cana-1834 292 13 one one NUM cana-1834 292 14 . . PUNCT cana-1834 293 1 clearly clearly ADV cana-1834 293 2 , , PUNCT cana-1834 293 3 ( ( PUNCT cana-1834 293 4 ψ,1Ξ³1Γ—Β·Β·Β·Γ—Ξ³n)is ψ,1Ξ³1Γ—Β·Β·Β·Γ—Ξ³n)is ADJ cana-1834 293 5 onto onto ADP cana-1834 293 6 . . PUNCT cana-1834 294 1 now now ADV cana-1834 294 2 , , PUNCT cana-1834 294 3 we we PRON cana-1834 294 4 prove prove VERB cana-1834 294 5 that that SCONJ cana-1834 294 6 ( ( PUNCT cana-1834 294 7 ψ ψ X cana-1834 294 8 , , PUNCT cana-1834 294 9 Ξ³1Γ—Β·Β·Β·Γ—Ξ³n Ξ³1Γ—Β·Β·Β·Γ—Ξ³n NOUN cana-1834 294 10 ) ) PUNCT cana-1834 294 11 is be AUX cana-1834 294 12 a a DET cana-1834 294 13 homomorphism homomorphism NOUN cana-1834 294 14 . . PUNCT cana-1834 295 1 we we PRON cana-1834 295 2 have have VERB cana-1834 295 3 ψ((ΞΈ1(x1),Β·Β·Β·,ΞΈn(xn))βˆ“(ΞΈ1(y1),Β·Β·Β·,ΞΈn(yn)))=ψ(ΞΈ1(x1)βŠ•ΞΈ1(y1),Β·Β·Β·,ΞΈn(xn)βŠ•ΞΈn(yn))= ψ((ΞΈ1(x1),Β·Β·Β·,ΞΈn(xn))βˆ“(ΞΈ1(y1),Β·Β·Β·,ΞΈn(yn)))=ψ(ΞΈ1(x1)βŠ•ΞΈ1(y1),Β·Β·Β·,ΞΈn(xn)βŠ•ΞΈn(yn))= VERB cana-1834 295 4 ψ(ΞΈ1(x1+y1),Β·Β·Β·,ΞΈn(xn+yn))= ψ(ΞΈ1(x1+y1),Β·Β·Β·,ΞΈn(xn+yn))= PROPN cana-1834 295 5 ΞΈ((x1 ΞΈ((x1 NOUN cana-1834 295 6 + + CCONJ cana-1834 295 7 y1 y1 NOUN cana-1834 295 8 ) ) PUNCT cana-1834 295 9 , , PUNCT cana-1834 295 10 Β· Β· PUNCT cana-1834 295 11 Β· Β· PUNCT cana-1834 295 12 Β· Β· PUNCT cana-1834 295 13 , , PUNCT cana-1834 295 14 ( ( PUNCT cana-1834 295 15 xn+ xn+ X cana-1834 295 16 yn))= yn))= X cana-1834 295 17 ΞΈ(x1Β·Β·Β·,xn)βŠ•ΞΈ(y1Β·Β·Β·,yn)= ΞΈ(x1Β·Β·Β·,xn)βŠ•ΞΈ(y1Β·Β·Β·,yn)= ADP cana-1834 295 18 ψ(ΞΈ1(x1),Β·Β·Β·,ΞΈn(xn))βŠ•Οˆ(ΞΈ1(y1),Β·Β·Β·,ΞΈn(yn ψ(ΞΈ1(x1),Β·Β·Β·,ΞΈn(xn))βŠ•Οˆ(ΞΈ1(y1),Β·Β·Β·,ΞΈn(yn NOUN cana-1834 295 19 ) ) PUNCT cana-1834 295 20 ) ) PUNCT cana-1834 295 21 . . PUNCT cana-1834 296 1 also also ADV cana-1834 296 2 , , PUNCT cana-1834 296 3 we we PRON cana-1834 296 4 have have VERB cana-1834 296 5 ψ((ΞΈ1(x1),Β·Β·Β·,ΞΈn(xn))o(𝛾1,Β·Β·Β·,𝛾n)o(ΞΈ1(y1),Β·Β·Β·,ΞΈn(yn)))=ψ(ΞΈ1(x1)βŠ™Ξ³βŠ™π›Ύ1βŠ™Ξ³βŠ™ΞΈ1(y1),Β·Β·Β·,ΞΈn(xn)βŠ™Ξ³βŠ™π›ΎnβŠ™Ξ³ ψ((ΞΈ1(x1),Β·Β·Β·,ΞΈn(xn))o(𝛾1,Β·Β·Β·,𝛾n)o(ΞΈ1(y1),Β·Β·Β·,ΞΈn(yn)))=ψ(ΞΈ1(x1)βŠ™Ξ³βŠ™π›Ύ1βŠ™Ξ³βŠ™ΞΈ1(y1),Β·Β·Β·,ΞΈn(xn)βŠ™Ξ³βŠ™π›ΎnβŠ™Ξ³ PROPN cana-1834 296 6 βŠ™ΞΈn(yn))=ψ(ΞΈ1[x1Ξ³1y1],Β·Β·Β·,ΞΈn[xn𝛾nyn])=ΞΈ[x1𝛾1y1,Β·Β·Β·,xn𝛾nyn)=ΞΈ([x1,Β·Β·Β·xn])βŠ™Ξ³βŠ™(𝛾1,Β·Β·Β·,𝛾1)βŠ™Ξ³βŠ™ΞΈ([y1 βŠ™ΞΈn(yn))=ψ(ΞΈ1[x1Ξ³1y1],Β·Β·Β·,ΞΈn[xn𝛾nyn])=ΞΈ[x1𝛾1y1,Β·Β·Β·,xn𝛾nyn)=ΞΈ([x1,Β·Β·Β·xn])βŠ™Ξ³βŠ™(𝛾1,Β·Β·Β·,𝛾1)βŠ™Ξ³βŠ™ΞΈ([y1 PROPN cana-1834 296 7 , , PUNCT cana-1834 296 8 Β· Β· PUNCT cana-1834 296 9 Β· Β· PUNCT cana-1834 296 10 Β· Β· PUNCT cana-1834 296 11 , , PUNCT cana-1834 296 12 yn])= yn])= PROPN cana-1834 296 13 ψ(ΞΈ1(x1),Β·Β·Β·,ΞΈn(xn))βŠ™Ξ³βŠ™1𝛾1×···×𝛾n(𝛾1,Β·Β·Β·,𝛾n)βŠ™Ξ³βŠ™Οˆ(ΞΈ1(y1),Β·Β·Β·,ΞΈn(yn ψ(ΞΈ1(x1),Β·Β·Β·,ΞΈn(xn))βŠ™Ξ³βŠ™1𝛾1×···×𝛾n(𝛾1,Β·Β·Β·,𝛾n)βŠ™Ξ³βŠ™Οˆ(ΞΈ1(y1),Β·Β·Β·,ΞΈn(yn PROPN cana-1834 296 14 ) ) PUNCT cana-1834 296 15 ) ) PUNCT cana-1834 296 16 . . PUNCT cana-1834 297 1 therefore therefore ADV cana-1834 297 2 , , PUNCT cana-1834 297 3 ( ( PUNCT cana-1834 297 4 ψ,1𝛾1×···×𝛾n)is ψ,1𝛾1×···×𝛾n)is X cana-1834 297 5 an an DET cana-1834 297 6 isomorphism isomorphism NOUN cana-1834 297 7 . . PUNCT cana-1834 298 1 in in ADP cana-1834 298 2 the the DET cana-1834 298 3 next next ADJ cana-1834 298 4 theorems theorem NOUN cana-1834 298 5 , , PUNCT cana-1834 298 6 we we PRON cana-1834 298 7 consider consider VERB cana-1834 298 8 the the DET cana-1834 298 9 congruence congruence NOUN cana-1834 298 10 on on ADP cana-1834 298 11 the the DET cana-1834 298 12 ternary ternary ADJ cana-1834 298 13 𝛾-semiringst 𝛾-semiringst NOUN cana-1834 298 14 induced induce VERB cana-1834 298 15 by by ADP cana-1834 298 16 the the DET cana-1834 298 17 homomorphisms homomorphism NOUN cana-1834 298 18 and and CCONJ cana-1834 298 19 investigate investigate VERB cana-1834 298 20 the the DET cana-1834 298 21 corresponding corresponding ADJ cana-1834 298 22 results result NOUN cana-1834 298 23 and and CCONJ cana-1834 298 24 properties property NOUN cana-1834 298 25 associated associate VERB cana-1834 298 26 with with ADP cana-1834 298 27 this this DET cana-1834 298 28 congruence congruence NOUN cana-1834 298 29 on on ADP cana-1834 298 30 r. r. PROPN cana-1834 298 31 communications communication NOUN cana-1834 298 32 on on ADP cana-1834 298 33 applied apply VERB cana-1834 298 34 nonlinear nonlinear ADJ cana-1834 298 35 analysis analysis NOUN cana-1834 298 36 issn issn NOUN cana-1834 298 37 : : PUNCT cana-1834 298 38 1074 1074 NUM cana-1834 298 39 - - PUNCT cana-1834 298 40 133x 133x NUM cana-1834 298 41 vol vol NOUN cana-1834 298 42 32 32 NUM cana-1834 298 43 no no NOUN cana-1834 298 44 . . NOUN cana-1834 298 45 2 2 NUM cana-1834 298 46 ( ( PUNCT cana-1834 298 47 2025 2025 NUM cana-1834 298 48 ) ) PUNCT cana-1834 298 49 588 588 NUM cana-1834 298 50 https://internationalpubls.com https://internationalpubls.com X cana-1834 298 51 22 22 NUM cana-1834 298 52 theorem6.2 theorem6.2 NOUN cana-1834 298 53 . . PUNCT cana-1834 299 1 let let VERB cana-1834 299 2 ( ( PUNCT cana-1834 299 3 Ο•,g Ο•,g NOUN cana-1834 299 4 ) ) PUNCT cana-1834 299 5 : : PUNCT cana-1834 299 6 ( ( PUNCT cana-1834 299 7 r1,Ξ³1 r1,Ξ³1 PROPN cana-1834 299 8 , , PUNCT cana-1834 299 9 [ [ PUNCT cana-1834 299 10 ] ] X cana-1834 299 11 ) ) PUNCT cana-1834 299 12 βˆ’β†’(r2,Ξ³2 βˆ’β†’(r2,Ξ³2 X cana-1834 299 13 , , PUNCT cana-1834 299 14 [ [ PUNCT cana-1834 299 15 ] ] X cana-1834 299 16 ) ) PUNCT cana-1834 299 17 be be AUX cana-1834 299 18 a a DET cana-1834 299 19 homomorphism homomorphism NOUN cana-1834 299 20 . . PUNCT cana-1834 300 1 define define VERB cana-1834 300 2 the the DET cana-1834 300 3 relation relation NOUN cana-1834 300 4 ΞΈ(Ο•,g ΞΈ(Ο•,g NUM cana-1834 300 5 ) ) PUNCT cana-1834 300 6 on on ADP cana-1834 300 7 ( ( PUNCT cana-1834 300 8 r1,Ξ³1 r1,Ξ³1 PROPN cana-1834 300 9 ) ) PUNCT cana-1834 300 10 as as ADP cana-1834 300 11 follows:[xΞ³1ΞΈ(Ο•,g)Ξ³1y] follows:[xΞ³1ΞΈ(Ο•,g)Ξ³1y] NOUN cana-1834 300 12 ⇐ ⇐ PROPN cana-1834 300 13 β‡’Ο•(x)=Ο•(y β‡’Ο•(x)=Ο•(y PROPN cana-1834 300 14 ) ) PUNCT cana-1834 300 15 . . PUNCT cana-1834 301 1 then then ADV cana-1834 301 2 ΞΈ(Ο•,g ΞΈ(Ο•,g NUM cana-1834 301 3 ) ) PUNCT cana-1834 301 4 is be AUX cana-1834 301 5 a a DET cana-1834 301 6 congruence congruence NOUN cana-1834 301 7 on on ADP cana-1834 301 8 ( ( PUNCT cana-1834 301 9 r1Ξ³1 r1Ξ³1 ADV cana-1834 301 10 , , PUNCT cana-1834 301 11 [ [ PUNCT cana-1834 301 12 ] ] X cana-1834 301 13 ) ) PUNCT cana-1834 301 14 . . PUNCT cana-1834 302 1 proof proof NOUN cana-1834 302 2 . . PUNCT cana-1834 303 1 clearly clearly ADV cana-1834 303 2 , , PUNCT cana-1834 303 3 ΞΈ(Ο•,g ΞΈ(Ο•,g NUM cana-1834 303 4 ) ) PUNCT cana-1834 303 5 is be AUX cana-1834 303 6 an an DET cana-1834 303 7 equivalence equivalence NOUN cana-1834 303 8 relation relation NOUN cana-1834 303 9 . . PUNCT cana-1834 304 1 suppose suppose VERB cana-1834 304 2 that that SCONJ cana-1834 304 3 xΞΈ(Ο•,g)y xΞΈ(Ο•,g)y PROPN cana-1834 304 4 . . PUNCT cana-1834 305 1 we we PRON cana-1834 305 2 have have VERB cana-1834 305 3 Ο•(x)=Ο•(y Ο•(x)=Ο•(y NUM cana-1834 305 4 ) ) PUNCT cana-1834 306 1 = = NOUN cana-1834 306 2 β‡’Ο•(x)+Ο•(z)= β‡’Ο•(x)+Ο•(z)= X cana-1834 306 3 Ο•(y)+Ο•(z Ο•(y)+Ο•(z NOUN cana-1834 306 4 ) ) PUNCT cana-1834 307 1 = = NOUN cana-1834 307 2 β‡’Ο•(x+z)= β‡’Ο•(x+z)= NOUN cana-1834 307 3 Ο•(y+z Ο•(y+z NUM cana-1834 307 4 ) ) PUNCT cana-1834 307 5 for for ADP cana-1834 307 6 all all DET cana-1834 307 7 z∈r1.thus z∈r1.thus NUM cana-1834 307 8 [ [ X cana-1834 307 9 ( ( PUNCT cana-1834 307 10 x+z)Ξ³1ΞΈ(Ο•,g)Ξ³1(y+z x+z)Ξ³1ΞΈ(Ο•,g)Ξ³1(y+z ADJ cana-1834 307 11 ) ) PUNCT cana-1834 307 12 ] ] PUNCT cana-1834 307 13 . . PUNCT cana-1834 308 1 also also ADV cana-1834 308 2 , , PUNCT cana-1834 308 3 we we PRON cana-1834 308 4 have have VERB cana-1834 308 5 Ο•(x)=Ο•(y)=β‡’[Ο•(x)Ξ³1g(𝛾)Ξ³1Ο•(z)]=[Ο•(y)Ξ³1g(𝛾)Ξ³1Ο•(z)]=β‡’Ο•([xΞ³1𝛾γ1z])=Ο•([yΞ³1𝛾γ1z Ο•(x)=Ο•(y)=β‡’[Ο•(x)Ξ³1g(𝛾)Ξ³1Ο•(z)]=[Ο•(y)Ξ³1g(𝛾)Ξ³1Ο•(z)]=β‡’Ο•([xΞ³1𝛾γ1z])=Ο•([yΞ³1𝛾γ1z NUM cana-1834 308 6 ] ] PUNCT cana-1834 308 7 ) ) PUNCT cana-1834 308 8 for for ADP cana-1834 308 9 all all DET cana-1834 308 10 z∈r1 z∈r1 NOUN cana-1834 308 11 and and CCONJ cana-1834 308 12 π›ΎβˆˆΞ³1 π›ΎβˆˆΞ³1 NOUN cana-1834 308 13 . . PUNCT cana-1834 309 1 therefore therefore ADV cana-1834 309 2 , , PUNCT cana-1834 309 3 ΞΈ(Ο•,g ΞΈ(Ο•,g NUM cana-1834 309 4 ) ) PUNCT cana-1834 309 5 is be AUX cana-1834 309 6 a a DET cana-1834 309 7 congruence congruence NOUN cana-1834 309 8 on(r1,Ξ³1 on(r1,Ξ³1 NOUN cana-1834 309 9 , , PUNCT cana-1834 309 10 [ [ PUNCT cana-1834 309 11 ] ] X cana-1834 309 12 ) ) PUNCT cana-1834 309 13 . . PUNCT cana-1834 310 1 theorem6.3 theorem6.3 NOUN cana-1834 310 2 : : PUNCT cana-1834 310 3 let let AUX cana-1834 310 4 ( ( PUNCT cana-1834 310 5 Ο•,g):(r1Ξ³1 Ο•,g):(r1Ξ³1 NOUN cana-1834 310 6 , , PUNCT cana-1834 310 7 [ [ PUNCT cana-1834 310 8 ] ] X cana-1834 310 9 ) ) PUNCT cana-1834 310 10 βˆ’β†’(r2,Ξ³2 βˆ’β†’(r2,Ξ³2 X cana-1834 310 11 , , PUNCT cana-1834 310 12 [ [ PUNCT cana-1834 310 13 ] ] X cana-1834 310 14 ) ) PUNCT cana-1834 310 15 be be AUX cana-1834 310 16 a a DET cana-1834 310 17 homomorphism homomorphism NOUN cana-1834 310 18 . . PUNCT cana-1834 311 1 set set VERB cana-1834 311 2 a= a= PROPN cana-1834 311 3 { { PUNCT cana-1834 311 4 iβŠ†r1|ΞΈ(Ο•,g)βŠ†iΓ—i}and iβŠ†r1|ΞΈ(Ο•,g)βŠ†iΓ—i}and NOUN cana-1834 311 5 b= b= NOUN cana-1834 311 6 { { PUNCT cana-1834 311 7 j|jβŠ†r2}.then j|jβŠ†r2}.then NOUN cana-1834 311 8 , , PUNCT cana-1834 311 9 there there PRON cana-1834 311 10 exists exist VERB cana-1834 311 11 an an DET cana-1834 311 12 1 1 NUM cana-1834 311 13 - - SYM cana-1834 311 14 1 1 NUM cana-1834 311 15 mapping mapping NOUN cana-1834 311 16 from from ADP cana-1834 311 17 a a PRON cana-1834 311 18 to to ADP cana-1834 311 19 b. b. PROPN cana-1834 311 20 proof proof NOUN cana-1834 311 21 .define .define ADJ cana-1834 311 22 ψ ψ X cana-1834 311 23 : : PUNCT cana-1834 311 24 aβ†’b aβ†’b NUM cana-1834 311 25 by by ADP cana-1834 311 26 ψ(i)=Ο•(i ψ(i)=Ο•(i PROPN cana-1834 311 27 ) ) PUNCT cana-1834 311 28 . . PUNCT cana-1834 312 1 clearly clearly ADV cana-1834 312 2 , , PUNCT cana-1834 312 3 ψ ψ X cana-1834 312 4 is be AUX cana-1834 312 5 well well ADV cana-1834 312 6 - - PUNCT cana-1834 312 7 defined define VERB cana-1834 312 8 . . PUNCT cana-1834 313 1 suppose suppose VERB cana-1834 313 2 that that SCONJ cana-1834 313 3 