id sid tid token lemma pos iajs-2433 1 1 microsoft microsoft PROPN iajs-2433 1 2 word word NOUN iajs-2433 1 3 120 120 NUM iajs-2433 1 4 - - SYM iajs-2433 1 5 127 127 NUM iajs-2433 1 6     SPACE iajs-2433 1 7 120 120 NUM iajs-2433 1 8     SPACE iajs-2433 1 9 ibn ibn PROPN iajs-2433 1 10 al al PROPN iajs-2433 1 11 - - PUNCT iajs-2433 1 12 haitham haitham PROPN iajs-2433 1 13 jour jour X iajs-2433 1 14 . . PROPN iajs-2433 2 1 for for ADP iajs-2433 2 2 pure pure ADJ iajs-2433 2 3 & & CCONJ iajs-2433 2 4 appl appl PROPN iajs-2433 2 5 . . PUNCT iajs-2433 3 1 sci sci PROPN iajs-2433 3 2 . . PROPN iajs-2433 4 1 33 33 NUM iajs-2433 4 2 ( ( PUNCT iajs-2433 4 3 2 2 NUM iajs-2433 4 4 ) ) PUNCT iajs-2433 4 5 2020 2020 NUM iajs-2433 4 6             SPACE iajs-2433 4 7 on on ADP iajs-2433 4 8 sah sah NOUN iajs-2433 4 9 – – PUNCT iajs-2433 4 10 ideal ideal NOUN iajs-2433 4 11 of of ADP iajs-2433 4 12 bh bh NOUN iajs-2433 4 13 – – PUNCT iajs-2433 4 14 algebra algebra NOUN iajs-2433 4 15 alaa alaa PROPN iajs-2433 4 16 saleh saleh PROPN iajs-2433 4 17 abed abe VERB iajs-2433 4 18 alaas.abed@uokufa.edu.iq alaas.abed@uokufa.edu.iq ADJ iajs-2433 4 19 article article NOUN iajs-2433 4 20 history history NOUN iajs-2433 4 21 : : PUNCT iajs-2433 4 22 received receive VERB iajs-2433 4 23 23 23 NUM iajs-2433 4 24 june june PROPN iajs-2433 4 25 2019 2019 NUM iajs-2433 4 26 , , PUNCT iajs-2433 4 27 accepted accept VERB iajs-2433 4 28 8 8 NUM iajs-2433 4 29 september september PROPN iajs-2433 4 30 2019 2019 NUM iajs-2433 4 31 , , PUNCT iajs-2433 4 32 published publish VERB iajs-2433 4 33 in in ADP iajs-2433 4 34 april april PROPN iajs-2433 4 35 2020 2020 NUM iajs-2433 4 36 . . PUNCT iajs-2433 5 1 abstract abstract ADJ iajs-2433 5 2 the the DET iajs-2433 5 3 aim aim NOUN iajs-2433 5 4 of of ADP iajs-2433 5 5 this this DET iajs-2433 5 6 investigation investigation NOUN iajs-2433 5 7 is be AUX iajs-2433 5 8 to to PART iajs-2433 5 9 present present VERB iajs-2433 5 10 the the DET iajs-2433 5 11 idea idea NOUN iajs-2433 5 12 of of ADP iajs-2433 5 13 sah sah NOUN iajs-2433 5 14 – – PUNCT iajs-2433 5 15 ideal ideal ADJ iajs-2433 5 16 , , PUNCT iajs-2433 5 17 closed closed ADJ iajs-2433 5 18 sah sah NOUN iajs-2433 5 19 – – PUNCT iajs-2433 5 20 ideal ideal ADJ iajs-2433 5 21 and and CCONJ iajs-2433 5 22 closed closed ADJ iajs-2433 5 23 sah sah NOUN iajs-2433 5 24 – – PUNCT iajs-2433 5 25 ideal ideal ADJ iajs-2433 5 26 with with ADP iajs-2433 5 27 respect respect NOUN iajs-2433 5 28 to to ADP iajs-2433 5 29 an an DET iajs-2433 5 30 element element NOUN iajs-2433 5 31 , , PUNCT iajs-2433 5 32 𝑆𝐴𝐻 𝑆𝐴𝐻 PROPN iajs-2433 5 33 – – PUNCT iajs-2433 5 34 ideal ideal ADJ iajs-2433 5 35 and and CCONJ iajs-2433 5 36 s𝑆𝐴𝐻 s𝑆𝐴𝐻 PROPN iajs-2433 5 37 – – PUNCT iajs-2433 5 38 ideal ideal NOUN iajs-2433 5 39 of of ADP iajs-2433 5 40 bh bh NOUN iajs-2433 5 41 – – PUNCT iajs-2433 5 42 algebra algebra NOUN iajs-2433 5 43 . . PUNCT iajs-2433 6 1 we we PRON iajs-2433 6 2 detail detail VERB iajs-2433 6 3 and and CCONJ iajs-2433 6 4 show show VERB iajs-2433 6 5 theorems theorem NOUN iajs-2433 6 6 which which PRON iajs-2433 6 7 regulate regulate VERB iajs-2433 6 8 the the DET iajs-2433 6 9 relationship relationship NOUN iajs-2433 6 10 between between ADP iajs-2433 6 11 these these DET iajs-2433 6 12 ideas idea NOUN iajs-2433 6 13 and and CCONJ iajs-2433 6 14 provide provide VERB iajs-2433 6 15 some some DET iajs-2433 6 16 examples example NOUN iajs-2433 6 17 in in ADP iajs-2433 6 18 bh bh NOUN iajs-2433 6 19 – – PUNCT iajs-2433 6 20 algebra algebra NOUN iajs-2433 6 21 . . PUNCT iajs-2433 7 1 keywords keyword NOUN iajs-2433 7 2 : : PUNCT iajs-2433 7 3 bh bh NOUN iajs-2433 7 4 – – PUNCT iajs-2433 7 5 algebra algebra NOUN iajs-2433 7 6 , , PUNCT iajs-2433 7 7 sah sah NOUN iajs-2433 7 8 – – PUNCT iajs-2433 7 9 ideal ideal NOUN iajs-2433 7 10 of of ADP iajs-2433 7 11 bh bh NOUN iajs-2433 7 12 – – PUNCT iajs-2433 7 13 algebra algebra NOUN iajs-2433 7 14 , , PUNCT iajs-2433 7 15 closed closed ADJ iajs-2433 7 16 sah sah NOUN iajs-2433 7 17 – – PUNCT iajs-2433 7 18 ideal ideal ADJ iajs-2433 7 19 with with ADP iajs-2433 7 20 respect respect NOUN iajs-2433 7 21 to to ADP iajs-2433 7 22 an an DET iajs-2433 7 23 element element NOUN iajs-2433 7 24 of of ADP iajs-2433 7 25 bh bh NOUN iajs-2433 7 26 – – PUNCT iajs-2433 7 27 algebra algebra PROPN iajs-2433 7 28 , , PUNCT iajs-2433 7 29 𝑆𝐴𝐻 𝑆𝐴𝐻 PROPN iajs-2433 7 30 – – PUNCT iajs-2433 7 31 ideal ideal ADJ iajs-2433 7 32 . . PUNCT iajs-2433 8 1 1 1 X iajs-2433 8 2 . . X iajs-2433 8 3 introduction introduction NOUN iajs-2433 8 4 after after ADP iajs-2433 8 5 founding found VERB iajs-2433 8 6 of of ADP iajs-2433 8 7 fuzzy fuzzy ADJ iajs-2433 8 8 subset subset NOUN iajs-2433 8 9 by by ADP iajs-2433 8 10 zadeh zadeh PROPN iajs-2433 8 11 l. l. PROPN iajs-2433 8 12 a a PRON iajs-2433 9 1 [ [ X iajs-2433 9 2 1 1 NUM iajs-2433 9 3 ] ] PUNCT iajs-2433 9 4 . . PUNCT iajs-2433 10 1 several several ADJ iajs-2433 10 2 researchers researcher NOUN iajs-2433 10 3 presented present VERB iajs-2433 10 4 the the DET iajs-2433 10 5 generalizations generalization NOUN iajs-2433 10 6 of of ADP iajs-2433 10 7 the the DET iajs-2433 10 8 idea idea NOUN iajs-2433 10 9 of of ADP iajs-2433 10 10 fuzzy fuzzy ADJ iajs-2433 10 11 subsets subset NOUN iajs-2433 10 12 . . PUNCT iajs-2433 11 1 imai imai PROPN iajs-2433 11 2 and and CCONJ iajs-2433 11 3 iseki iseki PROPN iajs-2433 11 4 k. k. PROPN iajs-2433 11 5 established establish VERB iajs-2433 11 6 two two NUM iajs-2433 11 7 classes class NOUN iajs-2433 11 8 bck bck VERB iajs-2433 11 9 algebra algebra NOUN iajs-2433 11 10 and and CCONJ iajs-2433 11 11 bci bci NOUN iajs-2433 11 12 – – PUNCT iajs-2433 11 13 algebra algebra NOUN iajs-2433 11 14 [ [ X iajs-2433 11 15 2 2 NUM iajs-2433 11 16 , , PUNCT iajs-2433 11 17 3 3 NUM iajs-2433 11 18 ] ] PUNCT iajs-2433 11 19 . . PUNCT iajs-2433 12 1 jun jun PROPN iajs-2433 12 2 y. y. PROPN iajs-2433 12 3 b. b. PROPN iajs-2433 12 4 , , PUNCT iajs-2433 12 5 rogh rogh PROPN iajs-2433 12 6 e. e. PROPN iajs-2433 12 7 h. h. PROPN iajs-2433 12 8 and and CCONJ iajs-2433 12 9 kin kin PROPN iajs-2433 12 10 h. h. PROPN iajs-2433 12 11 s. s. PROPN iajs-2433 12 12 produced produce VERB iajs-2433 12 13 a a DET iajs-2433 12 14 new new ADJ iajs-2433 12 15 concept concept NOUN iajs-2433 12 16 , , PUNCT iajs-2433 12 17 named name VERB iajs-2433 12 18 a a DET iajs-2433 12 19 bh bh NOUN iajs-2433 12 20 – – PUNCT iajs-2433 12 21 algebra algebra NOUN iajs-2433 12 22 [ [ X iajs-2433 12 23 4 4 NUM iajs-2433 12 24 ] ] PUNCT iajs-2433 12 25 . . PUNCT iajs-2433 13 1 in in ADP iajs-2433 13 2 this this DET iajs-2433 13 3 paper paper NOUN iajs-2433 13 4 , , PUNCT iajs-2433 13 5 we we PRON iajs-2433 13 6 will will AUX iajs-2433 13 7 recall recall VERB iajs-2433 13 8 some some DET iajs-2433 13 9 basic basic ADJ iajs-2433 13 10 definitions definition NOUN iajs-2433 13 11 . . PUNCT iajs-2433 14 1 a a DET iajs-2433 14 2 bh bh NOUN iajs-2433 14 3 – – PUNCT iajs-2433 14 4 algebra algebra NOUN iajs-2433 14 5 is be AUX iajs-2433 14 6 a a DET iajs-2433 14 7 nonempty nonempty ADJ iajs-2433 14 8 set set VERB iajs-2433 14 9 ψ ψ NOUN iajs-2433 14 10 with with ADP iajs-2433 14 11 a a DET iajs-2433 14 12 binary binary ADJ iajs-2433 14 13 operation operation NOUN iajs-2433 14 14 * * PUNCT iajs-2433 14 15 satisfies satisfy VERB iajs-2433 14 16 the the DET iajs-2433 14 17 conditions condition NOUN iajs-2433 14 18 : : PUNCT iajs-2433 14 19 ж ж X iajs-2433 14 20 ∗ ∗ X iajs-2433 14 21 ж ж X iajs-2433 14 22 0 0 NUM iajs-2433 14 23 , , PUNCT iajs-2433 14 24 for for ADP iajs-2433 14 25 all all DET iajs-2433 14 26 ж ж PRON iajs-2433 14 27 ∈ ∈ PROPN iajs-2433 14 28 ψ ψ NOUN iajs-2433 14 29 , , PUNCT iajs-2433 14 30 ж ж X iajs-2433 14 31 ∗ ∗ NOUN iajs-2433 14 32 ц ц NOUN iajs-2433 14 33 0 0 NUM iajs-2433 14 34 and and CCONJ iajs-2433 14 35 ц ц NOUN iajs-2433 14 36 ∗ ∗ NOUN iajs-2433 14 37 ж ж X iajs-2433 14 38 0→ 0→ NOUN iajs-2433 14 39 ж ж PRON iajs-2433 14 40 ц ц NOUN iajs-2433 14 41 for for ADP iajs-2433 14 42 all all DET iajs-2433 14 43 ж ж NOUN iajs-2433 14 44 , , PUNCT iajs-2433 14 45 ц ц PROPN iajs-2433 14 46 ∈ ∈ PROPN iajs-2433 14 47 ψ ψ NOUN iajs-2433 14 48 and and CCONJ iajs-2433 14 49 ж ж X iajs-2433 14 50 ∗ ∗ NOUN iajs-2433 14 51 0 0 NUM iajs-2433 14 52 ж ж X iajs-2433 14 53 , , PUNCT iajs-2433 14 54 for for ADP iajs-2433 14 55 all all DET iajs-2433 14 56 ж ж PRON iajs-2433 14 57 ∈ ∈ NOUN iajs-2433 14 58 ψ ψ NOUN iajs-2433 14 59 4 4 NUM iajs-2433 14 60 . . PUNCT iajs-2433 15 1 we we PRON iajs-2433 15 2 will will AUX iajs-2433 15 3 use use VERB iajs-2433 15 4 ψ ψ NOUN iajs-2433 15 5 for for ADP iajs-2433 15 6 representing represent VERB iajs-2433 15 7 a a DET iajs-2433 15 8 bh bh NOUN iajs-2433 15 9 – – PUNCT iajs-2433 15 10 algebra algebra NOUN iajs-2433 15 11 ψ ψ NOUN iajs-2433 15 12 ; ; PUNCT iajs-2433 15 13 ∗ ∗ NOUN iajs-2433 15 14 , , PUNCT iajs-2433 15 15 0 0 NUM iajs-2433 15 16 . . PUNCT iajs-2433 16 1 let let VERB iajs-2433 16 2 𝔖 𝔖 PRON iajs-2433 16 3 a a DET iajs-2433 16 4 nonempty nonempty ADJ iajs-2433 16 5 subset subset NOUN iajs-2433 16 6 of of ADP iajs-2433 16 7 ψ ψ PRON iajs-2433 16 8 . . PUNCT iajs-2433 17 1 then then ADV iajs-2433 17 2 𝔖 𝔖 PROPN iajs-2433 17 3 is be AUX iajs-2433 17 4 named name VERB iajs-2433 17 5 an an DET iajs-2433 17 6 ideal ideal NOUN iajs-2433 17 7 of of ADP iajs-2433 17 8 ψ ψ PRON iajs-2433 17 9 if if SCONJ iajs-2433 17 10 it it PRON iajs-2433 17 11 holds hold VERB iajs-2433 17 12 : : PUNCT iajs-2433 17 13 0 0 NUM iajs-2433 17 14 ∈ ∈ PROPN iajs-2433 17 15 𝔖 𝔖 NOUN iajs-2433 17 16 ; ; PUNCT iajs-2433 17 17 ж ж X iajs-2433 17 18 ∗ ∗ X iajs-2433 17 19 ц ц PROPN iajs-2433 17 20 ∈ ∈ PROPN iajs-2433 17 21 𝔖 𝔖 PROPN iajs-2433 17 22 and and CCONJ iajs-2433 17 23 ц ц NOUN iajs-2433 17 24 ∈ ∈ NOUN iajs-2433 17 25 𝔖 𝔖 PROPN iajs-2433 17 26 → → SYM iajs-2433 17 27 ж ж X iajs-2433 17 28 ∈ ∈ PROPN iajs-2433 17 29 𝔖 𝔖 PROPN iajs-2433 17 30 4 4 NUM iajs-2433 17 31 . . PUNCT iajs-2433 18 1 let let VERB iajs-2433 18 2 ψ ψ NOUN iajs-2433 18 3 and and CCONJ iajs-2433 18 4 φ φ PROPN iajs-2433 18 5 be be AUX iajs-2433 18 6 bh bh NOUN iajs-2433 18 7 – – PUNCT iajs-2433 18 8 algebras algebras X iajs-2433 18 9 . . PUNCT iajs-2433 19 1 a a DET iajs-2433 19 2 mapping mapping NOUN iajs-2433 19 3 δ δ NOUN iajs-2433 19 4 : : PUNCT iajs-2433 19 5 ψ ψ X iajs-2433 19 6 → → SYM iajs-2433 19 7 φ φ PROPN iajs-2433 19 8 is be AUX iajs-2433 19 9 named name VERB iajs-2433 19 10 ahomomorphism ahomomorphism NOUN iajs-2433 19 11 if if SCONJ iajs-2433 19 12 : : PUNCT iajs-2433 19 13 δ δ PROPN iajs-2433 19 14 ( ( PUNCT iajs-2433 19 15 ж ж X iajs-2433 19 16 ∗ ∗ X iajs-2433 19 17 ц ц NOUN iajs-2433 19 18 ) ) PUNCT iajs-2433 19 19 δ δ PROPN iajs-2433 19 20 ( ( PUNCT iajs-2433 19 21 ж ж X iajs-2433 19 22 ∗ ∗ X iajs-2433 19 23 δ δ PROPN iajs-2433 19 24 ц ц NOUN iajs-2433 19 25 , , PUNCT iajs-2433 19 26 ∀ ∀ X iajs-2433 19 27 ж ж X iajs-2433 19 28 , , PUNCT iajs-2433 19 29 ц ц PROPN iajs-2433 19 30 ∈ ∈ PROPN iajs-2433 19 31 ψ ψ NOUN iajs-2433 19 32 . . PUNCT iajs-2433 20 1 a a DET iajs-2433 20 2 homomorphism homomorphism PROPN iajs-2433 20 3 δ δ PROPN iajs-2433 20 4 is be AUX iajs-2433 20 5 titled title VERB iajs-2433 20 6 a a DET iajs-2433 20 7 monomerphism monomerphism NOUN iajs-2433 20 8 ( ( PUNCT iajs-2433 20 9 resp resp NOUN iajs-2433 20 10 , , PUNCT iajs-2433 20 11 epimorphism epimorphism NOUN iajs-2433 20 12 ) ) PUNCT iajs-2433 20 13 if if SCONJ iajs-2433 20 14 it it PRON iajs-2433 20 15 injective injective VERB iajs-2433 20 16 ( ( PUNCT iajs-2433 20 17 resp resp NOUN iajs-2433 20 18 . . PUNCT iajs-2433 20 19 , , PUNCT iajs-2433 20 20 surjective surjective PROPN iajs-2433 20 21 ) ) PUNCT iajs-2433 20 22 . . PUNCT iajs-2433 21 1 a a DET iajs-2433 21 2 bijective bijective ADJ iajs-2433 21 3 homomorphism homomorphism NOUN iajs-2433 21 4 is be AUX iajs-2433 21 5 titled title VERB iajs-2433 21 6 an an DET iajs-2433 21 7 isomorphism isomorphism NOUN iajs-2433 21 8 . . PUNCT iajs-2433 22 1 two two NUM iajs-2433 22 2 bh bh NOUN iajs-2433 22 3 – – PUNCT iajs-2433 22 4 algebras algebras PROPN iajs-2433 22 5 ψ ψ PROPN iajs-2433 22 6 and and CCONJ iajs-2433 22 7 φ φ PROPN iajs-2433 22 8 are be AUX iajs-2433 22 9 said say VERB iajs-2433 22 10 to to PART iajs-2433 22 11 be be AUX iajs-2433 22 12 isomorphic isomorphic ADJ iajs-2433 22 13 , , PUNCT iajs-2433 22 14 written write VERB iajs-2433 22 15 ψ ψ X iajs-2433 22 16 ≅ ≅ PROPN iajs-2433 22 17 φ φ PROPN iajs-2433 22 18 , , PUNCT iajs-2433 22 19 if if SCONJ iajs-2433 22 20 there there PRON iajs-2433 22 21 exists exist VERB iajs-2433 22 22 an an DET iajs-2433 22 23 isomorphism isomorphism NOUN iajs-2433 22 24 δ δ NOUN iajs-2433 22 25 : : PUNCT iajs-2433 22 26 ψ ψ X iajs-2433 22 27 → → SYM iajs-2433 22 28 φ φ PROPN iajs-2433 22 29 . . PUNCT iajs-2433 23 1 for for ADP iajs-2433 23 2 any any DET iajs-2433 23 3 homomorphism homomorphism NOUN iajs-2433 23 4 : : PUNCT iajs-2433 23 5 ψ ψ X iajs-2433 23 6 → → SYM iajs-2433 23 7 φ φ PROPN iajs-2433 23 8 , , PUNCT iajs-2433 23 9 the the DET iajs-2433 23 10 set set NOUN iajs-2433 23 11 ж ж X iajs-2433 23 12 ∈ ∈ PROPN iajs-2433 23 13 ψ ψ NOUN iajs-2433 23 14 : : PUNCT iajs-2433 23 15 δ δ PROPN iajs-2433 23 16 ( ( PUNCT iajs-2433 23 17 ж ж X iajs-2433 23 18 ) ) PUNCT iajs-2433 23 19 = = SYM iajs-2433 23 20 0 0 X iajs-2433 23 21 ' ' PUNCT iajs-2433 23 22 is be AUX iajs-2433 23 23 titled title VERB iajs-2433 23 24 the the DET iajs-2433 23 25 kernel kernel NOUN iajs-2433 23 26 of of ADP iajs-2433 23 27 δ δ PROPN iajs-2433 23 28 , , PUNCT iajs-2433 23 29 symbolized symbolize VERB iajs-2433 23 30 by by ADP iajs-2433 23 31 ker ker PROPN iajs-2433 23 32 δ δ PROPN iajs-2433 23 33 , , PUNCT iajs-2433 23 34 and and CCONJ iajs-2433 23 35 the the DET iajs-2433 23 36 set set NOUN iajs-2433 23 37 δ δ PROPN iajs-2433 23 38 ж ж X iajs-2433 23 39 : : PUNCT iajs-2433 23 40 ж ж X iajs-2433 23 41 ∈ ∈ PROPN iajs-2433 23 42 ψ ψ X iajs-2433 23 43 is be AUX iajs-2433 23 44 named name VERB iajs-2433 23 45 the the DET iajs-2433 23 46 image image NOUN iajs-2433 23 47 of of ADP iajs-2433 23 48 δ δ PROPN iajs-2433 23 49 , , PUNCT iajs-2433 23 50 represented represent VERB iajs-2433 23 51 by by ADP iajs-2433 23 52 i i PROPN iajs-2433 23 53 m m PROPN iajs-2433 23 54 δ δ PROPN iajs-2433 23 55 . . PUNCT iajs-2433 24 1 sign sign VERB iajs-2433 24 2 that that SCONJ iajs-2433 24 3 δ δ PROPN iajs-2433 24 4 ( ( PUNCT iajs-2433 24 5 0 0 NUM iajs-2433 24 6 ) ) PUNCT iajs-2433 24 7 = = SYM iajs-2433 24 8 0 0 X iajs-2433 24 9 ' ' PUNCT iajs-2433 24 10 , , PUNCT iajs-2433 24 11 ∀ ∀ NUM iajs-2433 24 12 homomorphism homomorphism PROPN iajs-2433 24 13 δ δ PROPN iajs-2433 25 1 [ [ X iajs-2433 25 2 5 5 NUM iajs-2433 25 3 ] ] PUNCT iajs-2433 25 4 . . PUNCT iajs-2433 26 1 an an DET iajs-2433 26 2 ideal ideal ADJ iajs-2433 26 3 𝔖 𝔖 PROPN iajs-2433 26 4 of of ADP iajs-2433 26 5 ψ ψ PROPN iajs-2433 26 6 is be AUX iajs-2433 26 7 known know VERB iajs-2433 26 8 as as ADP iajs-2433 26 9 closed close VERB iajs-2433 26 10 ideal ideal NOUN iajs-2433 26 11 of of ADP iajs-2433 26 12 ψ ψ PRON iajs-2433 26 13 if if SCONJ iajs-2433 26 14 : : PUNCT iajs-2433 26 15 for for ADP iajs-2433 26 16 each each DET iajs-2433 26 17 ж ж PROPN iajs-2433 26 18 ∈ ∈ PROPN iajs-2433 26 19 𝔖 𝔖 PROPN iajs-2433 26 20 . . PUNCT iajs-2433 27 1 ibn ibn PROPN iajs-2433 27 2 al al PROPN iajs-2433 27 3 haitham haitham PROPN iajs-2433 27 4 journal journal PROPN iajs-2433 27 5 for for ADP iajs-2433 27 6 pure pure ADJ iajs-2433 27 7 and and CCONJ iajs-2433 27 8 applied apply VERB iajs-2433 27 9 science science NOUN iajs-2433 27 10 journal journal PROPN iajs-2433 27 11 homepage homepage NOUN iajs-2433 27 12 : : PUNCT iajs-2433 27 13 http://jih.uobaghdad.edu.iq/index.php/j/index http://jih.uobaghdad.edu.iq/index.php/j/index NOUN iajs-2433 27 14 doi doi NOUN iajs-2433 27 15 : : PUNCT iajs-2433 27 16 10.30526/33.2.2433 10.30526/33.2.2433 PROPN iajs-2433 27 17 department department NOUN iajs-2433 27 18 of of ADP iajs-2433 27 19 mathematics mathematic NOUN iajs-2433 27 20 , , PUNCT iajs-2433 27 21 faculty faculty NOUN iajs-2433 27 22 of of ADP iajs-2433 27 23 education education NOUN iajs-2433 27 24 for for ADP iajs-2433 27 25 girls girl NOUN iajs-2433 27 26 , , PUNCT iajs-2433 27 27 university university PROPN iajs-2433 27 28 of of ADP iajs-2433 27 29 kufa kufa PROPN iajs-2433 27 30 , , PUNCT iajs-2433 27 31 najaf najaf PROPN iajs-2433 27 32 , , PUNCT iajs-2433 27 33 iraq iraq PROPN iajs-2433 27 34     SPACE iajs-2433 27 35 121 121 NUM iajs-2433 27 36     SPACE iajs-2433 27 37 ibn ibn PROPN iajs-2433 27 38 al al PROPN iajs-2433 27 39 - - PUNCT iajs-2433 27 40 haitham haitham PROPN iajs-2433 27 41 jour jour X iajs-2433 27 42 . . PROPN iajs-2433 28 1 for for ADP iajs-2433 28 2 pure pure ADJ iajs-2433 28 3 & & CCONJ iajs-2433 28 4 appl appl PROPN iajs-2433 28 5 . . PUNCT iajs-2433 29 1 sci sci PROPN iajs-2433 29 2 . . PROPN iajs-2433 30 1 33 33 NUM iajs-2433 30 2 ( ( PUNCT iajs-2433 30 3 2 2 NUM iajs-2433 30 4 ) ) PUNCT iajs-2433 30 5 2020 2020 NUM iajs-2433 30 6 we we PRON iajs-2433 30 7 requisite requisite VERB iajs-2433 30 8 0 0 NUM iajs-2433 30 9 ∗ ∗ NOUN iajs-2433 31 1 ж ж X iajs-2433 31 2 ∈ ∈ NOUN iajs-2433 31 3 𝔖 𝔖 NOUN iajs-2433 32 1 [ [ X iajs-2433 32 2 6 6 NUM iajs-2433 32 3 ] ] PUNCT iajs-2433 32 4 . . PUNCT iajs-2433 33 1 let let VERB iajs-2433 33 2 𝔖 𝔖 PRON iajs-2433 33 3 be be AUX iajs-2433 33 4 an an DET iajs-2433 33 5 ideal ideal NOUN iajs-2433 33 6 of of ADP iajs-2433 33 7 ψ ψ PRON iajs-2433 33 8 . . PUNCT iajs-2433 34 1 it it PRON iajs-2433 34 2 is be AUX iajs-2433 34 3 named name VERB iajs-2433 34 4 a a DET iajs-2433 34 5 closed closed ADJ iajs-2433 34 6 ideal ideal NOUN iajs-2433 34 7 with