ψ(i1)=ψ(i2).then ψ(i1)=ψ(i2).then PUNCT cana-1834 313 4 Ο•(i1)=Ο•(i2).also Ο•(i1)=Ο•(i2).also PROPN cana-1834 313 5 we we PRON cana-1834 313 6 can can AUX cana-1834 313 7 see see VERB cana-1834 313 8 that that DET cana-1834 313 9 x∈i1 x∈i1 NOUN cana-1834 313 10 = = VERB cana-1834 313 11 β‡’ β‡’ PROPN cana-1834 313 12 Ο•(x Ο•(x PROPN cana-1834 313 13 ) ) PUNCT cana-1834 313 14 βˆˆΟ•(i1)=Ο•(i2 βˆˆΟ•(i1)=Ο•(i2 NUM cana-1834 313 15 ) ) PUNCT cana-1834 314 1 = = VERB cana-1834 314 2 β‡’ β‡’ NOUN cana-1834 314 3 βˆƒy∈i2,Ο•(x)=Ο•(y βˆƒy∈i2,Ο•(x)=Ο•(y NOUN cana-1834 314 4 ) ) PUNCT cana-1834 315 1 = = VERB cana-1834 315 2 β‡’ β‡’ NOUN cana-1834 315 3 ( ( PUNCT cana-1834 315 4 x x X cana-1834 315 5 , , PUNCT cana-1834 315 6 y)∈θ(Ο•,g)βŠ†i2Γ—i2 y)∈θ(Ο•,g)βŠ†i2Γ—i2 NOUN cana-1834 315 7 = = NOUN cana-1834 315 8 β‡’ β‡’ VERB cana-1834 315 9 x∈i2 x∈i2 NOUN cana-1834 316 1 = = NOUN cana-1834 316 2 β‡’ β‡’ VERB cana-1834 316 3 i1βŠ†i2.similarly i1βŠ†i2.similarly ADV cana-1834 316 4 , , PUNCT cana-1834 316 5 i2βŠ†i1andsoi1 i2βŠ†i1andsoi1 PROPN cana-1834 316 6 = = PROPN cana-1834 316 7 i2and i2and PROPN cana-1834 316 8 hence hence ADV cana-1834 316 9 , , PUNCT cana-1834 316 10 ψisone ψisone NOUN cana-1834 316 11 - - PUNCT cana-1834 316 12 to to ADP cana-1834 316 13 - - PUNCT cana-1834 316 14 one one NUM cana-1834 316 15 . . PUNCT cana-1834 317 1 theorem6.4.let theorem6.4.let ADJ cana-1834 317 2 ( ( PUNCT cana-1834 317 3 r1,Ξ³1 r1,Ξ³1 PROPN cana-1834 317 4 , , PUNCT cana-1834 317 5 [ [ PUNCT cana-1834 317 6 ] ] X cana-1834 317 7 ) ) PUNCT cana-1834 317 8 ( ( PUNCT cana-1834 317 9 Ο• Ο• NOUN cana-1834 317 10 1 1 NUM cana-1834 317 11 , , PUNCT cana-1834 317 12 g g PROPN cana-1834 317 13 1(r2,Ξ³2 1(r2,Ξ³2 NUM cana-1834 317 14 , , PUNCT cana-1834 317 15 [ [ PUNCT cana-1834 317 16 ] ] X cana-1834 317 17 ) ) PUNCT cana-1834 317 18 ( ( PUNCT cana-1834 317 19 Ο• Ο• NOUN cana-1834 317 20 2 2 NUM cana-1834 317 21 , , PUNCT cana-1834 317 22 g g PROPN cana-1834 317 23 2 2 NUM cana-1834 317 24 ) ) PUNCT cana-1834 317 25 ( ( PUNCT cana-1834 317 26 r3,Ξ³3 r3,Ξ³3 PROPN cana-1834 317 27 , , PUNCT cana-1834 317 28 [ [ PUNCT cana-1834 317 29 ] ] X cana-1834 317 30 ) ) PUNCT cana-1834 317 31 be be AUX cana-1834 317 32 a a DET cana-1834 317 33 sequence sequence NOUN cana-1834 317 34 of of ADP cana-1834 317 35 homomorphisms homomorphism NOUN cana-1834 317 36 .then .then PUNCT cana-1834 317 37 ( ( PUNCT cana-1834 317 38 ψ ψ X cana-1834 317 39 , , PUNCT cana-1834 317 40 g):(r1Γ—r2Γ—r3,Ξ³1Γ—Ξ³2Γ—Ξ³3 g):(r1Γ—r2Γ—r3,Ξ³1Γ—Ξ³2Γ—Ξ³3 PROPN cana-1834 317 41 , , PUNCT cana-1834 317 42 [ [ PUNCT cana-1834 317 43 ] ] X cana-1834 317 44 ) ) PUNCT cana-1834 317 45 β†’(r1Γ—r2Γ—r3,Ξ³1Γ—Ξ³2Γ—Ξ³3 β†’(r1Γ—r2Γ—r3,Ξ³1Γ—Ξ³2Γ—Ξ³3 NOUN cana-1834 317 46 , , PUNCT cana-1834 317 47 [ [ PUNCT cana-1834 317 48 ] ] X cana-1834 317 49 ) ) PUNCT cana-1834 317 50 defined define VERB cana-1834 317 51 by by ADP cana-1834 317 52 ψ(x ψ(x PROPN cana-1834 317 53 , , PUNCT cana-1834 317 54 y)=Ο•1(x),Ο•1(y y)=Ο•1(x),Ο•1(y PROPN cana-1834 317 55 ) ) PUNCT cana-1834 317 56 and and CCONJ cana-1834 317 57 g(𝛾,Ξ²)=g1(𝛾),g1(Ξ² g(𝛾,Ξ²)=g1(𝛾),g1(Ξ² X cana-1834 317 58 ) ) PUNCT cana-1834 317 59 for for ADP cana-1834 317 60 allx allx NOUN cana-1834 317 61 , , PUNCT cana-1834 317 62 y∈r1 y∈r1 NOUN cana-1834 317 63 and and CCONJ cana-1834 317 64 𝛾,β∈γ1 𝛾,β∈γ1 PROPN cana-1834 317 65 , , PUNCT cana-1834 317 66 is be AUX cana-1834 317 67 a a DET cana-1834 317 68 homomorphism homomorphism NOUN cana-1834 317 69 such such ADJ cana-1834 317 70 that that PRON cana-1834 317 71 ψ(ΞΈ(Ο•1,g1))βŠ†ΞΈ(Ο•2,g2 ψ(ΞΈ(Ο•1,g1))βŠ†ΞΈ(Ο•2,g2 NOUN cana-1834 317 72 ) ) PUNCT cana-1834 317 73 . . PUNCT cana-1834 318 1 moreover moreover ADV cana-1834 318 2 , , PUNCT cana-1834 318 3 if if SCONJ cana-1834 318 4 ( ( PUNCT cana-1834 318 5 ψ1,g1 ψ1,g1 PROPN cana-1834 318 6 ) ) PUNCT cana-1834 318 7 is be AUX cana-1834 318 8 on on ADP cana-1834 318 9 to to ADP cana-1834 318 10 and and CCONJ cana-1834 318 11 ( ( PUNCT cana-1834 318 12 ψ2,g2 ψ2,g2 PROPN cana-1834 318 13 ) ) PUNCT cana-1834 318 14 is be AUX cana-1834 318 15 one one NUM cana-1834 318 16 to to ADP cana-1834 318 17 one one NUM cana-1834 318 18 , , PUNCT cana-1834 318 19 then then ADV cana-1834 318 20 ψ ψ X cana-1834 318 21 ( ( PUNCT cana-1834 318 22 ΞΈ(Ο•1,g1))=ΞΈ(Ο•2,g2 ΞΈ(Ο•1,g1))=ΞΈ(Ο•2,g2 NOUN cana-1834 318 23 ) ) PUNCT cana-1834 318 24 . . PUNCT cana-1834 319 1 proof proof NOUN cana-1834 319 2 . . PUNCT cana-1834 320 1 it it PRON cana-1834 320 2 is be AUX cana-1834 320 3 trivial trivial ADJ cana-1834 320 4 that that SCONJ cana-1834 320 5 ( ( PUNCT cana-1834 320 6 ψ ψ X cana-1834 320 7 , , PUNCT cana-1834 320 8 g g NOUN cana-1834 320 9 ) ) PUNCT cana-1834 320 10 is be AUX cana-1834 320 11 a a DET cana-1834 320 12 homomorphism.hence homomorphism.hence NOUN cana-1834 320 13 , , PUNCT cana-1834 320 14 we we PRON cana-1834 320 15 have have VERB cana-1834 320 16 ψ(a ψ(a PROPN cana-1834 320 17 , , PUNCT cana-1834 320 18 b)∈ψ(ΞΈ(Ο•1,g1)),(a b)∈ψ(ΞΈ(Ο•1,g1)),(a NOUN cana-1834 320 19 , , PUNCT cana-1834 320 20 b)∈θ(Ο•1,g1 b)∈θ(Ο•1,g1 PROPN cana-1834 320 21 ) ) PUNCT cana-1834 321 1 = = NOUN cana-1834 321 2 β‡’ β‡’ NOUN cana-1834 321 3 ψ1(a)=ψ1(b ψ1(a)=ψ1(b NOUN cana-1834 321 4 ) ) PUNCT cana-1834 322 1 = = NOUN cana-1834 322 2 β‡’ β‡’ NOUN cana-1834 322 3 ψ2(ψ1(a)=ψ2(ψ ψ2(ψ1(a)=ψ2(ψ NOUN cana-1834 322 4 1(b 1(b NUM cana-1834 322 5 ) ) PUNCT cana-1834 322 6 ) ) PUNCT cana-1834 323 1 = = VERB cana-1834 323 2 β‡’ β‡’ NOUN cana-1834 323 3 ( ( PUNCT cana-1834 323 4 ψ1(a),ψ1(b))∈θ(ψ2,g2 ψ1(a),ψ1(b))∈θ(ψ2,g2 NOUN cana-1834 323 5 ) ) PUNCT cana-1834 323 6 = = NOUN cana-1834 323 7 β‡’ β‡’ NOUN cana-1834 323 8 ψ(a ψ(a PROPN cana-1834 323 9 , , PUNCT cana-1834 323 10 b)∈θ(ψ2,g2 b)∈θ(ψ2,g2 PROPN cana-1834 323 11 ) ) PUNCT cana-1834 323 12 . . PUNCT cana-1834 324 1 thus thus ADV cana-1834 324 2 , , PUNCT cana-1834 324 3 ψ(ΞΈ(ψ1,g1))βŠ†ΞΈ(ψ2,g2).now ψ(ΞΈ(ψ1,g1))βŠ†ΞΈ(ψ2,g2).now ADV cana-1834 324 4 , , PUNCT cana-1834 324 5 if if SCONJ cana-1834 324 6 ( ( PUNCT cana-1834 324 7 ψ1,g1 ψ1,g1 PROPN cana-1834 324 8 ) ) PUNCT cana-1834 324 9 is be AUX cana-1834 324 10 surjective surjective ADJ cana-1834 324 11 and and CCONJ cana-1834 324 12 ( ( PUNCT cana-1834 324 13 ψ2,g2 ψ2,g2 PROPN cana-1834 324 14 ) ) PUNCT cana-1834 324 15 is be AUX cana-1834 324 16 in in ADP cana-1834 324 17 - - PUNCT cana-1834 324 18 jective jective ADJ cana-1834 324 19 , , PUNCT cana-1834 324 20 then then ADV cana-1834 324 21 we we PRON cana-1834 324 22 have have VERB cana-1834 324 23 ψ ψ VERB cana-1834 324 24 ( ( PUNCT cana-1834 324 25 ΞΈ(ψ1,g1))=ΞΈ(ψ2,g2)it ΞΈ(ψ1,g1))=ΞΈ(ψ2,g2)it PROPN cana-1834 324 26 suffices suffice VERB cana-1834 324 27 to to PART cana-1834 324 28 prove prove VERB cana-1834 324 29 that that SCONJ cana-1834 324 30 ΞΈ(ψ2,g2)βŠ†Οˆ ΞΈ(ψ2,g2)βŠ†Οˆ PUNCT cana-1834 324 31 ( ( PUNCT cana-1834 324 32 ΞΈ(Ο•1,g1 ΞΈ(Ο•1,g1 NOUN cana-1834 324 33 ) ) PUNCT cana-1834 324 34 ) ) PUNCT cana-1834 325 1 hence hence ADV cana-1834 325 2 , , PUNCT cana-1834 325 3 we we PRON cana-1834 325 4 have have VERB cana-1834 325 5 ( ( PUNCT cana-1834 325 6 t t NOUN cana-1834 325 7 , , PUNCT cana-1834 325 8 t')∈θ(ψ2,g2)=β‡’ t')∈θ(ψ2,g2)=β‡’ NOUN cana-1834 325 9 ψ2(t)= ψ2(t)= NOUN cana-1834 325 10 ψ2(t ψ2(t PRON cana-1834 325 11 ' ' PUNCT cana-1834 325 12 ) ) PUNCT cana-1834 326 1 = = VERB cana-1834 326 2 β‡’ β‡’ NOUN cana-1834 326 3 βˆƒa βˆƒa NOUN cana-1834 326 4 , , PUNCT cana-1834 326 5 b∈r b∈r NOUN cana-1834 326 6 , , PUNCT cana-1834 326 7 ψ1(a)=t ψ1(a)=t PROPN cana-1834 326 8 , , PUNCT cana-1834 326 9 ψ1(b)=t ψ1(b)=t ADJ cana-1834 326 10 ' ' PART cana-1834 326 11 = = ADJ cana-1834 326 12 β‡’ β‡’ NOUN cana-1834 326 13 ( ( PUNCT cana-1834 326 14 t t PROPN cana-1834 326 15 , , PUNCT cana-1834 326 16 t')= t')= PRON cana-1834 326 17 ψ(a ψ(a PROPN cana-1834 326 18 , , PUNCT cana-1834 326 19 b)=(ψ1(a),ψ1(b b)=(ψ1(a),ψ1(b NOUN cana-1834 326 20 ) ) PUNCT cana-1834 326 21 ) ) PUNCT cana-1834 327 1 = = VERB cana-1834 327 2 β‡’ β‡’ NOUN cana-1834 327 3 ( ( PUNCT cana-1834 327 4 t t PROPN cana-1834 327 5 , , PUNCT cana-1834 327 6 t')∈ψ t')∈ψ PROPN cana-1834 327 7 ( ( PUNCT cana-1834 327 8 ΞΈ(ψ1,g1 ΞΈ(ψ1,g1 NOUN cana-1834 327 9 ) ) PUNCT cana-1834 327 10 ) ) PUNCT cana-1834 327 11 communications communication NOUN cana-1834 327 12 on on ADP cana-1834 327 13 applied apply VERB cana-1834 327 14 nonlinear nonlinear ADJ cana-1834 327 15 analysis analysis NOUN cana-1834 327 16 issn issn NOUN cana-1834 327 17 : : PUNCT cana-1834 327 18 1074 1074 NUM cana-1834 327 19 - - PUNCT cana-1834 327 20 133x 133x NUM cana-1834 327 21 vol vol NOUN cana-1834 327 22 32 32 NUM cana-1834 327 23 no no NOUN cana-1834 327 24 . . NOUN cana-1834 327 25 2 2 NUM cana-1834 327 26 ( ( PUNCT cana-1834 327 27 2025 2025 NUM cana-1834 327 28 ) ) PUNCT cana-1834 328 1 589 589 NUM cana-1834 328 2 https://internationalpubls.com https://internationalpubls.com X cana-1834 328 3 this this PRON cana-1834 328 4 shows show VERB cana-1834 328 5 that that SCONJ cana-1834 328 6 ΞΈ(Ο•2,g2)βŠ†ΟˆΞΈ(Ο•1,g1 ΞΈ(Ο•2,g2)βŠ†ΟˆΞΈ(Ο•1,g1 PROPN cana-1834 328 7 ) ) PUNCT cana-1834 328 8 , , PUNCT cana-1834 328 9 and and CCONJ cana-1834 328 10 the the DET cana-1834 328 11 proof proof NOUN cana-1834 328 12 is be AUX cana-1834 328 13 completed complete VERB cana-1834 328 14 . . PUNCT cana-1834 329 1 theorem6.5 theorem6.5 NUM cana-1834 329 2 .. .. PUNCT cana-1834 329 3 let let VERB cana-1834 329 4 ( ( PUNCT cana-1834 329 5 r1,Ξ³1 r1,Ξ³1 PROPN cana-1834 329 6 , , PUNCT cana-1834 329 7 [ [ PUNCT cana-1834 329 8 ] ] X cana-1834 329 9 ) ) PUNCT cana-1834 329 10 ( ( PUNCT cana-1834 329 11 Ο• Ο• NOUN cana-1834 329 12 1 1 NUM cana-1834 329 13 , , PUNCT cana-1834 329 14 g g PROPN cana-1834 329 15 1(r2,Ξ³2 1(r2,Ξ³2 NUM cana-1834 329 16 , , PUNCT cana-1834 329 17 [ [ PUNCT cana-1834 329 18 ] ] X cana-1834 329 19 ) ) PUNCT cana-1834 329 20 ( ( PUNCT cana-1834 329 21 Ο• Ο• NOUN cana-1834 329 22 2 2 NUM cana-1834 329 23 , , PUNCT cana-1834 329 24 g g PROPN cana-1834 329 25 2 2 NUM cana-1834 329 26 ) ) PUNCT cana-1834 329 27 ( ( PUNCT cana-1834 329 28 r3,Ξ³3 r3,Ξ³3 PROPN cana-1834 329 29 , , PUNCT cana-1834 329 30 [ [ PUNCT cana-1834 329 31 ] ] X cana-1834 329 32 ) ) PUNCT cana-1834 329 33 be be AUX cana-1834 329 34 a a DET cana-1834 329 35 sequence sequence NOUN cana-1834 329 36 of of ADP cana-1834 329 37 homomorphism homomorphism NOUN cana-1834 329 38 ’s ’s PART cana-1834 329 39 . . PUNCT cana-1834 330 1 then then ADV cana-1834 330 2 imψ1Γ—imψ1βŠ†ΞΈ(ψ2,g2 imψ1Γ—imψ1βŠ†ΞΈ(ψ2,g2 NOUN cana-1834 330 3 ) ) PUNCT cana-1834 330 4 if if SCONJ cana-1834 330 5 and and CCONJ cana-1834 330 6 only only ADV cana-1834 330 7 if if SCONJ cana-1834 330 8 ψ2 ψ2 NOUN cana-1834 330 9 o o NOUN cana-1834 330 10 ψ1 ψ1 NOUN cana-1834 330 11 is be AUX cana-1834 330 12 constant constant ADJ cana-1834 330 13 . . PUNCT cana-1834 331 1 proof proof NOUN cana-1834 331 2 . . PUNCT cana-1834 332 1 the the DET cana-1834 332 2 proof proof NOUN cana-1834 332 3 of of ADP cana-1834 332 4 the the DET cana-1834 332 5 necessary necessary ADJ cana-1834 332 6 part part NOUN cana-1834 332 7 is be AUX cana-1834 332 8 routine routine ADJ cana-1834 332 9 and and CCONJ cana-1834 332 10 we we PRON cana-1834 332 11 only only ADV cana-1834 332 12 prove prove VERB cana-1834 332 13 the the DET cana-1834 332 14 sufficiency sufficiency NOUN cana-1834 332 15 part part NOUN cana-1834 332 16 . . PUNCT cana-1834 333 1 (= (= X cana-1834 333 2 β‡’):letx β‡’):letx ADV cana-1834 333 3 , , PUNCT cana-1834 333 4 y∈r1.then y∈r1.then ADV cana-1834 333 5 , , PUNCT cana-1834 333 6 ( ( PUNCT cana-1834 333 7 ψ1(x),ψ1(y))∈imψ1Γ—imψ1βŠ†ΞΈ(ψ2,g2 ψ1(x),ψ1(y))∈imψ1Γ—imψ1βŠ†ΞΈ(ψ2,g2 NOUN cana-1834 333 8 ) ) PUNCT cana-1834 333 9 . . PUNCT cana-1834 334 1 hence hence ADV cana-1834 334 2 , , PUNCT cana-1834 334 3 ψ2ψ1(x)=ψ2ψ1(y).this ψ2ψ1(x)=ψ2ψ1(y).this PRON cana-1834 334 4 show show NOUN cana-1834 334 5 that that PRON cana-1834 334 6 ψ2oψ1is ψ2oψ1i VERB cana-1834 334 7 a a DET cana-1834 334 8 constant constant ADJ cana-1834 334 9 . . PUNCT cana-1834 335 1 finally finally ADV cana-1834 335 2 , , PUNCT cana-1834 335 3 by by ADP cana-1834 335 4 the the DET cana-1834 335 5 congruence congruence NOUN cana-1834 335 6 on on ADP cana-1834 335 7 the the DET cana-1834 335 8 ternary ternary ADJ cana-1834 335 9 𝛾-semiring 𝛾-semiring NOUN cana-1834 335 10 induced induce VERB cana-1834 335 11 by by ADP cana-1834 335 12 homomorphism homomorphism PROPN cana-1834 335 13 , , PUNCT cana-1834 335 14 we we PRON cana-1834 335 15 are be AUX cana-1834 335 16 able able ADJ cana-1834 335 17 to to PART cana-1834 335 18 establish establish VERB cana-1834 335 19 some some DET cana-1834 335 20 isomorphism isomorphism NOUN cana-1834 335 21 theorems theorem NOUN cana-1834 335 22 and and CCONJ cana-1834 335 23 investigate investigate VERB cana-1834 335 24 the the DET cana-1834 335 25 commutativity commutativity NOUN cana-1834 335 26 of of ADP cana-1834 335 27 some some DET cana-1834 335 28 diagrams diagram NOUN cana-1834 335 29 . . PUNCT cana-1834 336 1 theorem6.6 theorem6.6 NUM cana-1834 336 2 . . PUNCT cana-1834 337 1 ( ( PUNCT cana-1834 337 2 isomorphismtheorem isomorphismtheorem VERB cana-1834 337 3 ) ) PUNCT cana-1834 337 4 if if SCONJ cana-1834 337 5 ( ( PUNCT cana-1834 337 6 ψ ψ X cana-1834 337 7 , , PUNCT cana-1834 337 8 g):(r1 g):(r1 PROPN cana-1834 337 9 , , PUNCT cana-1834 337 10 Ξ³1 Ξ³1 NOUN cana-1834 337 11 , , PUNCT cana-1834 337 12 [ [ PUNCT cana-1834 337 13 ] ] X cana-1834 337 14 ) ) PUNCT cana-1834 337 15 βˆ’β†’(r2,Ξ³2 βˆ’β†’(r2,Ξ³2 X cana-1834 337 16 , , PUNCT cana-1834 337 17 [ [ PUNCT cana-1834 337 18 ] ] X cana-1834 337 19 ) ) PUNCT cana-1834 337 20 is be AUX cana-1834 337 21 an an DET cana-1834 337 22 epimorphism epimorphism NOUN cana-1834 337 23 , , PUNCT cana-1834 337 24 then then ADV cana-1834 337 25 there there PRON cana-1834 337 26 exists exist VERB cana-1834 337 27 an an DET cana-1834 337 28 unique unique ADJ cana-1834 337 29 isomorphism isomorphism NOUN cana-1834 337 30 ( ( PUNCT cana-1834 337 31 ψ ψ NOUN cana-1834 337 32 , , PUNCT cana-1834 337 33 g):(r1 g):(r1 NOUN cana-1834 337 34 : : PUNCT cana-1834 337 35 ΞΈ(Ο•,g),Ξ³1 ΞΈ(Ο•,g),Ξ³1 NUM cana-1834 337 36 , , PUNCT cana-1834 337 37 [ [ PUNCT cana-1834 337 38 ] ] X cana-1834 337 39 ) ) PUNCT cana-1834 337 40 βˆ’β†’(r2,Ξ³2 βˆ’β†’(r2,Ξ³2 X cana-1834 337 41 , , PUNCT cana-1834 337 42 [ [ PUNCT cana-1834 337 43 ] ] X cana-1834 337 44 ) ) PUNCT cana-1834 337 45 such such ADJ cana-1834 337 46 that that SCONJ cana-1834 337 47 the the DET cana-1834 337 48 following follow VERB cana-1834 337 49 diagram diagram NOUN cana-1834 337 50 commutes commute NOUN cana-1834 337 51 : : PUNCT cana-1834 337 52 ( ( PUNCT cana-1834 337 53 ψ ψ X cana-1834 337 54 , , PUNCT cana-1834 337 55 g g NOUN cana-1834 337 56 ) ) PUNCT cana-1834 337 57 ( ( PUNCT cana-1834 337 58 r1,Ξ³1 r1,Ξ³1 PROPN cana-1834 337 59 , , PUNCT cana-1834 337 60 [ [ PUNCT cana-1834 337 61 ] ] X cana-1834 337 62 ) ) PUNCT cana-1834 337 63 ( ( PUNCT cana-1834 337 64 r1,Ξ³2 r1,Ξ³2 PROPN cana-1834 337 65 , , PUNCT cana-1834 337 66 [ [ PUNCT cana-1834 337 67 ] ] X cana-1834 337 68 ) ) PUNCT cana-1834 337 69 ( ( PUNCT cana-1834 337 70 𝛱r1,1r1 𝛱r1,1r1 NOUN cana-1834 337 71 ) ) PUNCT cana-1834 337 72 ( ( PUNCT cana-1834 337 73 ψ ψ X cana-1834 337 74 , , PUNCT cana-1834 337 75 g g NOUN cana-1834 337 76 ) ) PUNCT cana-1834 337 77 ( ( PUNCT cana-1834 337 78 r1,:ΞΈ(ψ r1,:ΞΈ(ψ NOUN cana-1834 337 79 , , PUNCT cana-1834 337 80 g),Ξ³1 g),Ξ³1 NOUN cana-1834 337 81 , , PUNCT cana-1834 337 82 ) ) PUNCT cana-1834 337 83 ) ) PUNCT cana-1834 337 84 where where SCONJ cana-1834 337 85 Ο€r1,:r1,β†’r1 Ο€r1,:r1,β†’r1 NOUN cana-1834 337 86 , , PUNCT cana-1834 337 87 : : PUNCT cana-1834 337 88 ΞΈ(Ο•,g ΞΈ(Ο•,g NUM cana-1834 337 89 ) ) PUNCT cana-1834 337 90 is be AUX cana-1834 337 91 defined define VERB cana-1834 337 92 by by ADP cana-1834 337 93 Ο€r1(x)=ΞΈ(Ο•,g Ο€r1(x)=ΞΈ(Ο•,g PROPN cana-1834 337 94 ) ) PUNCT cana-1834 337 95 ( ( PUNCT cana-1834 337 96 x x X cana-1834 337 97 ) ) PUNCT cana-1834 337 98 for for ADP cana-1834 337 99 all all DET cana-1834 337 100 x∈r1 x∈r1 NOUN cana-1834 337 101 , , PUNCT cana-1834 337 102 and1r1is and1r1i VERB cana-1834 337 103 identity identity NOUN cana-1834 337 104 . . PUNCT cana-1834 338 1 proof proof NOUN cana-1834 338 2 . . PUNCT cana-1834 339 1 define define VERB cana-1834 339 2 ψ:𝑅1 ψ:𝑅1 NOUN cana-1834 339 3 : : PUNCT cana-1834 339 4 ΞΈ(Ο•,g)βˆ’r2 ΞΈ(Ο•,g)βˆ’r2 PROPN cana-1834 339 5 by by ADP cana-1834 339 6 ψ(ΞΈ(Ο•,g)(x))=ψ(x ψ(ΞΈ(Ο•,g)(x))=ψ(x PROPN cana-1834 339 7 ) ) PUNCT cana-1834 339 8 for for ADP cana-1834 339 9 all all PRON cana-1834 339 10 x∈r1.then x∈r1.then X cana-1834 339 11 we we PRON cana-1834 339 12 haveΞΈ(Ο•,g)(x)= haveΞΈ(Ο•,g)(x)= VERB cana-1834 340 1 ΞΈ(Ο•,g)(y) ΞΈ(Ο•,g)(y) X cana-1834 340 2 ⇐ ⇐ ADJ cana-1834 340 3 β‡’xΞΈ(Ο•,g)y β‡’xΞΈ(Ο•,g)y NOUN cana-1834 340 4 ⇐ ⇐ ADJ cana-1834 340 5 β‡’Οˆ(x)=ψ(y),and β‡’Οˆ(x)=ψ(y),and PROPN cana-1834 340 6 hence hence ADV cana-1834 340 7 ψ ψ NOUN cana-1834 340 8 is be AUX cana-1834 340 9 well well ADV cana-1834 340 10 defined define VERB cana-1834 340 11 and and CCONJ cana-1834 340 12 is be AUX cana-1834 340 13 a a DET cana-1834 340 14 1 1 NUM cana-1834 340 15 - - SYM cana-1834 340 16 1 1 NUM cana-1834 340 17 mapping.now,(ψ mapping.now,(ψ NUM cana-1834 340 18 , , PUNCT cana-1834 340 19 g g NOUN cana-1834 340 20 ) ) PUNCT cana-1834 340 21 is be AUX cana-1834 340 22 a a DET cana-1834 340 23 homomorphism homomorphism NOUN cana-1834 340 24 . . PUNCT cana-1834 341 1 we we PRON cana-1834 341 2 have have VERB cana-1834 341 3 ψ(ΞΈ(Ο•,g)(x)βŠ•ΞΈ(Ο•,g)(y))=ψ(ΞΈ(Ο•,g)(x+y ψ(ΞΈ(Ο•,g)(x)βŠ•ΞΈ(Ο•,g)(y))=ψ(ΞΈ(Ο•,g)(x+y NOUN cana-1834 341 4 ) ) PUNCT cana-1834 341 5 ) ) PUNCT cana-1834 342 1 = = VERB cana-1834 342 2 ψ(x+y)=ψ(x)+ψ(y ψ(x+y)=ψ(x)+ψ(y VERB cana-1834 342 3 ) ) PUNCT cana-1834 342 4 = = SYM cana-1834 342 5 ψ(ΞΈ(Ο•,g)(x))+ψ(ΞΈ(Ο•,g)(y ψ(ΞΈ(Ο•,g)(x))+ψ(ΞΈ(Ο•,g)(y PROPN cana-1834 342 6 ) ) PUNCT cana-1834 342 7 ) ) PUNCT cana-1834 342 8 . . PUNCT cana-1834 343 1 we we PRON cana-1834 343 2 deduce deduce VERB cana-1834 343 3 that that SCONJ cana-1834 343 4 ψ(ΞΈ(Ο•,g)(x)βŠ™Ξ³βŠ™π›ΎβŠ™Ξ³βŠ™ΞΈ(Ο•,g)(y)=ψθ(Ο•,g)(xβŠ™Ξ³βŠ™π›ΎβŠ™Ξ³βŠ™y ψ(ΞΈ(Ο•,g)(x)βŠ™Ξ³βŠ™π›ΎβŠ™Ξ³βŠ™ΞΈ(Ο•,g)(y)=ψθ(Ο•,g)(xβŠ™Ξ³βŠ™π›ΎβŠ™Ξ³βŠ™y NOUN cana-1834 343 5 ) ) PUNCT cana-1834 343 6 = = NOUN cana-1834 343 7 Ο•(xβŠ™Ξ³βŠ™π›ΎβŠ™Ξ³βŠ™y)=Ο•(x)βŠ™Ξ³βŠ™g(𝛾)βŠ™Ξ³βŠ™Ο•(y)=ψ(ΞΈ(Ο•,g)(x))βŠ™Ξ³βŠ™g(𝛾)βŠ™Ξ³βŠ™Οˆ(ΞΈ(Ο•,g)(y Ο•(xβŠ™Ξ³βŠ™π›ΎβŠ™Ξ³βŠ™y)=Ο•(x)βŠ™Ξ³βŠ™g(𝛾)βŠ™Ξ³βŠ™Ο•(y)=ψ(ΞΈ(Ο•,g)(x))βŠ™Ξ³βŠ™g(𝛾)βŠ™Ξ³βŠ™Οˆ(ΞΈ(Ο•,g)(y PROPN cana-1834 343 8 ) ) PUNCT cana-1834 343 9 ) ) PUNCT cana-1834 343 10 . . PUNCT cana-1834 344 1 communications communication NOUN cana-1834 344 2 on on ADP cana-1834 344 3 applied apply VERB cana-1834 344 4 nonlinear nonlinear ADJ cana-1834 344 5 analysis analysis NOUN cana-1834 344 6 issn issn NOUN cana-1834 344 7 : : PUNCT cana-1834 344 8 1074 1074 NUM cana-1834 344 9 - - PUNCT cana-1834 344 10 133x 133x NUM cana-1834 344 11 vol vol NOUN cana-1834 344 12 32 32 NUM cana-1834 344 13 no no NOUN cana-1834 344 14 . . NOUN cana-1834 344 15 2 2 NUM cana-1834 344 16 ( ( PUNCT cana-1834 344 17 2025 2025 NUM cana-1834 344 18 ) ) PUNCT cana-1834 344 19 590 590 NUM cana-1834 344 20 https://internationalpubls.com https://internationalpubls.com X cana-1834 345 1 = = SYM cana-1834 345 2 ΞΈ(Ο•2,g2)Ο•1(x)βŠ•ΞΈ(Ο•2,g2)Ο•1(y ΞΈ(Ο•2,g2)Ο•1(x)βŠ•ΞΈ(Ο•2,g2)Ο•1(y ADJ cana-1834 345 3 ) ) PUNCT cana-1834 345 4 1 1 NUM cana-1834 345 5 , , PUNCT cana-1834 345 6 1 1 NUM cana-1834 345 7 1 1 NUM cana-1834 345 8 , , PUNCT cana-1834 345 9 1 1 NUM cana-1834 345 10 ( ( PUNCT cana-1834 345 11 ( ( PUNCT cana-1834 345 12 ( ( PUNCT cana-1834 345 13 ( ( PUNCT cana-1834 345 14 ) ) PUNCT cana-1834 345 15 ) ) PUNCT cana-1834 346 1 ( ( PUNCT cana-1834 346 2 ( ( PUNCT cana-1834 346 3 ) ) PUNCT cana-1834 346 4 g g PROPN cana-1834 346 5 gx gx PROPN cana-1834 346 6 yοͺ yοͺ PROPN cana-1834 347 1 οͺοͺ οͺοͺ PROPN cana-1834 347 2   PROPN cana-1834 347 3 οͺ οͺ PROPN cana-1834 347 4 = = PROPN cana-1834 347 5 οƒ… οƒ… PROPN cana-1834 347 6 ΞΈ(Ο•1,g1)(x)βŠ™g1(Ξ³)βŠ™ΟˆΞΈ(Ο•1,g1)(y ΞΈ(Ο•1,g1)(x)βŠ™g1(Ξ³)βŠ™ΟˆΞΈ(Ο•1,g1)(y PROPN cana-1834 347 7 ) ) PUNCT cana-1834 347 8 . . PUNCT cana-1834 348 1 therefore,(ψ therefore,(ψ PROPN cana-1834 348 2 , , PUNCT cana-1834 348 3 g g NOUN cana-1834 348 4 ) ) PUNCT cana-1834 348 5 is be AUX cana-1834 348 6 a a DET cana-1834 348 7 homomorphism homomorphism NOUN cana-1834 348 8 . . PUNCT cana-1834 349 1 also also ADV cana-1834 349 2 Ο•(x Ο•(x X cana-1834 349 3 ) ) PUNCT cana-1834 349 4 = = SYM cana-1834 349 5 ψθ(Ο•,g)(x)= ψθ(Ο•,g)(x)= NOUN