with ADP iajs-2433 34 8 respect respect NOUN iajs-2433 34 9 to to ADP iajs-2433 34 10 an an DET iajs-2433 34 11 element element NOUN iajs-2433 34 12 s s PART iajs-2433 34 13 ∈ ∈ PROPN iajs-2433 34 14 ψ ψ X iajs-2433 34 15 ( ( PUNCT iajs-2433 34 16 symbolized symbolize VERB iajs-2433 34 17 by by ADP iajs-2433 34 18 s s X iajs-2433 34 19 closed close VERB iajs-2433 34 20 ideal ideal ADJ iajs-2433 34 21 ) ) PUNCT iajs-2433 34 22 if if SCONJ iajs-2433 34 23 s s VERB iajs-2433 34 24 ∗ ∗ NOUN iajs-2433 34 25 0 0 NUM iajs-2433 34 26 ∗ ∗ NOUN iajs-2433 34 27 ж ж X iajs-2433 34 28 ∈ ∈ PROPN iajs-2433 34 29 𝔖 𝔖 PROPN iajs-2433 34 30 , , PUNCT iajs-2433 34 31 ∀ ∀ PUNCT iajs-2433 34 32 ж ж X iajs-2433 34 33 ∈ ∈ NOUN iajs-2433 34 34 𝔖 𝔖 NOUN iajs-2433 35 1 [ [ X iajs-2433 35 2 7 7 NUM iajs-2433 35 3 ] ] PUNCT iajs-2433 35 4 . . PUNCT iajs-2433 36 1 an an DET iajs-2433 36 2 ideal ideal ADJ iajs-2433 36 3 𝔖 𝔖 PROPN iajs-2433 36 4 of of ADP iajs-2433 36 5 ψ ψ PROPN iajs-2433 36 6 is be AUX iajs-2433 36 7 known know VERB iajs-2433 36 8 as as ADP iajs-2433 36 9 completely completely ADV iajs-2433 36 10 closed closed ADJ iajs-2433 36 11 ideal ideal ADJ iajs-2433 36 12 if if SCONJ iajs-2433 36 13 ж ж X iajs-2433 36 14 ∗ ∗ VERB iajs-2433 36 15 ц ц NOUN iajs-2433 36 16 ∈ ∈ PROPN iajs-2433 36 17 𝔖 𝔖 PROPN iajs-2433 36 18 , , PUNCT iajs-2433 36 19 ∀ ∀ NUM iajs-2433 36 20 , , PUNCT iajs-2433 36 21 ц ц PROPN iajs-2433 36 22 ∈ ∈ PROPN iajs-2433 36 23 𝔖 𝔖 PROPN iajs-2433 36 24 7 7 NUM iajs-2433 36 25 . . PUNCT iajs-2433 37 1 let let VERB iajs-2433 37 2 𝔖 𝔖 PRON iajs-2433 37 3 be be AUX iajs-2433 37 4 an an DET iajs-2433 37 5 ideal ideal NOUN iajs-2433 37 6 of of ADP iajs-2433 37 7 ψ ψ NOUN iajs-2433 37 8 and and CCONJ iajs-2433 37 9 s s NOUN iajs-2433 37 10 ∈ ∈ PROPN iajs-2433 37 11 𝔖 𝔖 PROPN iajs-2433 37 12 . . PUNCT iajs-2433 38 1 it it PRON iajs-2433 38 2 is be AUX iajs-2433 38 3 named name VERB iajs-2433 38 4 a a DET iajs-2433 38 5 completely completely ADV iajs-2433 38 6 closed close VERB iajs-2433 38 7 with with ADP iajs-2433 38 8 respect respect NOUN iajs-2433 38 9 to to ADP iajs-2433 38 10 an an DET iajs-2433 38 11 element element NOUN iajs-2433 38 12 s s PART iajs-2433 38 13 ( ( PUNCT iajs-2433 38 14 know know VERB iajs-2433 38 15 by by ADP iajs-2433 38 16 s s PRON iajs-2433 38 17 completely completely ADV iajs-2433 38 18 closed closed ADJ iajs-2433 38 19 ideal ideal ADJ iajs-2433 38 20 ) ) PUNCT iajs-2433 38 21 if if SCONJ iajs-2433 38 22 : : PUNCT iajs-2433 38 23 s s X iajs-2433 38 24 ∗ ∗ X iajs-2433 38 25 ж ж X iajs-2433 38 26 ∗ ∗ NOUN iajs-2433 38 27 ц ц NOUN iajs-2433 38 28 ∈ ∈ PROPN iajs-2433 38 29 𝔖 𝔖 PROPN iajs-2433 38 30 , , PUNCT iajs-2433 38 31 ∀ ∀ X iajs-2433 38 32 ж ж X iajs-2433 38 33 , , PUNCT iajs-2433 38 34 ц ц PROPN iajs-2433 38 35 ∈ ∈ NOUN iajs-2433 38 36 𝔖 𝔖 PROPN iajs-2433 39 1 [ [ X iajs-2433 39 2 7 7 NUM iajs-2433 39 3 ] ] PUNCT iajs-2433 39 4 . . PUNCT iajs-2433 40 1 in in ADP iajs-2433 40 2 the the DET iajs-2433 40 3 next next ADJ iajs-2433 40 4 parts part NOUN iajs-2433 40 5 of of ADP iajs-2433 40 6 our our PRON iajs-2433 40 7 research research NOUN iajs-2433 40 8 , , PUNCT iajs-2433 40 9 we we PRON iajs-2433 40 10 will will AUX iajs-2433 40 11 symbolize symbolize VERB iajs-2433 40 12 to to PART iajs-2433 40 13 bhalgebra bhalgebra VERB iajs-2433 40 14 € € NUM iajs-2433 40 15 ; ; PUNCT iajs-2433 40 16 ∗ ∗ NOUN iajs-2433 40 17 , , PUNCT iajs-2433 40 18 0 0 NUM iajs-2433 40 19 by by ADP iajs-2433 40 20 € € NOUN iajs-2433 40 21 . . PUNCT iajs-2433 41 1 2 2 X iajs-2433 41 2 . . NUM iajs-2433 41 3 closed close VERB iajs-2433 41 4 sah sah NOUN iajs-2433 41 5 – – PUNCT iajs-2433 41 6 ideal ideal ADJ iajs-2433 41 7 with with ADP iajs-2433 41 8 respect respect NOUN iajs-2433 41 9 to to ADP iajs-2433 41 10 an an DET iajs-2433 41 11 element element NOUN iajs-2433 41 12 of of ADP iajs-2433 41 13 bh bh NOUN iajs-2433 41 14 – – PUNCT iajs-2433 41 15 algebra algebra NOUN iajs-2433 41 16 definition definition NOUN iajs-2433 41 17 ( ( PUNCT iajs-2433 41 18 1 1 X iajs-2433 41 19 ) ) PUNCT iajs-2433 41 20 an an DET iajs-2433 41 21 ideal ideal ADJ iajs-2433 41 22 𝔜 𝔜 NOUN iajs-2433 41 23 of of ADP iajs-2433 41 24 € € NOUN iajs-2433 41 25 is be AUX iajs-2433 41 26 named name VERB iajs-2433 41 27 a a DET iajs-2433 41 28 sah sah NOUN iajs-2433 41 29 – – PUNCT iajs-2433 41 30 ideal ideal NOUN iajs-2433 41 31 of of ADP iajs-2433 41 32 € € NOUN iajs-2433 41 33 if if SCONJ iajs-2433 41 34 it it PRON iajs-2433 41 35 fillfulls fillfull VERB iajs-2433 41 36 the the DET iajs-2433 41 37 requirement requirement NOUN iajs-2433 41 38 : : PUNCT iajs-2433 41 39 ∀ς ∀ς ADV iajs-2433 41 40 , , PUNCT iajs-2433 41 41 ζ ζ NOUN iajs-2433 41 42 ∈ ∈ PROPN iajs-2433 41 43 𝔜 𝔜 NOUN iajs-2433 41 44 , , PUNCT iajs-2433 41 45 if if SCONJ iajs-2433 41 46 ς∗ ς∗ PROPN iajs-2433 41 47 ∗ ∗ VERB iajs-2433 41 48 ζ ζ NOUN iajs-2433 41 49 ∈ ∈ PROPN iajs-2433 41 50 𝔜 𝔜 PROPN iajs-2433 41 51 , , PUNCT iajs-2433 41 52 ζ∗ ζ∗ PROPN iajs-2433 41 53 ∈ ∈ PROPN iajs-2433 41 54 𝔜 𝔜 PROPN iajs-2433 41 55 → → SYM iajs-2433 41 56 ζ∗ ζ∗ PROPN iajs-2433 41 57 ∗ ∗ NOUN iajs-2433 41 58 ς ς PROPN iajs-2433 41 59 ∈ ∈ PROPN iajs-2433 41 60 𝔜 𝔜 PROPN iajs-2433 41 61 , , PUNCT iajs-2433 41 62 where where SCONJ iajs-2433 41 63 ς∗ ς∗ PROPN iajs-2433 41 64 e e PROPN iajs-2433 41 65 ∗ ∗ VERB iajs-2433 41 66 ς ς PROPN iajs-2433 41 67 , , PUNCT iajs-2433 41 68 and and CCONJ iajs-2433 41 69 e e PROPN iajs-2433 41 70 is be AUX iajs-2433 41 71 unit unit NOUN iajs-2433 41 72 number number NOUN iajs-2433 41 73 , , PUNCT iajs-2433 41 74 i.e i.e PROPN iajs-2433 41 75 : : PUNCT iajs-2433 41 76 ς ς PROPN iajs-2433 41 77 ∗ ∗ NOUN iajs-2433 41 78 e e NOUN iajs-2433 41 79 = = SYM iajs-2433 41 80 0 0 NUM iajs-2433 41 81 example example NOUN iajs-2433 41 82 ( ( PUNCT iajs-2433 41 83 2 2 X iajs-2433 41 84 ) ) PUNCT iajs-2433 41 85 assume assume VERB iajs-2433 41 86 € € SYM iajs-2433 41 87 0 0 NUM iajs-2433 41 88 , , PUNCT iajs-2433 41 89 𝔴 𝔴 PROPN iajs-2433 41 90 , , PUNCT iajs-2433 41 91 𝔳 𝔳 PROPN iajs-2433 41 92 with with ADP iajs-2433 41 93 the the DET iajs-2433 41 94 binary binary PROPN iajs-2433 41 95 operation operation NOUN iajs-2433 41 96 ∗ ∗ NOUN iajs-2433 41 97 symbolized symbolize VERB iajs-2433 41 98 by by ADP iajs-2433 41 99 the the DET iajs-2433 41 100 subsequent subsequent ADJ iajs-2433 41 101 table table NOUN iajs-2433 41 102 : : PUNCT iajs-2433 41 103 table table NOUN iajs-2433 41 104 1 1 NUM iajs-2433 41 105 . . PUNCT iajs-2433 42 1 * * PUNCT iajs-2433 42 2 0 0 NUM iajs-2433 42 3 𝔴 𝔴 NUM iajs-2433 42 4 𝔳 𝔳 PROPN iajs-2433 42 5 0 0 NUM iajs-2433 42 6 0 0 NUM iajs-2433 42 7 0 0 NUM iajs-2433 42 8 0 0 NUM iajs-2433 42 9 𝔴 𝔴 NUM iajs-2433 42 10 𝔴 𝔴 PROPN iajs-2433 42 11 0 0 NUM iajs-2433 42 12 0 0 NUM iajs-2433 42 13 𝔳 𝔳 PROPN iajs-2433 42 14 𝔳 𝔳 PRON iajs-2433 42 15 𝔳 𝔳 PROPN iajs-2433 42 16 0 0 NUM iajs-2433 43 1 then then ADV iajs-2433 43 2 the the DET iajs-2433 43 3 ideal ideal ADJ iajs-2433 43 4 𝔜 𝔜 PROPN iajs-2433 43 5 0 0 PROPN iajs-2433 43 6 , , PUNCT iajs-2433 43 7 𝔳 𝔳 PROPN iajs-2433 43 8 is be AUX iajs-2433 43 9 a a DET iajs-2433 43 10 sah sah NOUN iajs-2433 43 11 – – PUNCT iajs-2433 43 12 ideal ideal NOUN iajs-2433 43 13 of of ADP iajs-2433 43 14 € € SYM iajs-2433 43 15 . . PUNCT iajs-2433 44 1 definition definition NOUN iajs-2433 44 2 ( ( PUNCT iajs-2433 44 3 3 3 X iajs-2433 44 4 ) ) PUNCT iajs-2433 44 5 assume assume VERB iajs-2433 44 6 𝔜 𝔜 PROPN iajs-2433 44 7 is be AUX iajs-2433 44 8 sah sah NOUN iajs-2433 44 9 – – PUNCT iajs-2433 44 10 ideal ideal NOUN iajs-2433 44 11 of of ADP iajs-2433 44 12 € € NUM iajs-2433 44 13 , , PUNCT iajs-2433 44 14 then then ADV iajs-2433 44 15 𝔜 𝔜 PROPN iajs-2433 44 16 is be AUX iajs-2433 44 17 known know VERB iajs-2433 44 18 as as ADP iajs-2433 44 19 closed close VERB iajs-2433 44 20 sah sah NOUN iajs-2433 44 21 – – PUNCT iajs-2433 44 22 ideal ideal ADJ iajs-2433 44 23 if if SCONJ iajs-2433 44 24 it it PRON iajs-2433 44 25 fulfills fulfill VERB iajs-2433 44 26 the the DET iajs-2433 44 27 requirement requirement NOUN iajs-2433 44 28 : : PUNCT iajs-2433 44 29 ∀ ∀ PUNCT iajs-2433 44 30 ς ς PROPN iajs-2433 44 31 , , PUNCT iajs-2433 44 32 ζ ζ NOUN iajs-2433 44 33 ∈ ∈ PROPN iajs-2433 44 34 𝔜 𝔜 NOUN iajs-2433 44 35 if if SCONJ iajs-2433 44 36 0 0 NUM iajs-2433 44 37 ∗ ∗ NOUN iajs-2433 44 38 ς∗ ς∗ NOUN iajs-2433 44 39 ∗ ∗ NOUN iajs-2433 44 40 ζ ζ NOUN iajs-2433 44 41 ∈ ∈ PROPN iajs-2433 44 42 𝔜 𝔜 NOUN iajs-2433 44 43 ∧ ∧ PROPN iajs-2433 44 44 0 0 NUM iajs-2433 44 45 ∗ ∗ NOUN iajs-2433 44 46 ζ∗ ζ∗ PROPN iajs-2433 44 47 ∈ ∈ PROPN iajs-2433 44 48 𝔜 𝔜 PROPN iajs-2433 44 49 → → SYM iajs-2433 44 50 0 0 NUM iajs-2433 44 51 ∗ ∗ NOUN iajs-2433 44 52 ζ∗ ζ∗ PROPN iajs-2433 44 53 ∗ ∗ VERB iajs-2433 44 54 ς ς PROPN iajs-2433 44 55 ∈ ∈ PROPN iajs-2433 44 56 𝔜 𝔜 PROPN iajs-2433 44 57 example example NOUN iajs-2433 44 58 ( ( PUNCT