cana-1834 349 6 ΟˆΟ€r1(x ΟˆΟ€r1(x NUM cana-1834 349 7 ) ) PUNCT cana-1834 349 8 and and CCONJ cana-1834 349 9 go1𝛾1 go1𝛾1 PROPN cana-1834 349 10 = = PROPN cana-1834 349 11 g g PROPN cana-1834 349 12 which which PRON cana-1834 349 13 imply imply VERB cana-1834 349 14 that that SCONJ cana-1834 349 15 the the DET cana-1834 349 16 diagram diagram NOUN cana-1834 349 17 is be AUX cana-1834 349 18 commutative commutative ADJ cana-1834 349 19 . . PUNCT cana-1834 350 1 let let VERB cana-1834 350 2 ( ( PUNCT cana-1834 350 3 ψ ψ X cana-1834 350 4 , , PUNCT cana-1834 350 5 g):(r1 g):(r1 PROPN cana-1834 350 6 : : PUNCT cana-1834 350 7 ΞΈ(Ο•,g ΞΈ(Ο•,g NUM cana-1834 350 8 ) ) PUNCT cana-1834 350 9 , , PUNCT cana-1834 350 10 Ξ³1 Ξ³1 NOUN cana-1834 350 11 , , PUNCT cana-1834 350 12 [ [ PUNCT cana-1834 350 13 ] ] X cana-1834 350 14 ) ) PUNCT cana-1834 350 15 βˆ’β†’(r2 βˆ’β†’(r2 PROPN cana-1834 350 16 , , PUNCT cana-1834 350 17 Ξ³2 Ξ³2 ADJ cana-1834 350 18 , , PUNCT cana-1834 350 19 [ [ PUNCT cana-1834 350 20 ] ] X cana-1834 350 21 ) ) PUNCT cana-1834 350 22 be be AUX cana-1834 350 23 such such ADJ cana-1834 350 24 that that SCONJ cana-1834 350 25 ψoΟ€r1=Ο•.then ψoΟ€r1=Ο•.then NOUN cana-1834 350 26 , , PUNCT cana-1834 350 27 wehave wehave ADJ cana-1834 350 28 ψθ(Ο•,g)(x)=ΟˆΟ€r1(x)=Ο•(x)=ΟˆΟ€r1(x)=ψθ(Ο•,g)(x ψθ(Ο•,g)(x)=ΟˆΟ€r1(x)=Ο•(x)=ΟˆΟ€r1(x)=ψθ(Ο•,g)(x PROPN cana-1834 350 29 ) ) PUNCT cana-1834 350 30 . . PUNCT cana-1834 351 1 thus thus ADV cana-1834 351 2 , , PUNCT cana-1834 351 3 ( ( PUNCT cana-1834 351 4 ψ ψ X cana-1834 351 5 , , PUNCT cana-1834 351 6 g g NOUN cana-1834 351 7 ) ) PUNCT cana-1834 351 8 is be AUX cana-1834 351 9 unique unique ADJ cana-1834 351 10 and and CCONJ cana-1834 351 11 the the DET cana-1834 351 12 proof proof NOUN cana-1834 351 13 is be AUX cana-1834 351 14 completed complete VERB cana-1834 351 15 . . PUNCT cana-1834 352 1 theorem theorem VERB cana-1834 352 2 6.7 6.7 NUM cana-1834 352 3 . . PUNCT cana-1834 353 1 let let VERB cana-1834 353 2 ( ( PUNCT cana-1834 353 3 r1 r1 PROPN cana-1834 353 4 , , PUNCT cana-1834 353 5 Ξ³1 Ξ³1 NOUN cana-1834 353 6 , , PUNCT cana-1834 353 7 [ [ PUNCT cana-1834 353 8 ] ] X cana-1834 353 9 ) ) PUNCT cana-1834 353 10 ( ( PUNCT cana-1834 353 11 Ο•1,g1 Ο•1,g1 PROPN cana-1834 353 12 ) ) PUNCT cana-1834 353 13 , , PUNCT cana-1834 353 14 ( ( PUNCT cana-1834 353 15 r2,𝛾2 r2,𝛾2 PROPN cana-1834 353 16 , , PUNCT cana-1834 353 17 [ [ PUNCT cana-1834 353 18 ] ] X cana-1834 353 19 ) ) PUNCT cana-1834 353 20 ( ( PUNCT cana-1834 353 21 Ο•2,g2)(r3,𝛾3 Ο•2,g2)(r3,𝛾3 NOUN cana-1834 353 22 , , PUNCT cana-1834 353 23 [ [ PUNCT cana-1834 353 24 ] ] X cana-1834 353 25 ) ) PUNCT cana-1834 353 26 be be AUX cana-1834 353 27 a a DET cana-1834 353 28 sequence sequence NOUN cana-1834 353 29 of of ADP cana-1834 353 30 homomorphism homomorphism NOUN cana-1834 353 31 ’s ’s PART cana-1834 353 32 .then .then PUNCT cana-1834 353 33 , , PUNCT cana-1834 353 34 there there PRON cana-1834 353 35 exists exist VERB cana-1834 353 36 an an DET cana-1834 353 37 unique unique ADJ cana-1834 353 38 homomorphism homomorphism NOUN cana-1834 353 39 ( ( PUNCT cana-1834 353 40 ψ ψ NOUN cana-1834 353 41 , , PUNCT cana-1834 353 42 g1):(t1 g1):(t1 PROPN cana-1834 353 43 : : PUNCT cana-1834 353 44 ΞΈ(Ο•1,g1),𝛾1)β†’(t2 ΞΈ(Ο•1,g1),𝛾1)β†’(t2 NOUN cana-1834 353 45 : : PUNCT cana-1834 353 46 ΞΈ(Ο•2,g2),𝛾2 ΞΈ(Ο•2,g2),𝛾2 VERB cana-1834 353 47 ) ) PUNCT cana-1834 353 48 such such ADJ cana-1834 353 49 that that SCONJ cana-1834 353 50 the the DET cana-1834 353 51 following follow VERB cana-1834 353 52 diagram diagram NOUN cana-1834 353 53 is be AUX cana-1834 353 54 commutative commutative ADJ cana-1834 353 55 : : PUNCT cana-1834 353 56 ( ( PUNCT cana-1834 353 57 Ο•1,g1 Ο•1,g1 PROPN cana-1834 353 58 ) ) PUNCT cana-1834 353 59 ( ( PUNCT cana-1834 353 60 r1,𝛾1 r1,𝛾1 PROPN cana-1834 353 61 ) ) PUNCT cana-1834 353 62 ( ( PUNCT cana-1834 353 63 r2,𝛾2 r2,𝛾2 PROPN cana-1834 353 64 ) ) PUNCT cana-1834 353 65 ( ( PUNCT cana-1834 353 66 Ο€r1 Ο€r1 NOUN cana-1834 353 67 , , PUNCT cana-1834 353 68 1r1 1r1 NUM cana-1834 353 69 ) ) PUNCT cana-1834 353 70 ( ( PUNCT cana-1834 353 71 Ο€r2 Ο€r2 NOUN cana-1834 353 72 , , PUNCT cana-1834 353 73 1r2 1r2 NUM cana-1834 353 74 ) ) PUNCT cana-1834 353 75 ( ( PUNCT cana-1834 353 76 r1 r1 PROPN cana-1834 353 77 : : PUNCT cana-1834 353 78 ΞΈ(Ο•1,g1),𝛾1 ΞΈ(Ο•1,g1),𝛾1 NUM cana-1834 353 79 , , PUNCT cana-1834 353 80 [ [ PUNCT cana-1834 353 81 ] ] X cana-1834 353 82 ) ) PUNCT cana-1834 353 83 ( ( PUNCT cana-1834 353 84 r2 r2 NOUN cana-1834 353 85 : : PUNCT cana-1834 353 86 ΞΈ(Ο•2,g2),𝛾2 ΞΈ(Ο•2,g2),𝛾2 VERB cana-1834 353 87 , , PUNCT cana-1834 353 88 [ [ PUNCT cana-1834 353 89 ] ] X cana-1834 353 90 ) ) PUNCT cana-1834 353 91 moreover moreover ADV cana-1834 353 92 , , PUNCT cana-1834 353 93 if(Ο•1,g1 if(Ο•1,g1 NUM cana-1834 353 94 ) ) PUNCT cana-1834 353 95 is be AUX cana-1834 353 96 on on ADP cana-1834 353 97 to to ADP cana-1834 353 98 and and CCONJ cana-1834 353 99 ( ( PUNCT cana-1834 353 100 Ο•2,g2 Ο•2,g2 PROPN cana-1834 353 101 ) ) PUNCT cana-1834 353 102 is be AUX cana-1834 353 103 1 1 NUM cana-1834 353 104 - - SYM cana-1834 353 105 1,then(ψ 1,then(ψ NUM cana-1834 353 106 , , PUNCT cana-1834 353 107 g1 g1 PROPN cana-1834 353 108 ) ) PUNCT cana-1834 353 109 is be AUX cana-1834 353 110 an an DET cana-1834 353 111 isomorphism isomorphism NOUN cana-1834 353 112 . . PUNCT cana-1834 354 1 proof proof NOUN cana-1834 354 2 .define .define VERB cana-1834 355 1 ψ ψ X cana-1834 355 2 : : PUNCT cana-1834 355 3 r1 r1 NOUN cana-1834 355 4 : : PUNCT cana-1834 355 5 ΞΈ(Ο•1,g1)r2 ΞΈ(Ο•1,g1)r2 ADJ cana-1834 355 6 : : PUNCT cana-1834 355 7 ΞΈ(Ο•2,g2)by ΞΈ(Ο•2,g2)by VERB cana-1834 355 8 ψ(ΞΈ(Ο•1,g1)(x))= ψ(ΞΈ(Ο•1,g1)(x))= PROPN cana-1834 355 9 ΞΈ(Ο•2,g2)Ο•1(x ΞΈ(Ο•2,g2)Ο•1(x PROPN cana-1834 355 10 ) ) PUNCT cana-1834 355 11 . . PUNCT cana-1834 356 1 then then ADV cana-1834 356 2 , , PUNCT cana-1834 356 3 ψ ψ X cana-1834 356 4 is be AUX cana-1834 356 5 welldefined.finally welldefined.finally ADV cana-1834 356 6 , , PUNCT cana-1834 356 7 we we PRON cana-1834 356 8 need need AUX cana-1834 356 9 prove prove VERB cana-1834 356 10 that(ψ1,g1 that(ψ1,g1 NOUN cana-1834 356 11 ) ) PUNCT cana-1834 356 12 is be AUX cana-1834 356 13 a a DET cana-1834 356 14 homomorphism homomorphism NOUN cana-1834 356 15 . . PUNCT cana-1834 357 1 we we PRON cana-1834 357 2 have have VERB cana-1834 357 3 ψ(ΞΈ(Ο•1,g1)(x)βŠ•ΞΈ(Ο•1,g1)(y))=ψ(ΞΈ(Ο•1,g1)(x+y))=ΞΈ(Ο•2,g2)Ο•1(x+y)=ΞΈ(Ο•2,g2)Ο•1(x)+Ο•1(y ψ(ΞΈ(Ο•1,g1)(x)βŠ•ΞΈ(Ο•1,g1)(y))=ψ(ΞΈ(Ο•1,g1)(x+y))=ΞΈ(Ο•2,g2)Ο•1(x+y)=ΞΈ(Ο•2,g2)Ο•1(x)+Ο•1(y PROPN cana-1834 357 4 ) ) PUNCT cana-1834 357 5 also also ADV cana-1834 357 6 , , PUNCT cana-1834 357 7 we we PRON cana-1834 357 8 have have VERB cana-1834 357 9 ψ(ΞΈ(Ο•1,g1)(x)βŠ™Ξ³βŠ™π›ΎβŠ™Ξ³βŠ™ΞΈ(Ο•1,g1)(y))=ψ(ΞΈ(Ο•1,g1)(xβŠ™Ξ³βŠ™y ψ(ΞΈ(Ο•1,g1)(x)βŠ™Ξ³βŠ™π›ΎβŠ™Ξ³βŠ™ΞΈ(Ο•1,g1)(y))=ψ(ΞΈ(Ο•1,g1)(xβŠ™Ξ³βŠ™y NOUN cana-1834 357 10 ) ) PUNCT cana-1834 358 1 = = NOUN cana-1834 358 2 ΞΈ(Ο•2,g2)Ο•1(xβŠ™Ξ³βŠ™y ΞΈ(Ο•2,g2)Ο•1(xβŠ™Ξ³βŠ™y X cana-1834 358 3 ) ) PUNCT cana-1834 358 4 = = SYM cana-1834 358 5 ΞΈ(Ο•2,g2)Ο•1(x)g1(𝛾)Ο•1(y ΞΈ(Ο•2,g2)Ο•1(x)g1(𝛾)Ο•1(y NOUN cana-1834 358 6 ) ) PUNCT cana-1834 359 1 = = NOUN cana-1834 359 2 ΞΈ(Ο•2,g2)Ο•1(x)βŠ™g1(𝛾)βŠ™ΞΈ(Ο•2,g2)Ο•1(y ΞΈ(Ο•2,g2)Ο•1(x)βŠ™g1(𝛾)βŠ™ΞΈ(Ο•2,g2)Ο•1(y X cana-1834 359 3 ) ) PUNCT cana-1834 359 4 = = PUNCT cana-1834 359 5 therefore therefore ADV cana-1834 359 6 , , PUNCT cana-1834 359 7 ( ( PUNCT cana-1834 359 8 ψ ψ X cana-1834 359 9 , , PUNCT cana-1834 359 10 g1 g1 NOUN cana-1834 359 11 ) ) PUNCT cana-1834 359 12 is be AUX cana-1834 359 13 a a DET cana-1834 359 14 homomorphism homomorphism NOUN cana-1834 359 15 .also .also PUNCT cana-1834 359 16 , , PUNCT cana-1834 359 17 we we PRON cana-1834 359 18 have have VERB cana-1834 359 19 communications communication NOUN cana-1834 359 20 on on ADP cana-1834 359 21 applied apply VERB cana-1834 359 22 nonlinear nonlinear ADJ cana-1834 359 23 analysis analysis NOUN cana-1834 359 24 issn issn NOUN cana-1834 359 25 : : PUNCT cana-1834 359 26 1074 1074 NUM cana-1834 359 27 - - PUNCT cana-1834 359 28 133x 133x NUM cana-1834 359 29 vol vol NOUN cana-1834 359 30 32 32 NUM cana-1834 360 1 no no NOUN cana-1834 360 2 . . NOUN cana-1834 360 3 2 2 NUM cana-1834 360 4 ( ( PUNCT cana-1834 360 5 2025 2025 NUM cana-1834 360 6 ) ) PUNCT cana-1834 360 7 591 591 NUM cana-1834 360 8 https://internationalpubls.com https://internationalpubls.com X cana-1834 360 9 ΟˆΟ€r1(x)= ΟˆΟ€r1(x)= NOUN cana-1834 360 10 ψθ(Ο•1,g1)(x)= ψθ(Ο•1,g1)(x)= NOUN cana-1834 360 11 ΞΈ(Ο•2,g2)Ο•1(x)= ΞΈ(Ο•2,g2)Ο•1(x)= PROPN cana-1834 360 12 Ο€r2Ο•1(x Ο€r2Ο•1(x NOUN cana-1834 360 13 ) ) PUNCT cana-1834 360 14 , , PUNCT cana-1834 360 15 and and CCONJ cana-1834 360 16 g1o1𝛾1=1𝛾2og1.this g1o1𝛾1=1𝛾2og1.this PRON cana-1834 360 17 shows show VERB cana-1834 360 18 that that SCONJ cana-1834 360 19 the the DET cana-1834 360 20 diagram diagram NOUN cana-1834 360 21 is be AUX cana-1834 360 22 commutative commutative ADJ cana-1834 360 23 . . PUNCT cana-1834 361 1 let(ψ¯,g1):(t1 let(ψ¯,g1):(t1 PROPN cana-1834 361 2 : : PUNCT cana-1834 361 3 ΞΈ(Ο•1,g1),𝛾1 ΞΈ(Ο•1,g1),𝛾1 NUM cana-1834 361 4 , , PUNCT cana-1834 361 5 [ [ PUNCT cana-1834 361 6 ] ] X cana-1834 361 7 ) ) PUNCT cana-1834 361 8 βˆ’β†’(t2 βˆ’β†’(t2 NOUN cana-1834 361 9 : : PUNCT cana-1834 361 10 ΞΈ(Ο•2,g2),𝛾2 ΞΈ(Ο•2,g2),𝛾2 VERB cana-1834 361 11 , , PUNCT cana-1834 361 12 [ [ PUNCT cana-1834 361 13 ] ] X cana-1834 361 14 ) ) PUNCT cana-1834 361 15 be be AUX cana-1834 361 16 a a DET cana-1834 361 17 homomorphism homomorphism NOUN cana-1834 361 18 which which PRON cana-1834 361 19 makes make VERB cana-1834 361 20 the the DET cana-1834 361 21 diagram diagram NOUN cana-1834 361 22 commutative commutative ADJ cana-1834 361 23 . . PUNCT cana-1834 362 1 then then ADV cana-1834 362 2 , , PUNCT cana-1834 362 3 we we PRON cana-1834 362 4 have have VERB cana-1834 362 5 ψ¯θ(Ο•1,g1)(x)= ψ¯θ(Ο•1,g1)(x)= VERB cana-1834 362 6 ΟˆΒ―Ο€t1(x)=Ο€t2Ο•1(x)=ΞΈ(Ο•2,g2)Ο•1(x)=ψθ(Ο•1,g1)(x ΟˆΒ―Ο€t1(x)=Ο€t2Ο•1(x)=ΞΈ(Ο•2,g2)Ο•1(x)=ψθ(Ο•1,g1)(x NOUN cana-1834 362 7 ) ) PUNCT cana-1834 362 8 . . PUNCT cana-1834 363 1 thus thus ADV cana-1834 363 2 , , PUNCT cana-1834 363 3 ( ( PUNCT cana-1834 363 4 ψ ψ X cana-1834 363 5 , , PUNCT cana-1834 363 6 g1 g1 NOUN cana-1834 363 7 ) ) PUNCT cana-1834 363 8 is be AUX cana-1834 363 9 unique unique ADJ cana-1834 363 10 and and CCONJ cana-1834 363 11 the the DET cana-1834 363 12 proof proof NOUN cana-1834 363 13 is be AUX cana-1834 363 14 completed complete VERB cana-1834 363 15 . . PUNCT cana-1834 364 1 acknowledgements acknowledgement NOUN cana-1834 364 2 : : PUNCT cana-1834 364 3 the the DET cana-1834 364 4 first first ADJ cana-1834 364 5 and and CCONJ cana-1834 364 6 second second ADJ cana-1834 364 7 authors author NOUN cana-1834 364 8 express express VERB cana-1834 364 9 their their PRON cana-1834 364 10 warmest warm ADJ cana-1834 364 11 thanks thank NOUN cana-1834 364 12 to to ADP cana-1834 364 13 the the DET cana-1834 364 14 research research NOUN cana-1834 364 15 director director NOUN cana-1834 364 16 dr dr PROPN cana-1834 364 17 . . PROPN cana-1834 364 18 d. d. PROPN cana-1834 364 19 madhusudana madhusudana PROPN cana-1834 364 20 rao rao PROPN cana-1834 364 21 , , PUNCT cana-1834 364 22 department department NOUN cana-1834 364 23 of of ADP cana-1834 364 24 mathematics mathematic NOUN cana-1834 364 25 , , PUNCT cana-1834 364 26 govt govt NOUN cana-1834 364 27 . . PUNCT cana-1834 365 1 women woman NOUN cana-1834 365 2 degree degree PROPN cana-1834 365 3 college college PROPN cana-1834 365 4 , , PUNCT cana-1834 365 5 sambasivapet sambasivapet ADJ cana-1834 365 6 , , PUNCT cana-1834 365 7 guntur guntur PROPN cana-1834 365 8 . . PUNCT cana-1834 366 1 the the DET cana-1834 366 2 authors author NOUN cana-1834 366 3 would would AUX cana-1834 366 4 like like VERB cana-1834 366 5 to to PART cana-1834 366 6 thank thank VERB cana-1834 366 7 the the DET cana-1834 366 8 experts expert NOUN cana-1834 366 9 who who PRON cana-1834 366 10 have have AUX cana-1834 366 11 contributed contribute VERB cana-1834 366 12 towards towards ADP cana-1834 366 13 preparation preparation NOUN cana-1834 366 14 and and CCONJ cana-1834 366 15 development development NOUN cana-1834 366 16 of of ADP cana-1834 366 17 this this DET cana-1834 366 18 research research NOUN cana-1834 366 19 article article NOUN cana-1834 366 20 and and CCONJ cana-1834 366 21 the the DET cana-1834 366 22 referees referee NOUN cana-1834 366 23 , , PUNCT cana-1834 366 24 chief chief ADJ cana-1834 366 25 editor editor NOUN cana-1834 366 26 for for ADP cana-1834 366 27 the the DET cana-1834 366 28 valuable valuable ADJ cana-1834 366 29 suggestions suggestion NOUN cana-1834 366 30 and and CCONJ cana-1834 366 31 corrections correction NOUN cana-1834 366 32 for for ADP cana-1834 366 33 the the DET cana-1834 366 34 improvement improvement NOUN cana-1834 366 35 of of ADP cana-1834 366 36 this this DET cana-1834 366 37 research research NOUN cana-1834 366 38 article article NOUN cana-1834 366 39 . . PUNCT cana-1834 367 1 references reference NOUN cana-1834 367 2 [ [ X cana-1834 367 3 1 1 NUM cana-1834 367 4 ] ] X cana-1834 367 5 dutta dutta NOUN cana-1834 367 6 . . PUNCT cana-1834 368 1 t. t. PROPN cana-1834 368 2 k. k. PROPN cana-1834 368 3 and and CCONJ cana-1834 368 4 kar kar PROPN cana-1834 368 5 . . PUNCT cana-1834 369 1 s. s. PROPN cana-1834 369 2 ,on ,on PUNCT cana-1834 369 3 regular regular ADJ cana-1834 369 4 ternary ternary ADJ cana-1834 369 5 semiring semiring NOUN cana-1834 369 6 , , PUNCT cana-1834 369 7 advances advance NOUN cana-1834 369 8 in in ADP cana-1834 369 9 algebra algebra NOUN cana-1834 369 10 , , PUNCT cana-1834 369 11 proceedings proceeding NOUN cana-1834 369 12 of of ADP cana-1834 369 13 the the DET cana-1834 369 14 icm icm PROPN cana-1834 369 15 satellite satellite PROPN cana-1834 369 16 conference conference NOUN cana-1834 369 17 in in ADP cana-1834 369 18 algebra algebra PROPN cana-1834 369 19 and and CCONJ cana-1834 369 20 related related ADJ cana-1834 369 21 topics topic NOUN cana-1834 369 22 , , PUNCT cana-1834 369 23 world world NOUN cana-1834 369 24 scientific scientific ADJ cana-1834 369 25 ( ( PUNCT cana-1834 369 26 2003 2003 NUM cana-1834 369 27 ) ) PUNCT cana-1834 369 28 , , PUNCT cana-1834 369 29 343 343 NUM cana-1834 369 30 - - SYM cana-1834 369 31 355 355 NUM cana-1834 369 32 . . PUNCT cana-1834 370 1 2 2 X cana-1834 370 2 . . PUNCT cana-1834 371 1 [ [ X cana-1834 371 2 2 2 NUM cana-1834 371 3 ] ] X cana-1834 371 4 dutta dutta NOUN cana-1834 371 5 . . PUNCT cana-1834 372 1 t. t. PROPN cana-1834 372 2 k k PROPN cana-1834 372 3 and and CCONJ cana-1834 372 4 das das PROPN cana-1834 372 5 . . PROPN cana-1834 372 6 m. m. PROPN cana-1834 372 7 l. l. PROPN cana-1834 372 8 , , PUNCT cana-1834 372 9 on on ADP cana-1834 372 10 strongly strongly ADV cana-1834 372 11 primesemiring primesemiring NOUN cana-1834 372 12 , , PUNCT cana-1834 372 13 bull bull NOUN cana-1834 372 14 . . PUNCT cana-1834 373 1 malays malays PROPN cana-1834 373 2 . . PUNCT cana-1834 374 1 math math NOUN cana-1834 374 2 . . PUNCT cana-1834 375 1 sci sci PROPN cana-1834 375 2 . . PROPN cana-1834 375 3 soc soc PROPN cana-1834 375 4 . . PUNCT cana-1834 376 1 ( ( PUNCT cana-1834 376 2 2 2 NUM cana-1834 376 3 ) ) PUNCT cana-1834 376 4 30 30 NUM cana-1834 376 5 ( ( PUNCT cana-1834 376 6 2 2 NUM cana-1834 376 7 ) ) PUNCT cana-1834 376 8 ( ( PUNCT cana-1834 376 9 2007 2007 NUM cana-1834 376 10 ) ) PUNCT cana-1834 376 11 , , PUNCT cana-1834 376 12 135141 135141 NUM cana-1834 376 13 . . PUNCT cana-1834 377 1 [ [ X cana-1834 377 2 3 3 X cana-1834 377 3 ] ] PUNCT cana-1834 377 4 handelman handelman NOUN cana-1834 377 5 . . PUNCT cana-1834 378 1 d. d. PROPN cana-1834 378 2 and and CCONJ cana-1834 378 3 lawrence lawrence PROPN cana-1834 378 4 . . PUNCT cana-1834 379 1 j. j. PROPN cana-1834 379 2 ,strongly ,strongly PUNCT cana-1834 379 3 prime prime PROPN cana-1834 379 4 rings ring NOUN cana-1834 379 5 , , PUNCT cana-1834 379 6 trans trans PROPN cana-1834 379 7 . . PROPN cana-1834 380 1 amer amer PROPN cana-1834 380 2 . . PUNCT cana-1834 380 3 mat mat PROPN cana-1834 380 4 . . PUNCT cana-1834 380 5 soc soc PROPN cana-1834 380 6 . . PUNCT cana-1834 381 1 211 211 NUM cana-1834 381 2 ( ( PUNCT cana-1834 381 3 1975 1975 NUM cana-1834 381 4 ) ) PUNCT cana-1834 381 5 , , PUNCT cana-1834 381 6 209 209 NUM cana-1834 381 7 - - SYM cana-1834 381 8 223 223 NUM cana-1834 381 9 . . PUNCT cana-1834 382 1 [ [ X cana-1834 382 2 4 4 NUM cana-1834 382 3 ] ] X cana-1834 382 4 lister lister PROPN cana-1834 382 5 , , PUNCT cana-1834 382 6 w. w. PROPN cana-1834 382 7 g. g. PROPN cana-1834 382 8 ,ternary ,ternary PUNCT cana-1834 382 9 rings ring NOUN cana-1834 382 10 , , PUNCT cana-1834 382 11 trans trans PROPN cana-1834 382 12 . . PROPN cana-1834 382 13 amer amer PROPN cana-1834 382 14 . . PUNCT cana-1834 382 15 math math PROPN cana-1834 382 16 . . PUNCT cana-1834 383 1 soc soc PROPN cana-1834 383 2 . . PUNCT cana-1834 384 1 154 154 NUM cana-1834 384 2 ( ( PUNCT cana-1834 384 3 1971 1971 NUM cana-1834 384 4 ) ) PUNCT cana-1834 384 5 , , PUNCT cana-1834 384 6 37 37 NUM cana-1834 384 7 - - SYM cana-1834 384 8 55 55 NUM cana-1834 384 9 . . PUNCT cana-1834 385 1 5 5 NUM cana-1834 385 2 . . PUNCT cana-1834 386 1 [ [ X cana-1834 386 2 5 5 NUM cana-1834 386 3 ] ] SYM cana-1834 386 4 sajanilavanya sajanilavanya NOUN cana-1834 386 5 . . PUNCT cana-1834 387 1 m m PROPN cana-1834 387 2 , , PUNCT cana-1834 387 3 madhusudhana madhusudhana PROPN cana-1834 387 4 rao rao PROPN cana-1834 387 5 . . PUNCT cana-1834 388 1 d. d. PROPN cana-1834 388 2 and and CCONJ cana-1834 388 3 syam syam PROPN cana-1834 388 4 julius julius PROPN cana-1834 388 5 rajendra rajendra PROPN cana-1834 388 6 . . PUNCT cana-1834 389 1 v. v. ADP cana-1834 389 2 ,on ,on PUNCT cana-1834 389 3 lateral lateral ADJ cana-1834 389 4 ternary ternary ADJ cana-1834 389 5 Ξ³ Ξ³ NOUN cana-1834 389 6 - - NOUN cana-1834 389 7 ideals ideal NOUN cana-1834 389 8 of of ADP cana-1834 389 9 ternary ternary ADJ cana-1834 389 10 Ξ³semirings Ξ³semiring NOUN cana-1834 389 11 , , PUNCT cana-1834 389 12 american american PROPN cana-1834 389 13 international international PROPN cana-1834 389 14 journal journal PROPN cana-1834 389 15 of of ADP cana-1834 389 16 research research NOUN cana-1834 389 17 in in ADP cana-1834 389 18 science science NOUN cana-1834 389 19 , , PUNCT cana-1834 389 20 technology technology NOUN cana-1834 389 21 , , PUNCT cana-1834 389 22 engineering engineering NOUN cana-1834 389 23 & & CCONJ cana-1834 389 24 mathematics mathematics PROPN cana-1834 389 25 , , PUNCT cana-1834 389 26 12(1 12(1 NUM cana-1834 389 27 ) ) PUNCT cana-1834 389 28 , , PUNCT cana-1834 389 29 september september PROPN cana-1834 389 30 - - PUNCT cana-1834 389 31 november november PROPN cana-1834 389 32 , , PUNCT cana-1834 389 33 2015 2015 NUM cana-1834 389 34 , , PUNCT cana-1834 389 35 11 11 NUM cana-1834 389 36 - - SYM cana-1834 389 37 14 14 NUM cana-1834 389 38 . . PUNCT cana-1834 390 1 [ [ X cana-1834 390 2 6 6 NUM cana-1834 390 3 ] ] X cana-1834 390 4 sajanilavanya sajanilavanya NOUN cana-1834 390 5 . . PUNCT cana-1834 391 1 m m PROPN cana-1834 391 2 , , PUNCT cana-1834 391 3 madhusudhana madhusudhana PROPN cana-1834 391 4 rao rao PROPN cana-1834 391 5 . . PUNCT cana-1834 392 1 d. d. PROPN cana-1834 392 2 and and CCONJ cana-1834 392 3 syam syam PROPN cana-1834 392 4 julius julius PROPN cana-1834 392 5 rajendra rajendra PROPN cana-1834 392 6 . . PUNCT cana-1834 393 1 v. v. ADV cana-1834 393 2 , , PUNCT cana-1834 393 3 on on ADP cana-1834 393 4 quasi quasi ADJ cana-1834 393 5 - - ADJ cana-1834 393 6 ternary ternary ADJ cana-1834 393 7 -ideals -ideal NOUN cana-1834 393 8 and and CCONJ cana-1834 393 9 biternary biternary ADJ cana-1834 393 10 -ideals -ideal NOUN cana-1834 393 11 in in ADP cana-1834 393 12 ternary ternary ADJ cana-1834 393 13 -semirings -semiring NOUN cana-1834 393 14 , , PUNCT cana-1834 393 15 international international ADJ cana-1834 393 16 journal journal NOUN cana-1834 393 17 of of ADP cana-1834 393 18 mathematics mathematics PROPN cana-1834 393 19 and and CCONJ cana-1834 393 20 statistics statistic NOUN cana-1834 393 21 invention invention NOUN cana-1834 393 22 ( ( PUNCT cana-1834 393 23 ijmsi ijmsi NOUN cana-1834 393 24 ) ) PUNCT cana-1834 393 25 , , PUNCT cana-1834 393 26 volume volume NOUN cana-1834 393 27 3 3 NUM cana-1834 393 28 issue issue NOUN cana-1834 393 29 6 6 NUM cana-1834 393 30 , , PUNCT cana-1834 393 31 ( ( PUNCT cana-1834 393 32 september.2015 