iajs-2433 44 59 4 4 X iajs-2433 44 60 ) ) PUNCT iajs-2433 44 61 assume assume VERB iajs-2433 44 62 € € NOUN iajs-2433 44 63 0,1,2,3 0,1,2,3 NUM iajs-2433 44 64 with with ADP iajs-2433 44 65 the the DET iajs-2433 44 66 binary binary PROPN iajs-2433 44 67 operation operation NOUN iajs-2433 44 68 ∗ ∗ NOUN iajs-2433 44 69 definition definition NOUN iajs-2433 44 70 by by ADP iajs-2433 44 71 the the DET iajs-2433 44 72 ensuing ensue VERB iajs-2433 44 73 table table NOUN iajs-2433 44 74 : : PUNCT iajs-2433 44 75 table table NOUN iajs-2433 44 76 2 2 NUM iajs-2433 44 77 * * SYM iajs-2433 44 78 0 0 NUM iajs-2433 45 1 1 1 NUM iajs-2433 45 2 2 2 NUM iajs-2433 45 3 3 3 NUM iajs-2433 45 4 0 0 NUM iajs-2433 45 5 0 0 NUM iajs-2433 45 6 1 1 NUM iajs-2433 45 7 0 0 NUM iajs-2433 45 8 0 0 NUM iajs-2433 45 9 1 1 NUM iajs-2433 45 10 1 1 NUM iajs-2433 45 11 0 0 NUM iajs-2433 45 12 1 1 NUM iajs-2433 45 13 0 0 NUM iajs-2433 45 14 2 2 NUM iajs-2433 45 15 2 2 NUM iajs-2433 45 16 2 2 NUM iajs-2433 45 17 0 0 NUM iajs-2433 45 18 0 0 NUM iajs-2433 45 19 3 3 NUM iajs-2433 45 20 3 3 NUM iajs-2433 45 21 3 3 NUM iajs-2433 45 22 3 3 NUM iajs-2433 45 23 0 0 NUM iajs-2433 45 24 then then ADV iajs-2433 45 25 , , PUNCT iajs-2433 45 26 the the DET iajs-2433 45 27 ideal ideal ADJ iajs-2433 45 28 𝔜 𝔜 NOUN iajs-2433 45 29 0,3 0,3 NOUN iajs-2433 45 30 is be AUX iajs-2433 45 31 a a DET iajs-2433 45 32 closed closed ADJ iajs-2433 45 33 sah sah NOUN iajs-2433 45 34 – – PUNCT iajs-2433 45 35 ideal ideal NOUN iajs-2433 45 36 of of ADP iajs-2433 45 37 € € SYM iajs-2433 45 38 . . PUNCT iajs-2433 45 39     SPACE iajs-2433 45 40 122 122 NUM iajs-2433 45 41     SPACE iajs-2433 45 42 ibn ibn PROPN iajs-2433 45 43 al al PROPN iajs-2433 45 44 - - PUNCT iajs-2433 45 45 haitham haitham PROPN iajs-2433 45 46 jour jour X iajs-2433 45 47 . . PROPN iajs-2433 46 1 for for ADP iajs-2433 46 2 pure pure ADJ iajs-2433 46 3 & & CCONJ iajs-2433 46 4 appl appl PROPN iajs-2433 46 5 . . PUNCT iajs-2433 47 1 sci sci PROPN iajs-2433 47 2 . . PROPN iajs-2433 48 1 33 33 NUM iajs-2433 48 2 ( ( PUNCT iajs-2433 48 3 2 2 NUM iajs-2433 48 4 ) ) PUNCT iajs-2433 48 5 2020 2020 NUM iajs-2433 48 6 remark remark NOUN iajs-2433 48 7 ( ( PUNCT iajs-2433 48 8 5 5 NUM iajs-2433 48 9 ) ) PUNCT iajs-2433 48 10 we we PRON iajs-2433 48 11 know know VERB iajs-2433 48 12 that that SCONJ iajs-2433 48 13 every every DET iajs-2433 48 14 sah sah NOUN iajs-2433 48 15 – – PUNCT iajs-2433 48 16 ideal ideal NOUN iajs-2433 48 17 in in ADP iajs-2433 48 18 € € NOUN iajs-2433 48 19 is be AUX iajs-2433 48 20 closed close VERB iajs-2433 48 21 sah sah NOUN iajs-2433 48 22 – – PUNCT iajs-2433 48 23 ideal ideal ADJ iajs-2433 48 24 . . PUNCT iajs-2433 49 1 but but CCONJ iajs-2433 49 2 the the DET iajs-2433 49 3 converse converse NOUN iajs-2433 49 4 not not PART iajs-2433 49 5 correct correct ADJ iajs-2433 49 6 . . PUNCT iajs-2433 50 1 example example NOUN iajs-2433 50 2 ( ( PUNCT iajs-2433 50 3 6 6 X iajs-2433 50 4 ) ) PUNCT iajs-2433 50 5 consider consider VERB iajs-2433 50 6 € € SYM iajs-2433 50 7 0,1,2,3 0,1,2,3 NUM iajs-2433 50 8 with with ADP iajs-2433 50 9 a a DET iajs-2433 50 10 binary binary ADJ iajs-2433 50 11 operation operation NOUN iajs-2433 50 12 ∗ ∗ NOUN iajs-2433 50 13 connoted connote VERB iajs-2433 50 14 by by ADP iajs-2433 50 15 the the DET iajs-2433 50 16 ensuing ensue VERB iajs-2433 50 17 table table NOUN iajs-2433 50 18 : : PUNCT iajs-2433 50 19 table table NOUN iajs-2433 50 20 3 3 NUM iajs-2433 50 21 . . PUNCT iajs-2433 51 1 * * SYM iajs-2433 51 2 0 0 NUM iajs-2433 52 1 1 1 NUM iajs-2433 52 2 2 2 NUM iajs-2433 52 3 3 3 NUM iajs-2433 52 4 0 0 NUM iajs-2433 52 5 0 0 NUM iajs-2433 52 6 0 0 NUM iajs-2433 52 7 0 0 NUM iajs-2433 52 8 0 0 NUM iajs-2433 52 9 1 1 NUM iajs-2433 52 10 1 1 NUM iajs-2433 52 11 0 0 NUM iajs-2433 52 12 0 0 NUM iajs-2433 52 13 0 0 NUM iajs-2433 52 14 2 2 NUM iajs-2433 52 15 2 2 NUM iajs-2433 52 16 2 2 NUM iajs-2433 52 17 0 0 NUM iajs-2433 52 18 0 0 NUM iajs-2433 52 19 3 3 NUM iajs-2433 52 20 3 3 NUM iajs-2433 52 21 2 2 NUM iajs-2433 52 22 2 2 NUM iajs-2433 52 23 0 0 NUM iajs-2433 52 24 𝔜 𝔜 NOUN iajs-2433 52 25 0,3 0,3 NOUN iajs-2433 52 26 is be AUX iajs-2433 52 27 a a DET iajs-2433 52 28 closed closed ADJ iajs-2433 52 29 sah sah NOUN iajs-2433 52 30 – – PUNCT iajs-2433 52 31 ideal ideal NOUN iajs-2433 52 32 of of ADP iajs-2433 52 33 € € NUM iajs-2433 52 34 but but CCONJ iajs-2433 52 35 𝔜 𝔜 PROPN iajs-2433 52 36 does do AUX iajs-2433 52 37 n't not PART iajs-2433 52 38 sah sah VERB iajs-2433 52 39 – – PUNCT iajs-2433 52 40 ideal ideal ADJ iajs-2433 52 41 , , PUNCT iajs-2433 52 42 because because SCONJ iajs-2433 52 43 : : PUNCT iajs-2433 52 44 when when SCONJ iajs-2433 52 45 ς ς PROPN iajs-2433 52 46 2 2 NUM iajs-2433 52 47 , , PUNCT iajs-2433 52 48 ζ ζ NOUN iajs-2433 52 49 1 1 NUM iajs-2433 52 50 → → SYM iajs-2433 52 51 ς∗ ς∗ PROPN iajs-2433 52 52 2 2 NUM iajs-2433 52 53 , , PUNCT iajs-2433 52 54 ζ∗ ζ∗ PROPN iajs-2433 52 55 2 2 NUM iajs-2433 52 56 0 0 NUM iajs-2433 52 57 ∗ ∗ NOUN iajs-2433 52 58 2 2 NUM iajs-2433 52 59 0 0 NUM iajs-2433 52 60 ∈ ∈ PROPN iajs-2433 52 61 𝔜 𝔜 NOUN iajs-2433 52 62 , , PUNCT iajs-2433 52 63 0 0 NUM iajs-2433 52 64 ∗ ∗ NOUN iajs-2433 52 65 2 2 NUM iajs-2433 52 66 0 0 NUM iajs-2433 52 67 ∈ ∈ PROPN iajs-2433 52 68 𝔜 𝔜 PROPN iajs-2433 52 69 → → SYM iajs-2433 52 70 0 0 NUM iajs-2433 52 71 ∗ ∗ NOUN iajs-2433 52 72 0 0 NUM iajs-2433 52 73 0 0 NUM iajs-2433 52 74 ∈ ∈ PROPN iajs-2433 52 75 𝔜 𝔜 NOUN iajs-2433 52 76 , , PUNCT iajs-2433 52 77 while while SCONJ iajs-2433 52 78 2 2 NUM iajs-2433 52 79 ∗ ∗ NOUN iajs-2433 52 80 1 1 NUM iajs-2433 52 81 2 2 NUM iajs-2433 52 82 ∉ ∉ PROPN iajs-2433 52 83 𝔜 𝔜 PROPN iajs-2433 52 84 , , PUNCT iajs-2433 52 85 2 2 NUM iajs-2433 52 86 ∉ ∉ PROPN iajs-2433 52 87 𝔜 𝔜 PROPN iajs-2433 52 88 → → SYM iajs-2433 52 89 2 2 NUM iajs-2433 52 90 ∗ ∗ NOUN iajs-2433 52 91 2 2 NUM iajs-2433 52 92 0 0 NUM iajs-2433 52 93 ∈ ∈ PROPN iajs-2433 52 94 𝔜 𝔜 NOUN iajs-2433 52 95 theorem theorem NOUN iajs-2433 52 96 ( ( PUNCT iajs-2433 52 97 7 7 NUM iajs-2433 52 98 ) ) PUNCT iajs-2433 52 99 assume assume VERB iajs-2433 52 100 𝔜 𝔜 NOUN iajs-2433 52 101 , , PUNCT iajs-2433 52 102 λ λ PROPN iajs-2433 52 103 ∈ ∈ NOUN iajs-2433 52 104 λ λ NOUN iajs-2433 52 105 is be AUX iajs-2433 52 106 a a DET iajs-2433 52 107 collocation collocation NOUN iajs-2433 52 108 of of ADP iajs-2433 52 109 closed closed ADJ iajs-2433 52 110 sah sah NOUN iajs-2433 52 111 – – PUNCT iajs-2433 52 112 ideal ideal NOUN iajs-2433 52 113 of of ADP iajs-2433 52 114 € € SYM iajs-2433 52 115 . . PUNCT iajs-2433 53 1 then then ADV iajs-2433 53 2   VERB iajs-2433 53 3   PROPN iajs-2433 53 4 𝔜 𝔜 PROPN iajs-2433 53 5 is be AUX iajs-2433 53 6 a a DET iajs-2433 53 7 closed closed ADJ iajs-2433 53 8 sah sah NOUN iajs-2433 53 9 – – PUNCT iajs-2433 53 10 ideal ideal NOUN iajs-2433 53 11 of of ADP iajs-2433 53 12 € € X iajs-2433 53 13 . . PUNCT iajs-2433 54 1 proof proof NOUN iajs-2433 54 2 ∀ ∀ X iajs-2433 54 3 ς ς PROPN iajs-2433 54 4 , , PUNCT iajs-2433 54 5 ζ ζ PROPN iajs-2433 54 6 ∈ ∈ PROPN iajs-2433 54 7   ADP iajs-2433 54 8   PROPN iajs-2433 54 9 𝔜 𝔜 PROPN iajs-2433 54 10 ∴ ∴ NOUN iajs-2433 54 11 ς ς PROPN iajs-2433 54 12 , , PUNCT iajs-2433 54 13 ζ ζ NOUN iajs-2433 54 14 ∈ ∈ PROPN iajs-2433 54 15 𝔜 𝔜 NOUN iajs-2433 54 16 , , PUNCT iajs-2433 54 17 ∀ ∀ X iajs-2433 54 18 λ λ NOUN iajs-2433 54 19 ∈ ∈ NOUN iajs-2433 54 20 λ λ X iajs-2433 54 21 ⟹ ⟹ X iajs-2433 54 22 0 0 NUM iajs-2433 54 23 ∗ ∗ NOUN iajs-2433 54 24 ς∗ ς∗ PROPN iajs-2433 54 25 ∗ ∗ NOUN iajs-2433 54 26 ζ ζ NOUN iajs-2433 54 27 ∈ ∈ PROPN iajs-2433 54 28 𝔜 𝔜 NOUN iajs-2433 54 29 and and CCONJ iajs-2433 54 30 0 0 NUM iajs-2433 54 31 ∗ ∗ NOUN iajs-2433 54 32 ζ∗ ζ∗ PROPN iajs-2433 54 33 ∈ ∈ PROPN iajs-2433 54 34 𝔜 𝔜 PROPN iajs-2433 54 35 then then ADV iajs-2433 54 36 0 0 NUM iajs-2433 54 37 ∗ ∗ NOUN iajs-2433 54 38 ζ∗ ζ∗ PROPN iajs-2433 54 39 ∗ ∗ NOUN iajs-2433 54 40 ς ς PROPN iajs-2433 54 41 ∈ ∈ PROPN iajs-2433 54 42 𝔜 𝔜 PROPN iajs-2433 54 43 , , PUNCT iajs-2433 54 44 ∀λ ∀λ X iajs-2433 54 45 ∈ ∈ PROPN iajs-2433 54 46 λ λ PROPN iajs-2433 54 47 since since SCONJ iajs-2433 54 48 each each DET iajs-2433 54 49 𝔜 𝔜 PROPN iajs-2433 54 50 is be AUX iajs-2433 54 51 closed close VERB iajs-2433 54 52 sah sah NOUN iajs-2433 54 53 – – PUNCT iajs-2433 54 54 ideal ideal ADJ iajs-2433 54 55 ∀λ ∀λ NUM iajs-2433 54 56 ∈ ∈ PROPN iajs-2433 54 57 λ λ PROPN iajs-2433 54 58 ⇒ ⇒ NOUN iajs-2433 54 59 0 0 NUM iajs-2433 54 60 ∗ ∗ NOUN iajs-2433 54 61 ς∗ ς∗ PROPN iajs-2433 54 62 ∗ ∗ NOUN iajs-2433 54 63 ζ ζ PROPN iajs-2433 54 64 ∈ ∈ PROPN iajs-2433 54 65   ADP iajs-2433 54 66   PROPN iajs-2433 54 67 𝔜 𝔜 PROPN iajs-2433 54 68 and and CCONJ iajs-2433 54 69 0 0 NUM iajs-2433 54 70 ∗ ∗ NOUN iajs-2433 54 71 ζ∗ ζ∗ PROPN iajs-2433 54 72 ∈ ∈ PROPN iajs-2433 54 73   PART iajs-2433 54 74   PROPN iajs-2433 54 75 𝔜 𝔜 PROPN iajs-2433 54 76 then then ADV iajs-2433 