september.2015 NOUN cana-1834 393 33 ) ) PUNCT cana-1834 393 34 , , PUNCT cana-1834 393 35 pp-05 pp-05 NUM cana-1834 393 36 - - SYM cana-1834 393 37 14 14 NUM cana-1834 393 38 . . PUNCT cana-1834 394 1 [ [ X cana-1834 394 2 7 7 NUM cana-1834 394 3 ] ] X cana-1834 394 4 sajanilavanya sajanilavanya NOUN cana-1834 394 5 . . PUNCT cana-1834 395 1 m m PROPN cana-1834 395 2 , , PUNCT cana-1834 395 3 madhusudhana madhusudhana PROPN cana-1834 395 4 rao rao PROPN cana-1834 395 5 . . PUNCT cana-1834 396 1 d. d. PROPN cana-1834 396 2 and and CCONJ cana-1834 396 3 syam syam PROPN cana-1834 396 4 julius julius PROPN cana-1834 396 5 rajendra rajendra PROPN cana-1834 396 6 . . PUNCT cana-1834 397 1 v. v. ADP cana-1834 397 2 ,prime ,prime PUNCT cana-1834 397 3 bi bi ADJ cana-1834 397 4 - - ADJ cana-1834 397 5 ternary ternary ADJ cana-1834 397 6 -ideals -ideal NOUN cana-1834 397 7 in in ADP cana-1834 397 8 ternary ternary ADJ cana-1834 397 9 semirings semiring NOUN cana-1834 397 10 , , PUNCT cana-1834 397 11 british british ADJ cana-1834 397 12 journal journal PROPN cana-1834 397 13 of of ADP cana-1834 397 14 research research NOUN cana-1834 397 15 , , PUNCT cana-1834 397 16 ( ( PUNCT cana-1834 397 17 2 2 NUM cana-1834 397 18 ) ) PUNCT cana-1834 397 19 , , PUNCT cana-1834 397 20 ( ( PUNCT cana-1834 397 21 6 6 NUM cana-1834 397 22 ) ) PUNCT cana-1834 397 23 , , PUNCT cana-1834 397 24 ( ( PUNCT cana-1834 397 25 2015 2015 NUM cana-1834 397 26 ) ) PUNCT cana-1834 397 27 156 156 NUM cana-1834 397 28 - - SYM cana-1834 397 29 166 166 NUM cana-1834 397 30 . . PUNCT cana-1834 398 1 [ [ X cana-1834 398 2 8 8 NUM cana-1834 398 3 ] ] X cana-1834 398 4 sajanilavanya sajanilavanya NOUN cana-1834 398 5 . . PUNCT cana-1834 399 1 m m PROPN cana-1834 399 2 , , PUNCT cana-1834 399 3 madhusudhana madhusudhana PROPN cana-1834 399 4 rao rao PROPN cana-1834 399 5 . . PUNCT cana-1834 400 1 d. d. PROPN cana-1834 400 2 and and CCONJ cana-1834 400 3 syam syam PROPN cana-1834 400 4 julius julius PROPN cana-1834 400 5 rajendra rajendra PROPN cana-1834 400 6 . . PUNCT cana-1834 401 1 v. v. ADV cana-1834 401 2 ,a ,a PUNCT cana-1834 401 3 study study NOUN cana-1834 401 4 on on ADP cana-1834 401 5 the the DET cana-1834 401 6 jacobson jacobson PROPN cana-1834 401 7 radical radical PROPN cana-1834 401 8 of of ADP cana-1834 401 9 a a DET cana-1834 401 10 ternary ternary ADJ cana-1834 401 11 -semiring -semiring NOUN cana-1834 401 12 , , PUNCT cana-1834 401 13 international international ADJ cana-1834 401 14 journal journal NOUN cana-1834 401 15 of of ADP cana-1834 401 16 mathematics mathematics PROPN cana-1834 401 17 and and CCONJ cana-1834 401 18 computer computer NOUN cana-1834 401 19 applications application NOUN cana-1834 401 20 research research NOUN cana-1834 401 21 ( ( PUNCT cana-1834 401 22 ijmcar ijmcar NOUN cana-1834 401 23 ) ) PUNCT cana-1834 401 24 , , PUNCT cana-1834 401 25 vol vol NOUN cana-1834 401 26 . . PROPN cana-1834 401 27 6 6 NUM cana-1834 401 28 , , PUNCT cana-1834 401 29 issue1,(feb issue1,(feb PROPN cana-1834 401 30 2016 2016 NUM cana-1834 401 31 ) ) PUNCT cana-1834 401 32 , , PUNCT cana-1834 401 33 17 17 NUM cana-1834 401 34 - - SYM cana-1834 401 35 30 30 NUM cana-1834 401 36 . . PUNCT cana-1834 402 1 [ [ X cana-1834 402 2 9 9 X cana-1834 402 3 ] ] PUNCT cana-1834 402 4 p. p. NOUN cana-1834 402 5 a. a. NOUN cana-1834 402 6 grillet grillet PROPN cana-1834 402 7 , , PUNCT cana-1834 402 8 semigroups semigroup NOUN cana-1834 402 9 , , PUNCT cana-1834 402 10 an an DET cana-1834 402 11 introduction introduction NOUN cana-1834 402 12 to to ADP cana-1834 402 13 structure structure NOUN cana-1834 402 14 theory theory NOUN cana-1834 402 15 , , PUNCT cana-1834 402 16 dekker dekker PROPN cana-1834 402 17 new new PROPN cana-1834 402 18 york york PROPN cana-1834 402 19 1995 1995 NUM cana-1834 402 20 . . PUNCT cana-1834 403 1 [ [ X cana-1834 403 2 10 10 NUM cana-1834 403 3 ] ] PUNCT cana-1834 403 4 m. m. NOUN cana-1834 403 5 p. p. NOUN cana-1834 403 6 grillet grillet NOUN cana-1834 403 7 , , PUNCT cana-1834 403 8 p. p. NOUN cana-1834 403 9 a. a. NOUN cana-1834 403 10 grillet grillet PROPN cana-1834 403 11 , , PUNCT cana-1834 403 12 completely completely ADV cana-1834 403 13 0 0 NUM cana-1834 403 14 - - PUNCT cana-1834 403 15 simple simple ADJ cana-1834 403 16 semirings semiring NOUN cana-1834 403 17 , , PUNCT cana-1834 403 18 trans trans PROPN cana-1834 403 19 . . PROPN cana-1834 404 1 amer amer PROPN cana-1834 404 2 . . PUNCT cana-1834 404 3 math math PROPN cana-1834 404 4 . . PUNCT cana-1834 405 1 soc soc PROPN cana-1834 405 2 . . PUNCT cana-1834 406 1 155 155 NUM cana-1834 406 2 ( ( PUNCT cana-1834 406 3 1971 1971 NUM cana-1834 406 4 ) ) PUNCT cana-1834 406 5 19 19 NUM cana-1834 406 6 - - SYM cana-1834 406 7 33 33 NUM cana-1834 406 8 . . PUNCT cana-1834 407 1 [ [ X cana-1834 407 2 11 11 NUM cana-1834 407 3 ] ] X cana-1834 407 4 h. h. PROPN cana-1834 407 5 hedayati hedayati PROPN cana-1834 407 6 , , PUNCT cana-1834 407 7 generalized generalize VERB cana-1834 407 8 fuzzy fuzzy ADJ cana-1834 407 9 k k NOUN cana-1834 407 10 - - NOUN cana-1834 407 11 ideals ideal NOUN cana-1834 407 12 of of ADP cana-1834 407 13 semirings semiring NOUN cana-1834 407 14 with with ADP cana-1834 407 15 interval interval NOUN cana-1834 407 16 - - PUNCT cana-1834 407 17 valued value VERB cana-1834 407 18 membership membership NOUN cana-1834 407 19 functions function NOUN cana-1834 407 20 , , PUNCT cana-1834 407 21 bull bull NOUN cana-1834 407 22 . . PUNCT cana-1834 408 1 malays malays PROPN cana-1834 408 2 . . PUNCT cana-1834 409 1 math math NOUN cana-1834 409 2 . . PUNCT cana-1834 410 1 sci sci PROPN cana-1834 410 2 . . PROPN cana-1834 410 3 soc soc PROPN cana-1834 410 4 . . PUNCT cana-1834 411 1 2(32)3 2(32)3 NUM cana-1834 411 2 ( ( PUNCT cana-1834 411 3 2009 2009 NUM cana-1834 411 4 ) ) PUNCT cana-1834 411 5 409 409 NUM cana-1834 411 6 - - SYM cana-1834 411 7 424 424 NUM cana-1834 411 8 . . PUNCT cana-1834 412 1 [ [ X cana-1834 412 2 12 12 NUM cana-1834 412 3 ] ] X cana-1834 412 4 h. h. PROPN cana-1834 412 5 hedayati hedayati PROPN cana-1834 412 6 , , PUNCT cana-1834 412 7 r. r. PROPN cana-1834 412 8 ameri ameri PROPN cana-1834 412 9 , , PUNCT cana-1834 412 10 regular regular ADJ cana-1834 412 11 and and CCONJ cana-1834 412 12 fundamental fundamental ADJ cana-1834 412 13 relations relation NOUN cana-1834 412 14 on on ADP cana-1834 412 15 k k ADJ cana-1834 412 16 - - ADJ cana-1834 412 17 hyper hyper ADJ cana-1834 412 18 ideals ideal NOUN cana-1834 412 19 of of ADP cana-1834 412 20 semi semi ADJ cana-1834 412 21 - - ADJ cana-1834 412 22 hyper hyper ADJ cana-1834 412 23 rings ring NOUN cana-1834 412 24 , , PUNCT cana-1834 412 25 bull bull NOUN cana-1834 412 26 . . PUNCT cana-1834 413 1 calcutta calcutta PROPN cana-1834 413 2 math math PROPN cana-1834 413 3 . . PUNCT cana-1834 414 1 soc soc PROPN cana-1834 414 2 . . PUNCT cana-1834 415 1 101(2 101(2 NUM cana-1834 415 2 ) ) PUNCT cana-1834 415 3 ( ( PUNCT cana-1834 415 4 2009 2009 NUM cana-1834 415 5 ) ) PUNCT cana-1834 415 6 105 105 NUM cana-1834 415 7 - - SYM cana-1834 415 8 114 114 NUM cana-1834 415 9 . . PUNCT cana-1834 416 1 [ [ X cana-1834 416 2 13 13 NUM cana-1834 416 3 ] ] X cana-1834 416 4 h. h. PROPN cana-1834 416 5 hedayati hedayati PROPN cana-1834 416 6 , , PUNCT cana-1834 416 7 b. b. PROPN cana-1834 416 8 davvaz davvaz PROPN cana-1834 416 9 , , PUNCT cana-1834 416 10 fundamental fundamental ADJ cana-1834 416 11 relation relation NOUN cana-1834 416 12 on on ADP cana-1834 416 13 Ξ³ Ξ³ PROPN cana-1834 416 14 - - ADJ cana-1834 416 15 hyper hyper ADJ cana-1834 416 16 rings ring NOUN cana-1834 416 17 , , PUNCT cana-1834 416 18 accepted accept VERB cana-1834 416 19 by by ADP cana-1834 416 20 arts art NOUN cana-1834 416 21 combinatoria combinatoria NOUN cana-1834 416 22 . . PUNCT cana-1834 417 1 [ [ X cana-1834 417 2 14 14 NUM cana-1834 417 3 ] ] PUNCT cana-1834 417 4 s. s. PROPN cana-1834 417 5 kar kar PROPN cana-1834 417 6 , , PUNCT cana-1834 417 7 ideal ideal ADJ cana-1834 417 8 theory theory NOUN cana-1834 417 9 in in ADP cana-1834 417 10 the the DET cana-1834 417 11 ternary ternary ADJ cana-1834 417 12 semiring semire VERB cana-1834 417 13 zβˆ’ zβˆ’ PROPN cana-1834 417 14 0 0 NUM cana-1834 417 15 , , PUNCT cana-1834 417 16 tsukuba tsukuba PROPN cana-1834 417 17 j. j. PROPN cana-1834 417 18 math math PROPN cana-1834 417 19 . . PUNCT cana-1834 418 1 11(2 11(2 X cana-1834 418 2 ) ) PUNCT cana-1834 418 3 ( ( PUNCT cana-1834 418 4 1987 1987 NUM cana-1834 418 5 ) ) PUNCT cana-1834 418 6 371 371 NUM cana-1834 418 7 - - SYM cana-1834 418 8 382 382 NUM cana-1834 418 9 . . PUNCT cana-1834 419 1 [ [ X cana-1834 419 2 15 15 NUM cana-1834 419 3 ] ] X cana-1834 419 4 s. s. PROPN cana-1834 419 5 kyuno kyuno PROPN cana-1834 419 6 , , PUNCT cana-1834 419 7 prime prime ADJ cana-1834 419 8 ideals ideal NOUN cana-1834 419 9 in in ADP cana-1834 419 10 gamma gamma NOUN cana-1834 419 11 rings ring NOUN cana-1834 419 12 , , PUNCT cana-1834 419 13 pacific pacific PROPN cana-1834 419 14 j. j. PROPN cana-1834 419 15 math math PROPN cana-1834 419 16 . . PUNCT cana-1834 420 1 98(2 98(2 NUM cana-1834 420 2 ) ) PUNCT cana-1834 420 3 ( ( PUNCT cana-1834 420 4 1982 1982 NUM cana-1834 420 5 ) ) PUNCT cana-1834 420 6 375 375 NUM cana-1834 420 7 - - SYM cana-1834 420 8 379 379 NUM cana-1834 420 9 . . PUNCT cana-1834 421 1 [ [ X cana-1834 421 2 16 16 NUM cana-1834 421 3 ] ] PUNCT cana-1834 421 4 g. g. PROPN cana-1834 421 5 srinivasa srinivasa PROPN cana-1834 421 6 rao rao PROPN cana-1834 421 7 p. p. PROPN cana-1834 421 8 siva siva PROPN cana-1834 421 9 prasad prasad PROPN cana-1834 421 10 , , PUNCT cana-1834 421 11 m. m. NOUN cana-1834 421 12 vasantha vasantha PROPN cana-1834 421 13 , , PUNCT cana-1834 421 14 dr dr PROPN cana-1834 421 15 . . PROPN cana-1834 421 16 d. d. PROPN cana-1834 421 17 madhusudhana madhusudhana PROPN cana-1834 421 18 rao rao PROPN cana-1834 421 19 β€œ " PUNCT cana-1834 421 20 on on ADP cana-1834 421 21 strongly strongly ADV cana-1834 421 22 duo duo NOUN cana-1834 421 23 and and CCONJ cana-1834 421 24 duo duo NOUN cana-1834 421 25 left leave VERB cana-1834 421 26 Ξ³ Ξ³ PROPN cana-1834 421 27 - - PUNCT cana-1834 421 28 ts ts NOUN cana-1834 421 29 - - PUNCT cana-1834 421 30 acts act NOUN cana-1834 