54 77 0 0 NUM iajs-2433 54 78 ∗ ∗ NOUN iajs-2433 54 79 ζ∗ ζ∗ PROPN iajs-2433 54 80 ∗ ∗ VERB iajs-2433 54 81 ς ς PROPN iajs-2433 54 82 ∈ ∈ PROPN iajs-2433 54 83   PART iajs-2433 54 84   PROPN iajs-2433 54 85 𝔜 𝔜 PROPN iajs-2433 54 86 ∴ ∴ NOUN iajs-2433 54 87   PART iajs-2433 54 88   PROPN iajs-2433 54 89 𝔜 𝔜 PROPN iajs-2433 54 90 is be AUX iajs-2433 54 91 closed close VERB iajs-2433 54 92 sah sah NOUN iajs-2433 54 93 – – PUNCT iajs-2433 54 94 ideal ideal NOUN iajs-2433 54 95 of of ADP iajs-2433 54 96 bh bh NOUN iajs-2433 54 97 – – PUNCT iajs-2433 54 98 algebra algebra NOUN iajs-2433 54 99 € € NOUN iajs-2433 54 100 . . PUNCT iajs-2433 55 1 ∎ ∎ ADJ iajs-2433 55 2     SPACE iajs-2433 55 3 123 123 NUM iajs-2433 55 4     SPACE iajs-2433 55 5 ibn ibn PROPN iajs-2433 55 6 al al PROPN iajs-2433 55 7 - - PUNCT iajs-2433 55 8 haitham haitham PROPN iajs-2433 55 9 jour jour X iajs-2433 55 10 . . PROPN iajs-2433 55 11 for for ADP iajs-2433 55 12 pure pure ADJ iajs-2433 55 13 & & CCONJ iajs-2433 55 14 appl appl PROPN iajs-2433 55 15 . . PUNCT iajs-2433 56 1 sci sci PROPN iajs-2433 56 2 . . PROPN iajs-2433 57 1 33 33 NUM iajs-2433 57 2 ( ( PUNCT iajs-2433 57 3 2 2 NUM iajs-2433 57 4 ) ) PUNCT iajs-2433 57 5 2020 2020 NUM iajs-2433 57 6 theorem theorem NOUN iajs-2433 57 7 ( ( PUNCT iajs-2433 57 8 8) 8) NUM iajs-2433 57 9 assume assume VERB iajs-2433 57 10 𝔜 𝔜 NOUN iajs-2433 57 11 , , PUNCT iajs-2433 57 12 λ λ PROPN iajs-2433 57 13 ∈ ∈ NOUN iajs-2433 57 14 λ λ NOUN iajs-2433 57 15 is be AUX iajs-2433 57 16 a a DET iajs-2433 57 17 collocation collocation NOUN iajs-2433 57 18 of of ADP iajs-2433 57 19 closed closed ADJ iajs-2433 57 20 sah sah NOUN iajs-2433 57 21 – – PUNCT iajs-2433 57 22 ideals ideal NOUN iajs-2433 57 23 of of ADP iajs-2433 57 24 € € NOUN iajs-2433 57 25 . . PUNCT iajs-2433 58 1 then then ADV iajs-2433 58 2   PROPN iajs-2433 58 3   PROPN iajs-2433 58 4 𝔜 𝔜 NOUN iajs-2433 58 5 is be AUX iajs-2433 58 6 a a DET iajs-2433 58 7 closed closed ADJ iajs-2433 58 8 sah sah NOUN iajs-2433 58 9 – – PUNCT iajs-2433 58 10 ideal ideal NOUN iajs-2433 58 11 of of ADP iajs-2433 58 12 € € X iajs-2433 58 13 . . PUNCT iajs-2433 59 1 proof proof NOUN iajs-2433 59 2 to to PART iajs-2433 59 3 prove prove VERB iajs-2433 59 4 that that SCONJ iajs-2433 59 5   PROPN iajs-2433 59 6   NOUN iajs-2433 59 7 𝔜 𝔜 PROPN iajs-2433 59 8 is be AUX iajs-2433 59 9 closed close VERB iajs-2433 59 10 sah sah NOUN iajs-2433 59 11 – – PUNCT iajs-2433 59 12 ideal ideal ADJ iajs-2433 59 13 ∀ ∀ X iajs-2433 59 14 ς ς PROPN iajs-2433 59 15 , , PUNCT iajs-2433 59 16 ζ ζ NOUN iajs-2433 59 17 ∈ ∈ PROPN iajs-2433 59 18   NOUN iajs-2433 59 19   NOUN iajs-2433 59 20 𝔜 𝔜 PROPN iajs-2433 59 21 ⇒ ⇒ VERB iajs-2433 59 22 ∃ ∃ PROPN iajs-2433 59 23 𝔜 𝔜 PROPN iajs-2433 59 24 ∈ ∈ PROPN iajs-2433 59 25 𝔜 𝔜 NOUN iajs-2433 59 26 ∈ ∈ NOUN iajs-2433 59 27 is be AUX iajs-2433 59 28 a a DET iajs-2433 59 29 c c NOUN iajs-2433 59 30 sah sah NOUN iajs-2433 59 31 – – PUNCT iajs-2433 59 32 ideal ideal NOUN iajs-2433 59 33 such such ADJ iajs-2433 59 34 that that SCONJ iajs-2433 59 35 ∀ ∀ NOUN iajs-2433 59 36 ς ς NOUN iajs-2433 59 37 , , PUNCT iajs-2433 59 38 ζ ζ NOUN iajs-2433 59 39 ∈ ∈ PROPN iajs-2433 59 40 𝔜 𝔜 NOUN iajs-2433 59 41 ⟹ ⟹ SYM iajs-2433 59 42 0 0 NUM iajs-2433 59 43 ∗ ∗ NOUN iajs-2433 59 44 ς∗ ς∗ PROPN iajs-2433 59 45 ∗ ∗ NOUN iajs-2433 59 46 ζ ζ NOUN iajs-2433 59 47 ∈ ∈ PROPN iajs-2433 59 48 𝔜 𝔜 NOUN iajs-2433 59 49 and and CCONJ iajs-2433 59 50 0 0 NUM iajs-2433 59 51 ∗ ∗ NOUN iajs-2433 59 52 ζ∗ ζ∗ PROPN iajs-2433 59 53 ∈ ∈ PROPN iajs-2433 59 54 𝔜 𝔜 NOUN iajs-2433 60 1 so so ADV iajs-2433 60 2 0 0 NUM iajs-2433 60 3 ∗ ∗ NOUN iajs-2433 60 4 ζ∗ ζ∗ PROPN iajs-2433 60 5 ∗ ∗ VERB iajs-2433 60 6 ς ς PROPN iajs-2433 60 7 ∈ ∈ PROPN iajs-2433 60 8 𝔜 𝔜 PROPN iajs-2433 60 9 ⇒ ⇒ NOUN iajs-2433 60 10 0 0 NUM iajs-2433 61 1 ∗ ∗ NOUN iajs-2433 61 2 ζ∗ ζ∗ PROPN iajs-2433 61 3 ∗ ∗ VERB iajs-2433 61 4 ς ς PROPN iajs-2433 61 5 ∈ ∈ PROPN iajs-2433 61 6   NOUN iajs-2433 61 7   NOUN iajs-2433 61 8 𝔜 𝔜 PROPN iajs-2433 61 9 ⇒ ⇒ VERB iajs-2433 61 10   NOUN iajs-2433 61 11   NOUN iajs-2433 61 12 𝔜 𝔜 NOUN iajs-2433 61 13 is be AUX iajs-2433 61 14 closed close VERB iajs-2433 61 15 sah sah NOUN iajs-2433 61 16 – – PUNCT iajs-2433 61 17 ideal ideal NOUN iajs-2433 61 18 of of ADP iajs-2433 61 19 € € X iajs-2433 61 20 . . PUNCT iajs-2433 62 1 ∎ ∎ PROPN iajs-2433 62 2 theorem theorem ADJ iajs-2433 62 3 ( ( PUNCT iajs-2433 62 4 9 9 NUM iajs-2433 62 5 ) ) PUNCT iajs-2433 62 6 assume assume VERB iajs-2433 62 7 € € X iajs-2433 62 8 ∈ ∈ PROPN iajs-2433 62 9 is be AUX iajs-2433 62 10 a a DET iajs-2433 62 11 collocation collocation NOUN iajs-2433 62 12 of of ADP iajs-2433 62 13 € € NUM iajs-2433 62 14 and and CCONJ iajs-2433 62 15 𝔜 𝔜 PROPN iajs-2433 62 16 be be VERB iajs-2433 62 17 a a DET iajs-2433 62 18 closed closed ADJ iajs-2433 62 19 sah sah NOUN iajs-2433 62 20 – – PUNCT iajs-2433 62 21 ideal ideal NOUN iajs-2433 62 22 of of ADP iajs-2433 62 23 € € NUM iajs-2433 62 24 , , PUNCT iajs-2433 62 25 ∀λ ∀λ X iajs-2433 62 26 ∈ ∈ PROPN iajs-2433 62 27 λ λ PROPN iajs-2433 62 28 . . PUNCT iajs-2433 63 1 then then ADV iajs-2433 63 2   PROPN iajs-2433 63 3   PROPN iajs-2433 63 4 𝔜 𝔜 NOUN iajs-2433 63 5 is be AUX iajs-2433 63 6 a a DET iajs-2433 63 7 closed closed ADJ iajs-2433 63 8 sah sah NOUN iajs-2433 63 9 – – PUNCT iajs-2433 63 10 ideal ideal NOUN iajs-2433 63 11 of of ADP iajs-2433 63 12 the the DET iajs-2433 63 13 direct direct ADJ iajs-2433 63 14 product product NOUN iajs-2433 63 15 of of ADP iajs-2433 63 16 € € SYM iajs-2433 63 17 . . PUNCT iajs-2433 64 1 proof proof NOUN iajs-2433 64 2 ∀ ∀ X iajs-2433 64 3 ς ς PROPN iajs-2433 64 4 , , PUNCT iajs-2433 64 5 ζ ζ NOUN iajs-2433 64 6 ∈ ∈ PROPN iajs-2433 64 7 𝔜 𝔜 PROPN iajs-2433 64 8 0 0 NUM iajs-2433 64 9 ς∗ ς∗ NOUN iajs-2433 64 10 ζ𝛌 ζ𝛌 ADP iajs-2433 64 11 ∈ ∈ PROPN iajs-2433 64 12   PROPN iajs-2433 64 13   NOUN iajs-2433 64 14 𝔜 𝔜 PROPN iajs-2433 64 15 ∧ ∧ PROPN iajs-2433 64 16 0 0 NUM iajs-2433 64 17 ζ∗ ζ∗ PROPN iajs-2433 64 18 ∈ ∈ PROPN iajs-2433 64 19   PROPN iajs-2433 64 20   NOUN iajs-2433 64 21 𝔜 𝔜 PROPN iajs-2433 64 22 ⇒ ⇒ NOUN iajs-2433 64 23 0 0 NUM iajs-2433 64 24 ∗ ∗ NOUN iajs-2433 64 25 ς∗ ς∗ PROPN iajs-2433 64 26 ∗ ∗ NOUN iajs-2433 64 27 ζ ζ NOUN iajs-2433 64 28 ∈ ∈ PROPN iajs-2433 64 29   PROPN iajs-2433 64 30   NOUN iajs-2433 64 31 𝔜 𝔜 PROPN iajs-2433 64 32 ∧ ∧ PROPN iajs-2433 64 33 0 0 NUM iajs-2433 64 34 ∗ ∗ NOUN iajs-2433 64 35 ζ∗ ζ∗ PROPN iajs-2433 64 36 ∈ ∈ PROPN iajs-2433 64 37   PROPN iajs-2433 64 38   PROPN iajs-2433 64 39 𝔜 𝔜 SYM iajs-2433 64 40 0 0 NUM iajs-2433 64 41 ∗ ∗ NOUN iajs-2433 64 42 ς∗ ς∗ PROPN iajs-2433 64 43 ∗ ∗ NOUN iajs-2433 64 44 ζ ζ NOUN iajs-2433 64 45 ∈ ∈ PROPN iajs-2433 64 46 𝔜 𝔜 NOUN iajs-2433 64 47 ∧ ∧ PROPN iajs-2433 64 48 0 0 NUM iajs-2433 64 49 ∗ ∗ NOUN iajs-2433 64 50 ζ∗ ζ∗ PROPN iajs-2433 64 51 ∈ ∈ PROPN iajs-2433 64 52 𝔜 𝔜 PROPN iajs-2433 64 53 and and CCONJ iajs-2433 64 54 since since SCONJ iajs-2433 64 55 𝔜 𝔜 PROPN iajs-2433 64 56 is be AUX iajs-2433 64 57 closed close VERB iajs-2433 64 58 sah sah NOUN iajs-2433 64 59 – – PUNCT iajs-2433 64 60 ideal ideal ADJ iajs-2433 64 61 ∀λ ∀λ NUM iajs-2433 64 62 ∈ ∈ PROPN iajs-2433 64 63 λ λ PROPN iajs-2433 64 64 , , PUNCT iajs-2433 64 65 then then ADV iajs-2433 64 66 ∴ ∴ PROPN iajs-2433 64 67 0 0 NUM iajs-2433 64 68 ∗ ∗ NOUN iajs-2433 64 69 ζ∗ ζ∗ PROPN iajs-2433 64 70 ∗ ∗ NOUN iajs-2433 64 71 ς ς PROPN iajs-2433 64 72 ∈ ∈ PROPN iajs-2433 64 73 𝔜 𝔜 PROPN iajs-2433 64 74 , , PUNCT iajs-2433 64 75 ∀λ ∀λ X iajs-2433 64 76 ∈ ∈ PROPN iajs-2433 64 77 λ λ X iajs-2433 64 78 ⟹ ⟹ X iajs-2433 64 79 0 0 NUM iajs-2433 64 80 ∗ ∗ NOUN iajs-2433 64 81 ζ∗ ζ∗ PROPN iajs-2433 64 82 ∗ ∗ NOUN iajs-2433 64 83 ς ς PROPN iajs-2433 64 84 ∈ ∈ PROPN iajs-2433 64 85   PROPN iajs-2433 64 86   PROPN iajs-2433 64 87 𝔜 𝔜 NOUN iajs-2433 64 88 ⟹ ⟹ NUM iajs-2433 64 89   PROPN iajs-2433 64 90   NOUN iajs-2433 64 91 𝔜 𝔜 NOUN iajs-2433 64 92 is be AUX iajs-2433 64 93 closed close VERB iajs-2433 64 94 sah sah NOUN iajs-2433 64 95 – – PUNCT iajs-2433 64 96 ideal ideal NOUN iajs-2433 64 97 of of ADP iajs-2433 64 98 € € X iajs-2433 64 99 . . PUNCT iajs-2433 65 1 ∎ ∎ ADJ iajs-2433 65 2     SPACE iajs-2433 65 3 124 124 NUM iajs-2433 65 4     SPACE iajs-2433 65 5 ibn ibn PROPN iajs-2433 65 6 al al PROPN iajs-2433 65 7 - - PUNCT iajs-2433 65 8 haitham haitham PROPN iajs-2433 65 9 jour jour X iajs-2433 65 10 . . PROPN iajs-2433 65 11 for for ADP iajs-2433 65 12 pure pure ADJ iajs-2433 65 13 & & CCONJ iajs-2433 65 14 appl appl PROPN iajs-2433 65 15 . . PUNCT iajs-2433 66 1 sci sci PROPN iajs-2433 66 2 . . PROPN iajs-2433 67 1 33 33 NUM iajs-2433 67 2 ( ( PUNCT iajs-2433 67 3 2 2 NUM iajs-2433 67 4 ) ) PUNCT iajs-2433 67 5 2020 2020 NUM iajs-2433 67 6 definition definition NOUN iajs-2433 67 7 ( ( PUNCT iajs-2433 67 8 10 10 NUM iajs-2433 67 9 ) ) PUNCT iajs-2433 67 10 assume assume VERB iajs-2433 67 11 𝔜 𝔜 PROPN iajs-2433 67 12 is be AUX iajs-2433 67 13 a a DET iajs-2433 67 14 closed closed ADJ iajs-2433 67 15 sah sah NOUN iajs-2433 67 16 – – PUNCT iajs-2433 67 17 ideal ideal NOUN iajs-2433 67 18 of of ADP iajs-2433 67 19 € € X iajs-2433 67 20 . . PUNCT iajs-2433 68 1 then then ADV iajs-2433 68 2 𝔜 𝔜 PROPN iajs-2433 68 3 is be AUX iajs-2433 68 4 named name VERB iajs-2433 68 5 closed closed ADJ iajs-2433 68 6 sah sah NOUN iajs-2433 68 7 – – PUNCT iajs-2433 68 8 ideal ideal ADJ iajs-2433 68 9 with with ADP iajs-2433 68 10 respect respect NOUN iajs-2433 68 11 to to ADP iajs-2433 68 12 an an DET iajs-2433 68 13 element element NOUN iajs-2433 68 14 s s PART iajs-2433 68 15 ∈ ∈ NOUN iajs-2433 68 16 € € NOUN iajs-2433 68 17 ( ( PUNCT iajs-2433 68 18 represented represent VERB iajs-2433 68 19 by by ADP iajs-2433 68 20 s s PRON iajs-2433 68 21 closed closed ADJ iajs-2433 68 22 sah sah NOUN iajs-2433 68 23 – – PUNCT iajs-2433 68 24 ideal ideal ADJ iajs-2433 68 25 ) ) PUNCT iajs-2433 68 26 if if SCONJ iajs-2433 68 27 : : PUNCT iajs-2433 68 28 s s NOUN iajs-2433 68 29 ∗ ∗ NOUN iajs-2433 68 30 0 0 NUM iajs-2433 68 31 ∗ ∗ NOUN iajs-2433 68 32 ς∗ ς∗ PROPN iajs-2433 68 33 ∗ ∗ NOUN iajs-2433 68 34 ζ ζ NOUN iajs-2433 68 35 ∈ ∈ PROPN iajs-2433 68 36 𝔜 𝔜 NOUN iajs-2433 68 37 ∧ ∧ NOUN iajs-2433 68 38 s s PART iajs-2433 68 39 ∗ ∗ NOUN iajs-2433 68 40 0 0 NUM iajs-2433 68 41 ∗ ∗ NOUN iajs-2433 68 42 ζ∗ ζ∗ PROPN iajs-2433 68 43 ∈ ∈ PROPN iajs-2433 68 44 𝔜 𝔜 PROPN iajs-2433 68 45 . . PUNCT iajs-2433 69 1 then then ADV iajs-2433 69 2 s s AUX iajs-2433 69 3 ∗ ∗ NOUN iajs-2433 69 4 0 0 NUM iajs-2433 69 5 ∗ ∗ NOUN iajs-2433 69 6 ζ∗ ζ∗ PROPN iajs-2433 69 7 ∗ ∗ NOUN iajs-2433 69 8 ς ς PROPN iajs-2433 69 9 ∈ ∈ PROPN iajs-2433 69 10 𝔜 𝔜 PROPN iajs-2433 69 11 example example NOUN iajs-2433 69 12 ( ( PUNCT iajs-2433 69 13 11 11 NUM iajs-2433 69 14 ) ) PUNCT iajs-2433 69 15 consider consider VERB iajs-2433 69 16 € € NOUN iajs-2433 69 17 0,1,2,3 0,1,2,3 NUM iajs-2433 69 18 with with ADP iajs-2433 69 19 binary binary ADJ iajs-2433 69 20 operation operation NOUN iajs-2433 69 21 ∗ ∗ NOUN iajs-2433 69 22 defined define VERB iajs-2433 69 23 by by ADP iajs-2433 69 24 the the DET iajs-2433 69 25 ensuing ensue VERB iajs-2433 69 26 table table NOUN iajs-2433 69 27 : : PUNCT iajs-2433 69 28 table table NOUN iajs-2433 69 29 4 4 NUM iajs-2433 69 30 . . PUNCT iajs-2433 70 1 * * SYM iajs-2433 70 2 0 0 NUM iajs-2433 71 1 1 1 NUM iajs-2433 71 2 2 2 NUM iajs-2433 71 3 3 3 NUM iajs-2433 71 4 0 0 NUM iajs-2433 71 5 0 0 NUM iajs-2433 71 6 0 0 NUM iajs-2433 71 7 0 0 NUM iajs-2433 71 8 0 0 NUM iajs-2433 71 9 1 1 NUM iajs-2433 71 10 1 1 NUM iajs-2433 71 11 0 0 NUM iajs-2433 71 12 0 0 NUM iajs-2433 71 13 1 1 NUM iajs-2433 71 14 2 2 NUM iajs-2433 71 15 2 2 NUM iajs-2433 71 16 3 3 NUM iajs-2433 71 17 0 0 NUM iajs-2433 71 18 3 3 NUM iajs-2433 71 19 3 3 NUM iajs-2433 71 20 3 3 NUM iajs-2433 71 21 0 0 NUM iajs-2433 71 22 0 0 NUM iajs-2433 71 23 0 0 NUM iajs-2433 71 24 𝔜 𝔜 NOUN iajs-2433 71 25 0,2 0,2 NUM iajs-2433 71 26 , , PUNCT iajs-2433 71 27 s s NOUN iajs-2433 71 28 3 3 NUM iajs-2433 71 29 and and CCONJ iajs-2433 71 30 𝔜 𝔜 PROPN iajs-2433 71 31 is be AUX iajs-2433 71 32 3 3 NUM iajs-2433 71 33 – – PUNCT iajs-2433 71 34 closed close VERB iajs-2433 71 35 sah sah NOUN iajs-2433 71 36 – – PUNCT iajs-2433 71 37 ideal ideal NOUN iajs-2433 71 38 of of ADP iajs-2433 71 39 € € SYM iajs-2433 71 40 . . PUNCT iajs-2433 72 1 3 3 X iajs-2433 72 2 . . X iajs-2433 72 3 completely completely ADV iajs-2433 72 4 closed closed ADJ iajs-2433 72 5 sah sah NOUN iajs-2433 72 6 – – PUNCT iajs-2433 72 7 ideal ideal ADJ iajs-2433 72 8 with with ADP iajs-2433 72 9 respect respect NOUN iajs-2433 72 10 to to ADP iajs-2433 72 11 an an DET iajs-2433 72 12 element element NOUN iajs-2433 72 13 of of ADP iajs-2433 72 14 bh bh NOUN iajs-2433 72 15 – – PUNCT iajs-2433 72 16 algebra algebra NOUN iajs-2433 72 17 definition definition NOUN iajs-2433 72 18 ( ( PUNCT iajs-2433 72 19 12 12 NUM iajs-2433 72 20 ) ) PUNCT iajs-2433 72 21 a a DET iajs-2433 72 22 sah sah NOUN iajs-2433 72 23 – – PUNCT iajs-2433 72 24 ideal ideal ADJ iajs-2433 72 25 𝔜 𝔜 NOUN iajs-2433 72 26 of of ADP iajs-2433 72 27 € € NOUN iajs-2433 72 28 is be AUX iajs-2433 72 29 known know VERB iajs-2433 72 30 as as ADP iajs-2433 72 31 completely completely ADV iajs-2433 72 32 closed closed ADJ iajs-2433 72 33 sah sah NOUN iajs-2433 72 34 – – PUNCT iajs-2433 72 35 ideal ideal ADJ iajs-2433 72 36 if if SCONJ iajs-2433 72 37 ς ς PROPN iajs-2433 72 38 ∗ ∗ VERB iajs-2433 72 39 ζ ζ NOUN iajs-2433 72 40 ∈ ∈ PROPN iajs-2433 72 41 𝔜 𝔜 NOUN iajs-2433 72 42 , , PUNCT iajs-2433 72 43 ∀ ∀ X iajs-2433 72 44 ς ς PROPN iajs-2433 72 45 , , PUNCT iajs-2433 72 46 ζ ζ NOUN iajs-2433 72 47 ∈ ∈ PROPN iajs-2433 72 48 𝔜 𝔜 PROPN iajs-2433 72 49 ( ( PUNCT iajs-2433 72 50 represented represent VERB iajs-2433 72 51 by by ADP iajs-2433 72 52 𝑆𝐴𝐻 𝑆𝐴𝐻 PROPN iajs-2433 72 53 – – PUNCT iajs-2433 72 54 ideal ideal ADJ iajs-2433 72 55 ) ) PUNCT iajs-2433 72 56 . . PUNCT iajs-2433 73 1 example example NOUN iajs-2433 73 2 ( ( PUNCT iajs-2433 73 3 13 13 NUM iajs-2433 73 4 ) ) PUNCT iajs-2433 73 5 in in ADP iajs-2433 73 6 example example NOUN iajs-2433 73 7 ( ( PUNCT iajs-2433 73 8 11 11 NUM iajs-2433 73 9 ) ) PUNCT iajs-2433 73 10 , , PUNCT iajs-2433 73 11 we we PRON iajs-2433 73 12 have have VERB iajs-2433 73 13 𝔜 𝔜 PROPN iajs-2433 73 14 is be AUX iajs-2433 73 15 𝑆𝐴𝐻 𝑆𝐴𝐻 PROPN iajs-2433 73 16 – – PUNCT iajs-2433 73 17 ideal ideal NOUN iajs-2433 73 18 of of ADP iajs-2433 73 19 € € NOUN iajs-2433 73 20 since since SCONJ iajs-2433 73 21 : : PUNCT iajs-2433 73 22 0 0 NUM iajs-2433 73 23 ∗ ∗ NOUN iajs-2433 73 24 0 0 NUM iajs-2433 73 25 0 0 NUM iajs-2433 73 26 ∈ ∈ PROPN iajs-2433 73 27 𝔜 𝔜 NOUN iajs-2433 73 28 , , PUNCT iajs-2433 73 29 0 0 NUM iajs-2433 73 30 ∗ ∗ NOUN iajs-2433 73 31 2 2 NUM iajs-2433 73 32 0 0 NUM iajs-2433 73 33 ∈ ∈ PROPN iajs-2433 73 34 𝔜 𝔜 PROPN iajs-2433 73 35 2 2 NUM iajs-2433 73 36 ∗ ∗ NOUN iajs-2433 73 37 0 0 NUM iajs-2433 73 38 2 2 NUM iajs-2433 73 39 ∈ ∈ PROPN iajs-2433 73 40 𝔜 𝔜 NOUN iajs-2433 73 41 , , PUNCT iajs-2433 73 42 2 2 NUM iajs-2433 73 43 ∗ ∗ NOUN iajs-2433 73 44 2 2 NUM iajs-2433 73 45 0 0 NUM iajs-2433 73 46 ∈ ∈ NOUN iajs-2433 73 47 𝔜 𝔜 NOUN iajs-2433 73 48 definition definition NOUN iajs-2433 73 49 ( ( PUNCT iajs-2433 73 50 14 14 NUM iajs-2433 73 51 ) ) PUNCT iajs-2433 73 52 a a DET iajs-2433 73 53 sah sah NOUN iajs-2433 73 54 – – PUNCT iajs-2433 73 55 ideal ideal ADJ iajs-2433 73 56 𝔜 𝔜 PROPN iajs-2433 73 57 of of ADP iajs-2433 73 58 € € NUM iajs-2433 73 59 and and CCONJ iajs-2433 73 60 s s NOUN iajs-2433 73 61 ∈ ∈ NOUN iajs-2433 73 62 € € NOUN iajs-2433 73 63 , , PUNCT iajs-2433 73 64 then then ADV iajs-2433 73 65 𝔜 𝔜 PROPN iajs-2433 73 66 is be AUX iajs-2433 73 67 named name VERB iajs-2433 73 68 a a DET iajs-2433 73 69 completely completely ADV iajs-2433 73 70 closed closed ADJ iajs-2433 73 71 sah sah NOUN iajs-2433 73 72 – – PUNCT iajs-2433 73 73 ideal ideal ADJ iajs-2433 73 74 with with ADP iajs-2433 73 75 respect respect NOUN iajs-2433 73 76 to to ADP iajs-2433 73 77 an an DET iajs-2433 73 78 element element NOUN iajs-2433 73 79 s s PART iajs-2433 73 80 ∈ ∈ NOUN iajs-2433 73 81 € € NOUN iajs-2433 73 82 ( ( PUNCT iajs-2433 73 83 represented represent VERB iajs-2433 73 84 by by ADP iajs-2433 73 85 s s PROPN iajs-2433 73 86 𝑆𝐴𝐻 𝑆𝐴𝐻 PROPN iajs-2433 73 87 – – PUNCT iajs-2433 73 88 ideal ideal ADJ iajs-2433 73 89 ) ) PUNCT iajs-2433 73 90 if if SCONJ iajs-2433 73 91 s s VERB iajs-2433 73 92 ∗ ∗ NOUN iajs-2433 73 93 0 0 NUM iajs-2433 73 94 ∗ ∗ NOUN iajs-2433 73 95 ς ς PROPN iajs-2433 73 96 ∗ ∗ NOUN iajs-2433 73 97 ζ ζ NOUN iajs-2433 73 98 ∈ ∈ PROPN iajs-2433 73 99 𝔜 𝔜 NOUN iajs-2433 73 100 , , PUNCT iajs-2433 73 101 ∀ ∀ X iajs-2433 73 102 ς ς PROPN iajs-2433 73 103 , , PUNCT iajs-2433 74 1 ζ ζ NOUN iajs-2433 74 2 ∈ ∈ PROPN iajs-2433 74 3 𝔜 𝔜 PROPN iajs-2433 74 4 example example NOUN iajs-2433 74 5 ( ( PUNCT iajs-2433 74 6 15 15 NUM iajs-2433 74 7 ) ) PUNCT iajs-2433 74 8 in in ADP iajs-2433 74 9 example example NOUN iajs-2433 74 10 ( ( PUNCT iajs-2433 74 11 11 11 NUM iajs-2433 74 12 ) ) PUNCT iajs-2433 74 13 , , PUNCT iajs-2433 74 14 we we PRON iajs-2433 74 15 have have VERB iajs-2433 74 16 : : PUNCT iajs-2433 74 17 𝔜 𝔜 NOUN iajs-2433 74 18 0,2 0,2 NUM iajs-2433 74 19 and and CCONJ iajs-2433 74 20 s s NOUN iajs-2433 74 21 2 2 NUM iajs-2433 74 22 , , PUNCT iajs-2433 74 23 then then ADV iajs-2433 74 24 𝔜 𝔜 PROPN iajs-2433 74 25 is be AUX iajs-2433 74 26 2 2 NUM iajs-2433 74 27 – – PUNCT iajs-2433 74 28 𝑆𝐴𝐻 𝑆𝐴𝐻 PROPN iajs-2433 74 29 – – PUNCT iajs-2433 74 30 ideal ideal ADJ iajs-2433 74 31 since since SCONJ iajs-2433 74 32 : : PUNCT iajs-2433 74 33 2 2 NUM iajs-2433 74 34 ∗ ∗ NOUN iajs-2433 74 35 0 0 NUM iajs-2433 74 36 ∗ ∗ NOUN iajs-2433 74 37 0 0 NUM iajs-2433 74 38 ∗ ∗ NOUN iajs-2433 74 39 0 0 NUM