421 31 over over ADP cana-1834 421 32 ternary ternary NOUN cana-1834 421 33 - - PUNCT cana-1834 421 34 semigroups semigroup NOUN cana-1834 421 35 ” " PUNCT cana-1834 421 36 , , PUNCT cana-1834 421 37 international international ADJ cana-1834 421 38 journal journal NOUN cana-1834 421 39 of of ADP cana-1834 421 40 pure pure ADJ cana-1834 421 41 and and CCONJ cana-1834 421 42 applied applied ADJ cana-1834 421 43 mathematics mathematic NOUN cana-1834 421 44 volume volume NOUN cana-1834 421 45 113 113 NUM cana-1834 421 46 no no NOUN cana-1834 421 47 . . PROPN cana-1834 421 48 6 6 NUM cana-1834 421 49 2017 2017 NUM cana-1834 421 50 , , PUNCT cana-1834 421 51 65 65 NUM cana-1834 421 52 – – SYM cana-1834 421 53 73 73 NUM cana-1834 421 54 , , PUNCT cana-1834 421 55 issn issn NOUN cana-1834 421 56 : : PUNCT cana-1834 421 57 1311 1311 NUM cana-1834 421 58 - - SYM cana-1834 421 59 8080 8080 NUM cana-1834 421 60 ( ( PUNCT cana-1834 421 61 printed print VERB cana-1834 421 62 version version NOUN cana-1834 421 63 ) ) PUNCT cana-1834 421 64 ; ; PUNCT cana-1834 421 65 issn issn PROPN cana-1834 421 66 : : PUNCT cana-1834 421 67 1314 1314 NUM cana-1834 421 68 - - SYM cana-1834 421 69 3395 3395 NUM cana-1834 421 70 ( ( PUNCT cana-1834 421 71 on on ADP cana-1834 421 72 - - PUNCT cana-1834 421 73 line line NOUN cana-1834 421 74 version version NOUN cana-1834 421 75 ) ) PUNCT cana-1834 421 76 pp pp ADP cana-1834 421 77 67 67 NUM cana-1834 421 78 - - SYM cana-1834 421 79 73 73 NUM cana-1834 421 80 . . PUNCT cana-1834 422 1 communications communication NOUN cana-1834 422 2 on on ADP cana-1834 422 3 applied apply VERB cana-1834 422 4 nonlinear nonlinear ADJ cana-1834 422 5 analysis analysis NOUN cana-1834 422 6 issn issn NOUN cana-1834 422 7 : : PUNCT cana-1834 422 8 1074 1074 NUM cana-1834 422 9 - - PUNCT cana-1834 422 10 133x 133x NUM cana-1834 422 11 vol vol NOUN cana-1834 422 12 32 32 NUM cana-1834 422 13 no no NOUN cana-1834 422 14 . . NOUN cana-1834 422 15 2 2 NUM cana-1834 422 16 ( ( PUNCT cana-1834 422 17 2025 2025 NUM cana-1834 422 18 ) ) PUNCT cana-1834 422 19 592 592 NUM cana-1834 422 20 https://internationalpubls.com https://internationalpubls.com X cana-1834 423 1 [ [ X cana-1834 423 2 17 17 NUM cana-1834 423 3 ] ] X cana-1834 423 4 ch ch NOUN cana-1834 423 5 . . PROPN cana-1834 423 6 manikyarao manikyarao PROPN cana-1834 423 7 , , PUNCT cana-1834 423 8 p. p. PROPN cana-1834 423 9 siva siva PROPN cana-1834 423 10 prasad prasad PROPN cana-1834 423 11 , , PUNCT cana-1834 423 12 d. d. PROPN cana-1834 423 13 madhusudhana madhusudhana PROPN cana-1834 423 14 rao rao PROPN cana-1834 423 15 , , PUNCT cana-1834 424 1 g. g. PROPN cana-1834 424 2 srinivasarao,”maximal srinivasarao,”maximal PROPN cana-1834 424 3 ideal ideal NOUN cana-1834 424 4 of of ADP cana-1834 424 5 compact compact ADJ cana-1834 424 6 connected connect VERB cana-1834 424 7 topological topological ADJ cana-1834 424 8 ternary ternary ADJ cana-1834 424 9 semigroups semigroup NOUN cana-1834 424 10 ” " PUNCT cana-1834 424 11 international international ADJ cana-1834 424 12 conference conference NOUN cana-1834 424 13 on on ADP cana-1834 424 14 mathematics mathematics PROPN cana-1834 424 15 2015,at 2015,at NOUN cana-1834 424 16 : : PUNCT cana-1834 424 17 kerela kerela PROPN cana-1834 424 18 , , PUNCT cana-1834 424 19 volume volume NOUN cana-1834 424 20 : : PUNCT cana-1834 424 21 volume volume NOUN cana-1834 424 22 4 4 NUM cana-1834 424 23 issue issue NOUN cana-1834 424 24 2 2 NUM cana-1834 424 25 [ [ SYM cana-1834 424 26 18 18 NUM cana-1834 424 27 ] ] PUNCT cana-1834 424 28 p. p. NOUN cana-1834 424 29 sivaprasad sivaprasad PROPN cana-1834 424 30 , , PUNCT cana-1834 424 31 dr dr PROPN cana-1834 424 32 . . PROPN cana-1834 424 33 d. d. PROPN cana-1834 424 34 madhusudhana madhusudhana PROPN cana-1834 424 35 rao rao PROPN cana-1834 424 36 , , PUNCT cana-1834 424 37 g. g. PROPN cana-1834 424 38 srinivasa srinivasa PROPN cana-1834 424 39 rao rao PROPN cana-1834 424 40 , , PUNCT cana-1834 424 41 ” " PUNCT cana-1834 424 42 a a DET cana-1834 424 43 study study NOUN cana-1834 424 44 on on ADP cana-1834 424 45 structure structure NOUN cana-1834 424 46 of of ADP cana-1834 424 47 po po NOUN cana-1834 424 48 - - ADJ cana-1834 424 49 ternary ternary ADJ cana-1834 424 50 semirings semiring NOUN cana-1834 424 51 ” " PUNCT cana-1834 424 52 , , PUNCT cana-1834 424 53 journal journal NOUN cana-1834 424 54 of of ADP cana-1834 424 55 advances advance NOUN cana-1834 424 56 in in ADP cana-1834 424 57 mathematics mathematic NOUN cana-1834 424 58 , , PUNCT cana-1834 424 59 vol vol NOUN cana-1834 424 60 .10 .10 NUM cana-1834 424 61 , , PUNCT cana-1834 424 62 no.8,pp:3717 no.8,pp:3717 NOUN cana-1834 424 63 - - PUNCT cana-1834 424 64 3724 3724 NUM cana-1834 424 65 . . PUNCT cana-1834 425 1 [ [ X cana-1834 425 2 19 19 NUM cana-1834 425 3 ] ] PUNCT cana-1834 425 4 p. p. NOUN cana-1834 425 5 sivaprasad sivaprasad PROPN cana-1834 425 6 , , PUNCT cana-1834 425 7 dr dr PROPN cana-1834 425 8 . . PROPN cana-1834 425 9 d. d. PROPN cana-1834 425 10 madhusudhana madhusudhana PROPN cana-1834 425 11 rao rao PROPN cana-1834 425 12 , , PUNCT cana-1834 425 13 mamidipalli mamidipalli PROPN cana-1834 425 14 . . PUNCT cana-1834 426 1 vasantha vasantha PROPN cana-1834 426 2 , , PUNCT cana-1834 426 3 , , PUNCT cana-1834 426 4 b. b. PROPN cana-1834 426 5 srinivasa srinivasa PROPN cana-1834 426 6 kumar kumar PROPN cana-1834 426 7 , , PUNCT cana-1834 426 8 ” " PUNCT cana-1834 426 9 on on ADP cana-1834 426 10 Ξ³ Ξ³ PROPN cana-1834 426 11 - - PUNCT cana-1834 426 12 ts ts NOUN cana-1834 426 13 - - PUNCT cana-1834 426 14 acts act NOUN cana-1834 426 15 over over ADP cana-1834 426 16 ternary ternary ADJ cana-1834 426 17 Ξ³ Ξ³ PROPN cana-1834 426 18 - - PUNCT cana-1834 426 19 semigroups semigroup NOUN cana-1834 426 20 ” " PUNCT cana-1834 426 21 international international ADJ cana-1834 426 22 journal journal NOUN cana-1834 426 23 of of ADP cana-1834 426 24 engineering engineering PROPN cana-1834 426 25 & & CCONJ cana-1834 426 26 technology technology PROPN cana-1834 426 27 , , PUNCT cana-1834 426 28 7 7 NUM cana-1834 426 29 ( ( PUNCT cana-1834 426 30 4.10 4.10 NUM cana-1834 426 31 ) ) PUNCT cana-1834 426 32 ( ( PUNCT cana-1834 426 33 2018)pp:812 2018)pp:812 NUM cana-1834 426 34 - - SYM cana-1834 426 35 815 815 NUM cana-1834 426 36 . . PUNCT cana-1834 427 1 [ [ X cana-1834 427 2 20 20 NUM cana-1834 427 3 ] ] PUNCT cana-1834 427 4 g. g. PROPN cana-1834 427 5 srinivasa srinivasa PROPN cana-1834 427 6 rao rao PROPN cana-1834 427 7 , , PUNCT cana-1834 427 8 d. d. PROPN cana-1834 427 9 madhusudhanarao madhusudhanarao PROPN cana-1834 427 10 and and CCONJ cana-1834 427 11 p. p. PROPN cana-1834 427 12 siva siva PROPN cana-1834 427 13 prasad prasad PROPN cana-1834 427 14 , , PUNCT cana-1834 427 15 simple simple ADJ cana-1834 427 16 ternary ternary ADJ cana-1834 427 17 semi semi NOUN cana-1834 427 18 - - NOUN cana-1834 427 19 rings ring NOUN cana-1834 427 20 , , PUNCT cana-1834 427 21 the the DET cana-1834 427 22 global global ADJ cana-1834 427 23 journal journal NOUN cana-1834 427 24 of of ADP cana-1834 427 25 mathematics mathematics PROPN cana-1834 427 26 & & CCONJ cana-1834 427 27 mathematical mathematical PROPN cana-1834 427 28 sciences sciences PROPN cana-1834 427 29 , , PUNCT cana-1834 427 30 9(2 9(2 NUM cana-1834 427 31 ) ) PUNCT cana-1834 427 32 ( ( PUNCT cana-1834 427 33 2016 2016 NUM cana-1834 427 34 ) ) PUNCT cana-1834 427 35 , , PUNCT cana-1834 427 36 185 185 NUM cana-1834 427 37 - - SYM cana-1834 427 38 196 196 NUM cana-1834 427 39 . . PUNCT cana-1834 428 1 [ [ X cana-1834 428 2 21 21 NUM cana-1834 428 3 ] ] X cana-1834 428 4 d. d. PROPN cana-1834 428 5 madhusudhana madhusudhana PROPN cana-1834 428 6 rao rao PROPN cana-1834 428 7 , , PUNCT cana-1834 428 8 g. g. PROPN cana-1834 428 9 srinivasa srinivasa PROPN cana-1834 428 10 rao rao PROPN cana-1834 428 11 , , PUNCT cana-1834 428 12 special special ADJ cana-1834 428 13 elements element NOUN cana-1834 428 14 in in ADP cana-1834 428 15 ternary ternary ADJ cana-1834 428 16 semi semi ADJ cana-1834 428 17 rings ring NOUN cana-1834 428 18 , , PUNCT cana-1834 428 19 international international ADJ cana-1834 428 20 journal journal NOUN cana-1834 428 21 of of ADP cana-1834 428 22 engineering engineering NOUN cana-1834 428 23 research research NOUN cana-1834 428 24 and and CCONJ cana-1834 428 25 applications application NOUN cana-1834 428 26 , , PUNCT cana-1834 428 27 4(11 4(11 NUM cana-1834 428 28 ) ) PUNCT cana-1834 428 29 ( ( PUNCT cana-1834 428 30 2014 2014 NUM cana-1834 428 31 ) ) PUNCT cana-1834 428 32 , , PUNCT cana-1834 428 33 123 123 NUM cana-1834 428 34 - - SYM cana-1834 428 35 130 130 NUM cana-1834 428 36 . . PUNCT cana-1834 429 1 [ [ X cana-1834 429 2 22 22 NUM cana-1834 429 3 ] ] PUNCT cana-1834 429 4 g. g. PROPN cana-1834 429 5 srinivasa srinivasa PROPN cana-1834 429 6 rao rao PROPN cana-1834 429 7 , , PUNCT cana-1834 429 8 d. d. PROPN cana-1834 429 9 madhusudhana madhusudhana PROPN cana-1834 429 10 rao rao PROPN cana-1834 429 11 , , PUNCT cana-1834 429 12 structure structure NOUN cana-1834 429 13 of of ADP cana-1834 429 14 certain certain ADJ cana-1834 429 15 ideals ideal NOUN cana-1834 429 16 in in ADP cana-1834 429 17 ternary ternary ADJ cana-1834 429 18 semi semi ADJ cana-1834 429 19 rings ring NOUN cana-1834 429 20 , , PUNCT cana-1834 429 21 int int NOUN cana-1834 429 22 . . PUNCT cana-1834 430 1 j. j. PROPN cana-1834 430 2 of of ADP cana-1834 430 3 innovative innovative ADJ cana-1834 430 4 science science NOUN cana-1834 430 5 and and CCONJ cana-1834 430 6 modern modern ADJ cana-1834 430 7 engg engg PROPN cana-1834 430 8 . . PUNCT cana-1834 430 9 , , PUNCT cana-1834 430 10 3(3 3(3 NUM cana-1834 430 11 ) ) PUNCT cana-1834 430 12 ( ( PUNCT cana-1834 430 13 2015 2015 NUM cana-1834 430 14 ) ) PUNCT cana-1834 430 15 , , PUNCT cana-1834 430 16 49 49 NUM cana-1834 430 17 - - SYM cana-1834 430 18 56 56 NUM cana-1834 430 19 . . PUNCT cana-1834 431 1 [ [ X cana-1834 431 2 23 23 NUM cana-1834 431 3 ] ] PUNCT cana-1834 431 4 g. g. PROPN cana-1834 431 5 srinivasa srinivasa PROPN cana-1834 431 6 rao rao PROPN cana-1834 431 7 , , PUNCT cana-1834 431 8 d. d. PROPN cana-1834 431 9 madhusudhana madhusudhana PROPN cana-1834 431 10 rao rao PROPN