iajs-2433 74 40 2 2 NUM iajs-2433 74 41 ∈ ∈ PROPN iajs-2433 74 42 𝔜 𝔜 NOUN iajs-2433 74 43 , , PUNCT iajs-2433 74 44 2 2 NUM iajs-2433 74 45 ∗ ∗ NOUN iajs-2433 74 46 0 0 NUM iajs-2433 74 47 ∗ ∗ NOUN iajs-2433 74 48 0 0 NUM iajs-2433 74 49 ∗ ∗ NOUN iajs-2433 74 50 2 2 NUM iajs-2433 74 51 2 2 NUM iajs-2433 74 52 ∈ ∈ NOUN iajs-2433 74 53 𝔜 𝔜 NOUN iajs-2433 74 54 2 2 NUM iajs-2433 74 55 ∗ ∗ NOUN iajs-2433 74 56 0 0 NUM iajs-2433 74 57 ∗ ∗ NOUN iajs-2433 74 58 2 2 NUM iajs-2433 74 59 ∗ ∗ NOUN iajs-2433 74 60 0 0 NUM iajs-2433 74 61 2 2 NUM iajs-2433 74 62 ∈ ∈ PROPN iajs-2433 74 63 𝔜 𝔜 NOUN iajs-2433 74 64 , , PUNCT iajs-2433 74 65 2 2 NUM iajs-2433 74 66 ∗ ∗ NOUN iajs-2433 74 67 2 2 NUM iajs-2433 74 68 ∗ ∗ NOUN iajs-2433 74 69 2 2 NUM iajs-2433 74 70 ∗ ∗ NOUN iajs-2433 74 71 2 2 NUM iajs-2433 74 72 2 2 NUM iajs-2433 74 73 ∈ ∈ NOUN iajs-2433 74 74 𝔜 𝔜 NOUN iajs-2433 74 75 remark remark NOUN iajs-2433 74 76 ( ( PUNCT iajs-2433 74 77 16 16 NUM iajs-2433 74 78 ) ) PUNCT iajs-2433 74 79 in in ADP iajs-2433 74 80 € € SYM iajs-2433 74 81 every every PRON iajs-2433 74 82 s s X iajs-2433 74 83 𝑆𝐴𝐻 𝑆𝐴𝐻 PROPN iajs-2433 74 84 – – PUNCT iajs-2433 74 85 ideal ideal ADJ iajs-2433 74 86 is be AUX iajs-2433 74 87 a a DET iajs-2433 74 88 s s NOUN iajs-2433 74 89 closed closed ADJ iajs-2433 74 90 sah sah NOUN iajs-2433 74 91 – – PUNCT iajs-2433 74 92 ideal ideal ADJ iajs-2433 74 93 . . PUNCT iajs-2433 74 94     SPACE iajs-2433 75 1 125 125 NUM iajs-2433 75 2     SPACE iajs-2433 75 3 ibn ibn PROPN iajs-2433 75 4 al al PROPN iajs-2433 75 5 - - PUNCT iajs-2433 75 6 haitham haitham PROPN iajs-2433 75 7 jour jour X iajs-2433 75 8 . . PROPN iajs-2433 75 9 for for ADP iajs-2433 75 10 pure pure ADJ iajs-2433 75 11 & & CCONJ iajs-2433 75 12 appl appl PROPN iajs-2433 75 13 . . PUNCT iajs-2433 76 1 sci sci PROPN iajs-2433 76 2 . . PROPN iajs-2433 77 1 33 33 NUM iajs-2433 77 2 ( ( PUNCT iajs-2433 77 3 2 2 NUM iajs-2433 77 4 ) ) PUNCT iajs-2433 77 5 2020 2020 NUM iajs-2433 77 6 proposition proposition NOUN iajs-2433 77 7 ( ( PUNCT iajs-2433 77 8 17 17 NUM iajs-2433 77 9 ) ) PUNCT iajs-2433 77 10 assume assume VERB iajs-2433 77 11 𝔜 𝔜 PROPN iajs-2433 77 12 is be AUX iajs-2433 77 13 a a DET iajs-2433 77 14 𝑆𝐴𝐻 𝑆𝐴𝐻 PROPN iajs-2433 77 15 – – PUNCT iajs-2433 77 16 ideal ideal NOUN iajs-2433 77 17 of of ADP iajs-2433 77 18 € € X iajs-2433 77 19 . . PUNCT iajs-2433 78 1 then then ADV iajs-2433 78 2 𝔜 𝔜 PROPN iajs-2433 78 3 is be AUX iajs-2433 78 4 a a DET iajs-2433 78 5 s s X iajs-2433 78 6 𝑆𝐴𝐻 𝑆𝐴𝐻 PROPN iajs-2433 78 7 – – PUNCT iajs-2433 78 8 ideal ideal ADJ iajs-2433 78 9 , , PUNCT iajs-2433 78 10 ∀s ∀s PROPN iajs-2433 78 11 ∈ ∈ PROPN iajs-2433 78 12 𝔜 𝔜 PROPN iajs-2433 78 13 . . PUNCT iajs-2433 79 1 proof proof NOUN iajs-2433 79 2 assume assume VERB iajs-2433 79 3 ∀ ∀ X iajs-2433 79 4 ς ς PROPN iajs-2433 79 5 , , PUNCT iajs-2433 79 6 ζ ζ NOUN iajs-2433 79 7 ∈ ∈ NOUN iajs-2433 79 8 𝔜 𝔜 NOUN iajs-2433 79 9 mean mean VERB iajs-2433 79 10 while while SCONJ iajs-2433 79 11 𝔜 𝔜 PROPN iajs-2433 79 12 is be AUX iajs-2433 79 13 𝑆𝐴𝐻 𝑆𝐴𝐻 PROPN iajs-2433 79 14 – – PUNCT iajs-2433 79 15 ideal ideal ADJ iajs-2433 79 16 and and CCONJ iajs-2433 79 17 s s NOUN iajs-2433 79 18 ∈ ∈ PROPN iajs-2433 79 19 𝔜 𝔜 PROPN iajs-2433 79 20 then then ADV iajs-2433 79 21 𝑠 𝑠 INTJ iajs-2433 79 22 ∗ ∗ X iajs-2433 79 23 0 0 NUM iajs-2433 79 24 ∗ ∗ NOUN iajs-2433 79 25 ς ς PROPN iajs-2433 79 26 ∗ ∗ NOUN iajs-2433 79 27 ζ ζ NOUN iajs-2433 79 28 ∈ ∈ PROPN iajs-2433 79 29 𝔜 𝔜 NOUN iajs-2433 79 30 . . PUNCT iajs-2433 80 1 ∎ ∎ PROPN iajs-2433 80 2 theorem theorem ADJ iajs-2433 80 3 ( ( PUNCT iajs-2433 80 4 18 18 NUM iajs-2433 80 5 ) ) PUNCT iajs-2433 80 6 assume assume VERB iajs-2433 80 7 € € NOUN iajs-2433 80 8 ; ; PUNCT iajs-2433 80 9 ∗ ∗ NOUN iajs-2433 80 10 , , PUNCT iajs-2433 80 11 0 0 NUM iajs-2433 80 12 and and CCONJ iajs-2433 80 13 ₡ ₡ NUM iajs-2433 80 14 ; ; PUNCT iajs-2433 80 15 ⊛ ⊛ NUM iajs-2433 80 16 , , PUNCT iajs-2433 80 17 0 0 NUM iajs-2433 80 18 are be AUX iajs-2433 80 19 bh bh NOUN iajs-2433 80 20 – – PUNCT iajs-2433 80 21 algebras algebra NOUN iajs-2433 80 22 and and CCONJ iajs-2433 80 23 𝔥 𝔥 NOUN iajs-2433 80 24 : : PUNCT iajs-2433 80 25 € € NOUN iajs-2433 80 26 → → SYM iajs-2433 80 27 ₡ ₡ NOUN iajs-2433 80 28 is be AUX iajs-2433 80 29 a a DET iajs-2433 80 30 bh bh NOUN iajs-2433 80 31 – – PUNCT iajs-2433 80 32 epimorphism epimorphism NOUN iajs-2433 80 33 and and CCONJ iajs-2433 80 34 𝔜 𝔜 PROPN iajs-2433 80 35 is be AUX iajs-2433 80 36 a a DET iajs-2433 80 37 sah sah NOUN iajs-2433 80 38 – – PUNCT iajs-2433 80 39 ideal ideal NOUN iajs-2433 80 40 in in ADP iajs-2433 80 41 € € NOUN iajs-2433 80 42 , , PUNCT iajs-2433 80 43 then then ADV iajs-2433 80 44 𝔥 𝔥 PROPN iajs-2433 80 45 𝔜 𝔜 PROPN iajs-2433 80 46 is be AUX iajs-2433 80 47 a a DET iajs-2433 80 48 sah sah NOUN iajs-2433 80 49 – – PUNCT iajs-2433 80 50 ideal ideal ADJ iajs-2433 80 51 in in ADP iajs-2433 80 52 ₡ ₡ NOUN iajs-2433 80 53 . . PUNCT iajs-2433 81 1 proof proof NOUN iajs-2433 81 2 assume assume VERB iajs-2433 81 3 ς∗ ς∗ PROPN iajs-2433 81 4 ⊛ ⊛ ADJ iajs-2433 81 5 ζ ζ NOUN iajs-2433 81 6 ∈ ∈ PROPN iajs-2433 81 7 𝔥 𝔥 ADP iajs-2433 81 8 𝔜 𝔜 PROPN iajs-2433 81 9 ∧ ∧ PROPN iajs-2433 81 10 ζ∗ ζ∗ PROPN iajs-2433 81 11 ∈ ∈ PROPN iajs-2433 81 12 𝔥 𝔥 ADP iajs-2433 81 13 𝔜 𝔜 PROPN iajs-2433 81 14 to to PART iajs-2433 81 15 prove prove VERB iajs-2433 81 16 ζ∗ ζ∗ PROPN iajs-2433 81 17 ⊛ ⊛ NUM iajs-2433 81 18 ς ς PROPN iajs-2433 81 19 ∈ ∈ PROPN iajs-2433 81 20 𝔥 𝔥 ADP iajs-2433 81 21 𝔜 𝔜 PROPN iajs-2433 81 22 , , PUNCT iajs-2433 81 23 ∀ς ∀ς ADV iajs-2433 81 24 , , PUNCT iajs-2433 81 25 ζ ζ PROPN iajs-2433 81 26 ∈ ∈ PROPN iajs-2433 81 27 𝔜 𝔜 PROPN iajs-2433 81 28 ⟹ ⟹ NUM iajs-2433 81 29 ∃ ∃ PROPN iajs-2433 81 30 a a PROPN iajs-2433 81 31 , , PUNCT iajs-2433 81 32 b b X iajs-2433 81 33 ∈ ∈ PROPN iajs-2433 81 34 𝔜 𝔜 NOUN iajs-2433 81 35 such such ADJ iajs-2433 81 36 that that SCONJ iajs-2433 81 37 𝔥 𝔥 ADP iajs-2433 81 38 a a DET iajs-2433 81 39 ς ς PROPN iajs-2433 81 40 , , PUNCT iajs-2433 81 41 𝔥 𝔥 PROPN iajs-2433 81 42 b b PROPN iajs-2433 81 43 ζ ζ PROPN iajs-2433 81 44 , , PUNCT iajs-2433 81 45 𝔥 𝔥 PROPN iajs-2433 81 46 a a DET iajs-2433 81 47 ∗ ∗ NOUN iajs-2433 81 48 ⊛ ⊛ NUM iajs-2433 81 49 𝔥 𝔥 DET iajs-2433 81 50 b b PROPN iajs-2433 81 51 ∈ ∈ PROPN iajs-2433 81 52 𝔥 𝔥 ADP iajs-2433 81 53 𝔜 𝔜 PROPN iajs-2433 81 54 ∧ ∧ PROPN iajs-2433 81 55 𝔥 𝔥 PROPN iajs-2433 81 56 b b PROPN iajs-2433 81 57 ∗ ∗ X iajs-2433 81 58 ∈ ∈ NOUN iajs-2433 81 59 𝔥 𝔥 ADP iajs-2433 81 60 𝔜 𝔜 PROPN iajs-2433 81 61 𝔥 𝔥 PROPN iajs-2433 81 62 a a DET iajs-2433 81 63 ∗ ∗ NOUN iajs-2433 81 64 ⊛ ⊛ NUM iajs-2433 81 65 𝔥 𝔥 DET iajs-2433 81 66 b b PROPN iajs-2433 81 67 ∈ ∈ PROPN iajs-2433 81 68 𝔥 𝔥 ADP iajs-2433 81 69 𝔜 𝔜 PROPN iajs-2433 81 70 ∧ ∧ PROPN iajs-2433 81 71 𝔥 𝔥 PROPN iajs-2433 81 72 b b PROPN iajs-2433 81 73 ∗ ∗ X iajs-2433 81 74 ∈ ∈ NOUN iajs-2433 81 75 𝔥 𝔥 ADP iajs-2433 81 76 𝔜 𝔜 PROPN iajs-2433 81 77 𝔥 𝔥 ADP iajs-2433 81 78 a∗ a∗ PROPN iajs-2433 81 79 ∗ ∗ X iajs-2433 81 80 b b X iajs-2433 81 81 ∈ ∈ PROPN iajs-2433 81 82 𝔥 𝔥 ADP iajs-2433 81 83 𝔜 𝔜 PROPN iajs-2433 81 84 ∧ ∧ PROPN iajs-2433 81 85 𝔥 𝔥 ADP iajs-2433 81 86 b∗ b∗ ADJ iajs-2433 81 87 ∈ ∈ PROPN iajs-2433 81 88 𝔥 𝔥 ADP iajs-2433 81 89 𝔜 𝔜 PROPN iajs-2433 81 90 ⟹ ⟹ PROPN iajs-2433 81 91 a∗ a∗ PROPN iajs-2433 81 92 ∗ ∗ NOUN iajs-2433 81 93 b b PROPN iajs-2433 81 94 ∈ ∈ PROPN iajs-2433 81 95 𝔜 𝔜 PROPN iajs-2433 81 96 ∧ ∧ NOUN iajs-2433 81 97 b∗ b∗ ADJ iajs-2433 81 98 ∈ ∈ PROPN iajs-2433 81 99 𝔜 𝔜 PROPN iajs-2433 81 100 → → SYM iajs-2433 81 101 b∗ b∗ ADJ iajs-2433 81 102 ∗ ∗ VERB iajs-2433 81 103 a a DET iajs-2433 81 104 ∈ ∈ PROPN iajs-2433 81 105 𝔜 𝔜 PROPN iajs-2433 81 106 → → SYM iajs-2433 81 107 𝔥 𝔥 PROPN iajs-2433 81 108 b∗ b∗ ADJ iajs-2433 81 109 ∗ ∗ VERB iajs-2433 81 110 a a DET iajs-2433 81 111 ∈ ∈ NOUN iajs-2433 81 112 𝔥 𝔥 ADP iajs-2433 81 113 𝔜 𝔜 NOUN iajs-2433 81 114 ∵ ∵ NOUN iajs-2433 81 115 𝔥 𝔥 PROPN iajs-2433 82 1 is be AUX iajs-2433 82 2 epimorphism epimorphism NOUN iajs-2433 82 3 ⟹ ⟹ NOUN iajs-2433 82 4 𝔥 𝔥 ADP iajs-2433 82 5 b∗ b∗ PROPN iajs-2433 82 6 ⊛ ⊛ NUM iajs-2433 82 7 𝔥 𝔥 ADP iajs-2433 82 8 a a DET iajs-2433 82 9 ∈ ∈ PROPN iajs-2433 82 10 𝔥 𝔥 ADP iajs-2433 82 11 𝔜 𝔜 PROPN iajs-2433 82 12 ⟹ ⟹ NUM iajs-2433 82 13 𝔥 𝔥 PROPN iajs-2433 82 14 b b NOUN iajs-2433 82 15 ∗ ∗ NOUN iajs-2433 82 16 ⊛ ⊛ NUM iajs-2433 82 17 𝔥 𝔥 ADP iajs-2433 82 18 a a DET iajs-2433 82 19 ∈ ∈ PROPN iajs-2433 82 20 𝔥 𝔥 ADP iajs-2433 82 21 𝔜 𝔜 NOUN iajs-2433 82 22 ζ∗ ζ∗ PROPN iajs-2433 82 23 ⊛ ⊛ NUM iajs-2433 82 24 ς ς PROPN iajs-2433 82 25 ∈ ∈ PROPN iajs-2433 82 26 𝔥 𝔥 ADP iajs-2433 82 27 𝔜 𝔜 PROPN iajs-2433 82 28 ∴ ∴ NOUN iajs-2433 82 29 𝔥 𝔥 PROPN iajs-2433 82 30 𝔜 𝔜 PROPN iajs-2433 82 31 is be AUX iajs-2433 82 32 sah sah NOUN iajs-2433 82 33 – – PUNCT iajs-2433 82 34 ideal ideal ADJ iajs-2433 82 35 in in ADP iajs-2433 82 36 ₡ ₡ NOUN iajs-2433 82 37 . . PUNCT iajs-2433 