cana-1834 431 11 , , PUNCT cana-1834 431 12 a a DET cana-1834 431 13 study study NOUN cana-1834 431 14 on on ADP cana-1834 431 15 ternary ternary ADJ cana-1834 431 16 semi semi ADJ cana-1834 431 17 rings ring NOUN cana-1834 431 18 , , PUNCT cana-1834 431 19 int int NOUN cana-1834 431 20 . . PUNCT cana-1834 432 1 j. j. PROPN cana-1834 432 2 of of ADP cana-1834 432 3 math math PROPN cana-1834 432 4 . . PUNCT cana-1834 433 1 archive archive NOUN cana-1834 433 2 , , PUNCT cana-1834 433 3 5(12 5(12 NUM cana-1834 433 4 ) ) PUNCT cana-1834 433 5 ( ( PUNCT cana-1834 433 6 2014 2014 NUM cana-1834 433 7 ) ) PUNCT cana-1834 433 8 , , PUNCT cana-1834 433 9 2430 2430 NUM cana-1834 433 10 . . PUNCT cana-1834 434 1 [ [ X cana-1834 434 2 24 24 NUM cana-1834 434 3 ] ] X cana-1834 434 4 g. g. PROPN cana-1834 434 5 srinivasa srinivasa PROPN cana-1834 434 6 rao rao PROPN cana-1834 434 7 , , PUNCT cana-1834 434 8 d. d. PROPN cana-1834 434 9 madhusudhana madhusudhana PROPN cana-1834 434 10 rao rao PROPN cana-1834 434 11 , , PUNCT cana-1834 434 12 characteristics characteristic NOUN cana-1834 434 13 of of ADP cana-1834 434 14 ternary ternary ADJ cana-1834 434 15 semi semi ADJ cana-1834 434 16 rings ring NOUN cana-1834 434 17 , , PUNCT cana-1834 434 18 int.j int.j PROPN cana-1834 434 19 . . PROPN cana-1834 434 20 of of ADP cana-1834 434 21 engg engg PROPN cana-1834 434 22 . . PUNCT cana-1834 435 1 res re NOUN cana-1834 435 2 . . PUNCT cana-1834 435 3 and and CCONJ cana-1834 435 4 mgt mgt PROPN cana-1834 435 5 . . PUNCT cana-1834 435 6 , , PUNCT cana-1834 435 7 2(1 2(1 NUM cana-1834 435 8 ) ) PUNCT cana-1834 435 9 ( ( PUNCT cana-1834 435 10 2015 2015 NUM cana-1834 435 11 ) ) PUNCT cana-1834 435 12 , , PUNCT cana-1834 435 13 3 3 NUM cana-1834 435 14 - - SYM cana-1834 435 15 6 6 NUM cana-1834 435 16 . . PUNCT cana-1834 436 1 [ [ X cana-1834 436 2 25 25 NUM cana-1834 436 3 ] ] PUNCT cana-1834 436 4 g. g. PROPN cana-1834 436 5 srinivasa srinivasa PROPN cana-1834 436 6 rao rao PROPN cana-1834 436 7 , , PUNCT cana-1834 436 8 a. a. PROPN cana-1834 436 9 nagamalleswara nagamalleswara PROPN cana-1834 436 10 rao rao PROPN cana-1834 436 11 , , PUNCT cana-1834 436 12 p.l.n p.l.n PROPN cana-1834 436 13 . . PROPN cana-1834 436 14 varma varma PROPN cana-1834 436 15 , , PUNCT cana-1834 436 16 d.madhusudhana d.madhusudhana PROPN cana-1834 436 17 rao rao PROPN cana-1834 436 18 , , PUNCT cana-1834 436 19 ch ch NOUN cana-1834 436 20 . . PROPN cana-1834 436 21 ramprasad ramprasad ADJ cana-1834 436 22 , , PUNCT cana-1834 436 23 prime prime ADJ cana-1834 436 24 bi bi ADJ cana-1834 436 25 - - ADJ cana-1834 436 26 interior interior ADJ cana-1834 436 27 ideals ideal NOUN cana-1834 436 28 in in ADP cana-1834 436 29 tgsr tgsr ADJ cana-1834 436 30 , , PUNCT cana-1834 436 31 malaya malaya PROPN cana-1834 436 32 journal journal PROPN cana-1834 436 33 of of ADP cana-1834 436 34 mathematika mathematika NOUN cana-1834 436 35 , , PUNCT cana-1834 436 36 vol.9 vol.9 PROPN cana-1834 436 37 , , PUNCT cana-1834 436 38 no.1 no.1 NUM cana-1834 436 39 , , PUNCT cana-1834 436 40 pp:542 pp:542 ADV cana-1834 436 41 - - PUNCT cana-1834 436 42 546 546 NUM cana-1834 436 43 , , PUNCT cana-1834 436 44 2021 2021 NUM cana-1834 436 45 . . PUNCT cana-1834 437 1 [ [ X cana-1834 437 2 26 26 NUM cana-1834 437 3 ] ] PUNCT cana-1834 437 4 g. g. PROPN cana-1834 437 5 srinivasa srinivasa PROPN cana-1834 437 6 rao rao PROPN cana-1834 437 7 , , PUNCT cana-1834 437 8 a. a. PROPN cana-1834 437 9 nagamalleswara nagamalleswara PROPN cana-1834 437 10 rao rao PROPN cana-1834 437 11 , , PUNCT cana-1834 437 12 p.l.n p.l.n PROPN cana-1834 437 13 . . PROPN cana-1834 437 14 varma varma PROPN cana-1834 437 15 , , PUNCT cana-1834 437 16 d. d. PROPN cana-1834 437 17 madhusudhana madhusudhana PROPN cana-1834 437 18 rao rao PROPN cana-1834 437 19 , , PUNCT cana-1834 437 20 ch ch NOUN cana-1834 437 21 . . PROPN cana-1834 437 22 ramprasad ramprasad ADJ cana-1834 437 23 , , PUNCT cana-1834 437 24 bi bi ADJ cana-1834 437 25 - - ADJ cana-1834 437 26 interior interior ADJ cana-1834 437 27 ideals ideal NOUN cana-1834 437 28 in in ADP cana-1834 437 29 tgsr tgsr ADJ cana-1834 437 30 , , PUNCT cana-1834 437 31 advances advance NOUN cana-1834 437 32 in in ADP cana-1834 437 33 mathematics mathematics NOUN cana-1834 437 34 scientific scientific ADJ cana-1834 437 35 journal journal NOUN cana-1834 437 36 , , PUNCT cana-1834 437 37 10 10 NUM cana-1834 437 38 ( ( PUNCT cana-1834 437 39 2021 2021 NUM cana-1834 437 40 ) ) PUNCT cana-1834 437 41 , , PUNCT cana-1834 437 42 no.3 no.3 VERB cana-1834 437 43 , , PUNCT cana-1834 437 44 pp pp CCONJ cana-1834 437 45 : : PUNCT cana-1834 437 46 1183 1183 NUM cana-1834 437 47 - - SYM cana-1834 437 48 1195 1195 NUM cana-1834 437 49 . . PUNCT cana-1834 438 1 [ [ X cana-1834 438 2 27 27 NUM cana-1834 438 3 ] ] X cana-1834 438 4 siva siva PROPN cana-1834 438 5 prasad.p prasad.p PROPN cana-1834 438 6 , , PUNCT cana-1834 438 7 revathi revathi PROPN cana-1834 438 8 . . PUNCT cana-1834 439 1 k k X cana-1834 439 2 , , PUNCT cana-1834 439 3 2 2 NUM cana-1834 439 4 , , PUNCT cana-1834 439 5 sundarayya sundarayya ADJ cana-1834 439 6 . . PUNCT cana-1834 440 1 p p NOUN cana-1834 440 2 , , PUNCT cana-1834 440 3 madhusudhana madhusudhana PROPN cana-1834 440 4 rao rao PROPN cana-1834 440 5 . . PUNCT cana-1834 441 1 d d INTJ cana-1834 441 2 , , PUNCT cana-1834 441 3 β€œ " PUNCT cana-1834 441 4 compositions composition NOUN cana-1834 441 5 of of ADP cana-1834 441 6 fuzzy fuzzy ADJ cana-1834 441 7 t t NOUN cana-1834 441 8 - - PUNCT cana-1834 441 9 ideals ideal NOUN cana-1834 441 10 in in ADP cana-1834 441 11 ternary ternary ADJ cana-1834 441 12 semi semi ADJ cana-1834 441 13 ring ring NOUN cana-1834 441 14 ” " PUNCT cana-1834 441 15 , , PUNCT cana-1834 441 16 international international ADJ cana-1834 441 17 journal journal NOUN cana-1834 441 18 of of ADP cana-1834 441 19 advanced advanced ADJ cana-1834 441 20 in in ADP cana-1834 441 21 management management NOUN cana-1834 441 22 , , PUNCT cana-1834 441 23 technology technology NOUN cana-1834 441 24 and and CCONJ cana-1834 441 25 engineering engineering NOUN cana-1834 441 26 sciences science NOUN cana-1834 441 27 volume volume NOUN cana-1834 441 28 7 7 NUM cana-1834 441 29 , , PUNCT cana-1834 441 30 issue issue NOUN cana-1834 441 31 12 12 NUM cana-1834 441 32 , , PUNCT cana-1834 441 33 2017 2017 NUM cana-1834 441 34 issn issn VERB cana-1834 441 35 no no DET cana-1834 441 36 : : PUNCT cana-1834 441 37 2249 2249 NUM cana-1834 441 38 - - PUNCT cana-1834 441 39 7455,pp:135 7455,pp:135 NUM cana-1834 441 40 - - NUM cana-1834 441 41 145 145 NUM cana-1834 441 42 . . PUNCT cana-1834 442 1 [ [ X cana-1834 442 2 28 28 NUM cana-1834 442 3 ] ] X cana-1834 442 4 p. p. PROPN cana-1834 442 5 sivaprasad sivaprasad PROPN cana-1834 442 6 , , PUNCT cana-1834 442 7 c. c. PROPN cana-1834 442 8 sreemannarayana sreemannarayana PROPN cana-1834 442 9 , , PUNCT cana-1834 442 10 d. d. PROPN cana-1834 442 11 madhusudhana madhusudhana PROPN cana-1834 442 12 rao rao PROPN cana-1834 442 13 , , PUNCT cana-1834 442 14 t. t. PROPN cana-1834 442 15 nageswara nageswara PROPN cana-1834 442 16 rao rao PROPN cana-1834 442 17 , , PUNCT cana-1834 442 18 k. k. PROPN cana-1834 442 19 anuradha anuradha PROPN cana-1834 442 20 , , PUNCT cana-1834 442 21 ” " PUNCT cana-1834 442 22 on on ADP cana-1834 442 23 leternary leternary ADJ cana-1834 442 24 semi semi ADJ cana-1834 442 25 groups group NOUN cana-1834 442 26 - - PUNCT cana-1834 442 27 i i PROPN cana-1834 442 28 ” " PUNCT cana-1834 442 29 international international ADJ cana-1834 442 30 journal journal NOUN cana-1834 442 31 of of ADP cana-1834 442 32 recent recent ADJ cana-1834 442 33 technology technology NOUN cana-1834 442 34 and and CCONJ cana-1834 442 35 engineering engineering NOUN cana-1834 442 36 ( ( PUNCT cana-1834 442 37 ijrte ijrte NOUN cana-1834 442 38 ) ) PUNCT cana-1834 442 39 issn issn PROPN cana-1834 442 40 : : PUNCT cana-1834 442 41 2277 2277 NUM cana-1834 442 42 - - SYM cana-1834 442 43 3878 3878 NUM cana-1834 442 44 , , PUNCT cana-1834 442 45 volume-7 volume-7 ADJ cana-1834 442 46 , , PUNCT cana-1834 442 47 issueicetesm issueicetesm NOUN cana-1834 442 48 , , PUNCT cana-1834 442 49 march march PROPN cana-1834 442 50 2019 2019 NUM cana-1834 442 51 , , PUNCT cana-1834 442 52 pp:165 pp:165 NOUN cana-1834 442 53 - - SYM cana-1834 442 54 167 167 NUM cana-1834 442 55 . . PUNCT cana-1834 443 1 [ [ X cana-1834 443 2 29 29 NUM cana-1834 443 3 ] ] PUNCT cana-1834 443 4 p. p. PROPN cana-1834 443 5 sivaprasad sivaprasad PROPN cana-1834 443 6 , , PUNCT cana-1834 443 7 c. c. PROPN cana-1834 443 8 sreemannarayana sreemannarayana PROPN cana-1834 443 9 , , PUNCT cana-1834 443 10 d. d. PROPN cana-1834 443 11 madhusudhana madhusudhana PROPN cana-1834 443 12 rao rao PROPN cana-1834 443 13 , , PUNCT cana-1834 443 14 t. t. PROPN cana-1834 443 15 nageswara nageswara PROPN cana-1834 443 16 rao rao PROPN cana-1834 443 17 , , PUNCT cana-1834 443 18 sajani sajani PROPN cana-1834 443 19 lavanya lavanya NOUN cana-1834 443 20 . . PUNCT cana-1834 444 1 m m PROPN cana-1834 444 2 , , PUNCT cana-1834 444 3 ” " PUNCT cana-1834 444 4 on on ADP cana-1834 444 5 leternary leternary ADJ cana-1834 444 6 semi semi ADJ cana-1834 444 7 groups groups PROPN cana-1834 444 8 - - PUNCT cana-1834 444 9 ii ii NOUN cana-1834 444 10 ” " PUNCT cana-1834 444 11 international international ADJ cana-1834 444 12 journal journal NOUN cana-1834 444 13 of of ADP cana-1834 444 14 recent recent ADJ cana-1834 444 15 technology technology NOUN cana-1834 444 16 and and CCONJ cana-1834 444 17 engineering engineering NOUN cana-1834 444 18 ( ( PUNCT cana-1834 444 19 ijrte ijrte NOUN cana-1834 444 20 ) ) PUNCT cana-1834 444 21 issn issn PROPN cana-1834 444 22 : : PUNCT cana-1834 444 23 2277 2277 NUM cana-1834 444 24 - - SYM cana-1834 444 25 3878 3878 NUM cana-1834 444 26 , , PUNCT cana-1834 444 27 volume-7 volume-7 ADJ cana-1834 444 28 , , PUNCT cana-1834 444 29 issue issue NOUN cana-1834 444 30 - - PUNCT cana-1834 444 31 icetesm icetesm NOUN cana-1834 444 32 , , PUNCT cana-1834 444 33 march march PROPN cana-1834 444 34 2019 2019 NUM cana-1834 444 35 , , PUNCT cana-1834 444 36 pp:168 pp:168 NOUN cana-1834 444 37 - - SYM cana-1834 444 38 170 170 NUM cana-1834 444 39 . . PUNCT