83 1 ∎ ∎ PROPN iajs-2433 83 2 theorem theorem ADJ iajs-2433 83 3 ( ( PUNCT iajs-2433 83 4 19 19 NUM iajs-2433 83 5 ) ) PUNCT iajs-2433 83 6 assume assume VERB iajs-2433 83 7 € € NOUN iajs-2433 83 8 ; ; PUNCT iajs-2433 83 9 ∗ ∗ NOUN iajs-2433 83 10 , , PUNCT iajs-2433 83 11 0 0 NUM iajs-2433 83 12 and and CCONJ iajs-2433 83 13 ₡ ₡ NUM iajs-2433 83 14 ; ; PUNCT iajs-2433 83 15 ⊛ ⊛ NUM iajs-2433 83 16 , , PUNCT iajs-2433 83 17 0 0 NUM iajs-2433 83 18 are be AUX iajs-2433 83 19 bh bh NOUN iajs-2433 83 20 – – PUNCT iajs-2433 83 21 algebras algebra NOUN iajs-2433 83 22 and and CCONJ iajs-2433 83 23 𝔥 𝔥 NOUN iajs-2433 83 24 : : PUNCT iajs-2433 83 25 € € X iajs-2433 83 26 → → SYM iajs-2433 83 27 ₡ ₡ NOUN iajs-2433 83 28 an an DET iajs-2433 83 29 epimorphism epimorphism NOUN iajs-2433 83 30 and and CCONJ iajs-2433 83 31 𝔜 𝔜 PROPN iajs-2433 83 32 is be AUX iajs-2433 83 33 a a DET iajs-2433 83 34 sah sah NOUN iajs-2433 83 35 – – PUNCT iajs-2433 83 36 ideal ideal NOUN iajs-2433 83 37 in in ADP iajs-2433 83 38 € € X iajs-2433 83 39 . . PUNCT iajs-2433 84 1 then then ADV iajs-2433 84 2 𝔥 𝔥 PROPN iajs-2433 84 3 𝔜 𝔜 PROPN iajs-2433 84 4 is be AUX iajs-2433 84 5 a a DET iajs-2433 84 6 closed closed ADJ iajs-2433 84 7 sah sah NOUN iajs-2433 84 8 – – PUNCT iajs-2433 84 9 ideal ideal ADJ iajs-2433 84 10 in in ADP iajs-2433 84 11 ₡ ₡ NOUN iajs-2433 84 12 . . PUNCT iajs-2433 85 1 proof proof NOUN iajs-2433 85 2 assume assume VERB iajs-2433 85 3 𝔜 𝔜 PROPN iajs-2433 85 4 is be AUX iajs-2433 85 5 a a DET iajs-2433 85 6 sah sah NOUN iajs-2433 85 7 – – PUNCT iajs-2433 85 8 ideal ideal NOUN iajs-2433 85 9 in in ADP iajs-2433 85 10 € € X iajs-2433 85 11 𝔥 𝔥 PRON iajs-2433 85 12 𝔜 𝔜 PROPN iajs-2433 85 13 is be AUX iajs-2433 85 14 sah sah NOUN iajs-2433 85 15 – – PUNCT iajs-2433 85 16 ideal ideal ADJ iajs-2433 85 17 ( ( PUNCT iajs-2433 85 18 theorem theorem NOUN iajs-2433 85 19 ( ( PUNCT iajs-2433 85 20 18 18 NUM iajs-2433 85 21 ) ) PUNCT iajs-2433 85 22 ) ) PUNCT iajs-2433 85 23 and and CCONJ iajs-2433 85 24 by by ADP iajs-2433 85 25 using use VERB iajs-2433 85 26 remark remark NOUN iajs-2433 85 27 ( ( PUNCT iajs-2433 85 28 5 5 NUM iajs-2433 85 29 ) ) PUNCT iajs-2433 85 30     SPACE iajs-2433 85 31 126 126 NUM iajs-2433 85 32     SPACE iajs-2433 85 33 ibn ibn PROPN iajs-2433 85 34 al al PROPN iajs-2433 85 35 - - PUNCT iajs-2433 85 36 haitham haitham PROPN iajs-2433 85 37 jour jour X iajs-2433 85 38 . . PROPN iajs-2433 86 1 for for ADP iajs-2433 86 2 pure pure ADJ iajs-2433 86 3 & & CCONJ iajs-2433 86 4 appl appl PROPN iajs-2433 86 5 . . PUNCT iajs-2433 87 1 sci sci PROPN iajs-2433 87 2 . . PROPN iajs-2433 88 1 33 33 NUM iajs-2433 88 2 ( ( PUNCT iajs-2433 88 3 2 2 NUM iajs-2433 88 4 ) ) PUNCT iajs-2433 88 5 2020 2020 NUM iajs-2433 89 1 𝔥 𝔥 ADP iajs-2433 89 2 𝔜 𝔜 PROPN iajs-2433 89 3 is be AUX iajs-2433 89 4 a a DET iajs-2433 89 5 closed closed ADJ iajs-2433 89 6 sah sah NOUN iajs-2433 89 7 – – PUNCT iajs-2433 89 8 ideal ideal ADJ iajs-2433 89 9 in in ADP iajs-2433 89 10 ₡ ₡ NOUN iajs-2433 89 11 . . PUNCT iajs-2433 90 1 ∎ ∎ PROPN iajs-2433 90 2 remark remark NOUN iajs-2433 90 3 ( ( PUNCT iajs-2433 90 4 20 20 NUM iajs-2433 90 5 ) ) PUNCT iajs-2433 90 6 now now ADV iajs-2433 90 7 each each DET iajs-2433 90 8 sah sah NOUN iajs-2433 90 9 – – PUNCT iajs-2433 90 10 ideal ideal NOUN iajs-2433 90 11 of of ADP iajs-2433 90 12 € € NUM iajs-2433 90 13 is be AUX iajs-2433 90 14 a a DET iajs-2433 90 15 s s X iajs-2433 90 16 𝑆𝐴𝐻 𝑆𝐴𝐻 PROPN iajs-2433 90 17 – – PUNCT iajs-2433 90 18 ideal ideal NOUN iajs-2433 90 19 of of ADP iajs-2433 90 20 € € NUM iajs-2433 90 21 , , PUNCT iajs-2433 90 22 ∀s ∀s NOUN iajs-2433 90 23 ∈ ∈ PROPN iajs-2433 90 24 𝔜 𝔜 PROPN iajs-2433 90 25 . . PUNCT iajs-2433 91 1 theorem theorem NOUN iajs-2433 91 2 ( ( PUNCT iajs-2433 91 3 21 21 NUM iajs-2433 91 4 ) ) PUNCT iajs-2433 91 5 assume assume VERB iajs-2433 91 6 € € NOUN iajs-2433 91 7 ; ; PUNCT iajs-2433 91 8 ∗ ∗ NOUN iajs-2433 91 9 , , PUNCT iajs-2433 91 10 0 0 NUM iajs-2433 91 11 and and CCONJ iajs-2433 91 12 ₡ ₡ NUM iajs-2433 91 13 ; ; PUNCT iajs-2433 91 14 ⊛ ⊛ NUM iajs-2433 91 15 , , PUNCT iajs-2433 91 16 0 0 NUM iajs-2433 91 17 are be AUX iajs-2433 91 18 bh bh NOUN iajs-2433 91 19 – – PUNCT iajs-2433 91 20 algebras algebra NOUN iajs-2433 91 21 and and CCONJ iajs-2433 91 22 : : PUNCT iajs-2433 91 23 € € NOUN iajs-2433 91 24 → → SYM iajs-2433 91 25 ₡ ₡ NOUN iajs-2433 91 26 is be AUX iajs-2433 91 27 a a DET iajs-2433 91 28 epimorphism epimorphism NOUN iajs-2433 91 29 , , PUNCT iajs-2433 91 30 if if SCONJ iajs-2433 91 31 𝔜 𝔜 PROPN iajs-2433 91 32 is be AUX iajs-2433 91 33 a a DET iajs-2433 91 34 s s X iajs-2433 91 35 𝑆𝐴𝐻 𝑆𝐴𝐻 PROPN iajs-2433 91 36 – – PUNCT iajs-2433 91 37 ideal ideal ADJ iajs-2433 91 38 in in ADP iajs-2433 91 39 € € NOUN iajs-2433 91 40 , , PUNCT iajs-2433 91 41 then then ADV iajs-2433 91 42 𝔥 𝔥 PROPN iajs-2433 91 43 𝔜 𝔜 PROPN iajs-2433 91 44 is be AUX iajs-2433 91 45 a a DET iajs-2433 91 46 𝔥 𝔥 PROPN iajs-2433 91 47 s s X iajs-2433 91 48 𝑆𝐴𝐻 𝑆𝐴𝐻 PROPN iajs-2433 91 49 – – PUNCT iajs-2433 91 50 ideal ideal ADJ iajs-2433 91 51 in in ADP iajs-2433 91 52 ₡ ₡ NOUN iajs-2433 91 53 . . PUNCT iajs-2433 92 1 proof proof NOUN iajs-2433 92 2 assume assume VERB iajs-2433 92 3 𝔜 𝔜 PROPN iajs-2433 92 4 is be AUX iajs-2433 92 5 a a DET iajs-2433 92 6 s s X iajs-2433 92 7 𝑆𝐴𝐻 𝑆𝐴𝐻 PROPN iajs-2433 92 8 – – PUNCT iajs-2433 92 9 ideal ideal ADJ iajs-2433 92 10 in in ADP iajs-2433 92 11 € € NOUN iajs-2433 92 12 , , PUNCT iajs-2433 92 13 then then ADV iajs-2433 92 14 s s VERB iajs-2433 92 15 ∗ ∗ NOUN iajs-2433 92 16 a a DET iajs-2433 92 17 ∗ ∗ NOUN iajs-2433 92 18 c c NOUN iajs-2433 92 19 ∈ ∈ PROPN iajs-2433 92 20 𝔜 𝔜 PROPN iajs-2433 92 21 , , PUNCT iajs-2433 92 22 ∀ ∀ X iajs-2433 92 23 a a PRON iajs-2433 92 24 , , PUNCT iajs-2433 92 25 c c PROPN iajs-2433 92 26 ∈ ∈ PROPN iajs-2433 92 27 𝔜 𝔜 PROPN iajs-2433 92 28 since since SCONJ iajs-2433 92 29 𝔜 𝔜 PROPN iajs-2433 92 30 is be AUX iajs-2433 92 31 sah sah NOUN iajs-2433 92 32 – – PUNCT iajs-2433 92 33 ideal ideal ADJ iajs-2433 92 34 , , PUNCT iajs-2433 92 35 then then ADV iajs-2433 92 36 𝔥 𝔥 PROPN iajs-2433 92 37 𝔜 𝔜 PROPN iajs-2433 92 38 is be AUX iajs-2433 92 39 a a DET iajs-2433 92 40 sah sah NOUN iajs-2433 92 41 – – PUNCT iajs-2433 92 42 ideal ideal NOUN iajs-2433 92 43 ( ( PUNCT iajs-2433 92 44 theorem theorem ADJ iajs-2433 92 45 18 18 NUM iajs-2433 92 46 ) ) PUNCT iajs-2433 92 47 assume assume VERB iajs-2433 92 48 ς ς PROPN iajs-2433 92 49 , , PUNCT iajs-2433 92 50 ζ ζ NOUN iajs-2433 92 51 ∈ ∈ PROPN iajs-2433 92 52 𝔥 𝔥 ADP iajs-2433 92 53 𝔜 𝔜 PROPN iajs-2433 92 54 ⟹ ⟹ NUM iajs-2433 92 55 ∃ ∃ PROPN iajs-2433 92 56 m m PROPN iajs-2433 92 57 , , PUNCT iajs-2433 92 58 n n NOUN iajs-2433 92 59 ∈ ∈ NOUN iajs-2433 92 60 𝔜 𝔜 NOUN iajs-2433 92 61 such such ADJ iajs-2433 92 62 that that PRON iajs-2433 92 63 𝔥 𝔥 PROPN iajs-2433 92 64 m m NOUN iajs-2433 92 65 ς ς PROPN iajs-2433 92 66 , , PUNCT iajs-2433 92 67 𝖍 𝖍 PROPN iajs-2433 92 68 n n PRON iajs-2433 92 69 ζ ζ NOUN iajs-2433 92 70 𝔥 𝔥 X iajs-2433 92 71 s s PROPN iajs-2433 92 72 ⊛ ⊛ NUM iajs-2433 92 73 ς ς PROPN iajs-2433 92 74 ⊛ ⊛ NUM iajs-2433 92 75 ζ ζ NOUN iajs-2433 92 76 𝖍 𝖍 PROPN iajs-2433 92 77 s s PART iajs-2433 92 78 ⊛ ⊛ NUM iajs-2433 92 79 𝔥 𝔥 X iajs-2433 92 80 m m PROPN iajs-2433 92 81 ⊛ ⊛ NUM iajs-2433 92 82 𝔥 𝔥 ADP iajs-2433 92 83 n n PROPN iajs-2433 92 84 𝔥 𝔥 X iajs-2433 92 85 s s PROPN iajs-2433 92 86 ⊛ ⊛ NUM iajs-2433 92 87 𝔥 𝔥 X iajs-2433 92 88 m m PROPN iajs-2433 92 89 ∗ ∗ NOUN iajs-2433 92 90 n n CCONJ iajs-2433 92 91 𝔥 𝔥 PROPN iajs-2433 92 92 s s PROPN iajs-2433 92 93 ∗ ∗ PROPN iajs-2433 92 94 m m PROPN iajs-2433 92 95 ∗ ∗ NOUN iajs-2433 92 96 n n NOUN iajs-2433 92 97 ∈ ∈ NOUN iajs-2433 92 98 𝔥 𝔥 X iajs-2433 92 99 𝔜 𝔜 PROPN iajs-2433 93 1 [ [ X iajs-2433 93 2 since since SCONJ iajs-2433 93 3 s s PROPN iajs-2433 93 4 ∗ ∗ NOUN iajs-2433 93 5 m m PROPN iajs-2433 93 6 ∗ ∗ NOUN iajs-2433 93 7 n n CCONJ iajs-2433 93 8 ∈ ∈ PROPN iajs-2433 93 9 𝔜 𝔜 PROPN iajs-2433 93 10 ] ] X iajs-2433 93 11 ∴ ∴ NOUN iajs-2433 93 12 𝔥 𝔥 PROPN iajs-2433 93 13 𝔜 𝔜 PROPN iajs-2433 93 14 is be AUX iajs-2433 93 15 a a DET iajs-2433 93 16 𝔥 𝔥 PROPN iajs-2433 93 17 s s X iajs-2433 93 18 𝑆𝐴𝐻 𝑆𝐴𝐻 PROPN iajs-2433 93 19 – – PUNCT iajs-2433 93 20 ideal ideal ADJ iajs-2433 93 21 . . PUNCT iajs-2433 94 1 ∎ ∎ NOUN iajs-2433 94 2 proposition proposition NOUN iajs-2433 94 3 ( ( PUNCT iajs-2433 94 4 22 22 NUM iajs-2433 94 5 ) ) PUNCT iajs-2433 94 6 assume assume VERB iajs-2433 94 7 𝔜 𝔜 PROPN iajs-2433 94 8 is be AUX iajs-2433 94 9 a a DET iajs-2433 94 10 sah sah NOUN iajs-2433 94 11 – – PUNCT iajs-2433 94 12 ideal ideal NOUN iajs-2433 94 13 of of ADP iajs-2433 94 14 € € SYM iajs-2433 94 15 such such ADJ iajs-2433 94 16 that that SCONJ iajs-2433 94 17 𝔜 𝔜 SYM iajs-2433 94 18 ⊆ ⊆ NUM iajs-2433 94 19 € € NOUN iajs-2433 94 20 . . PUNCT iajs-2433 95 1 then then ADV iajs-2433 95 2 𝔜 𝔜 PROPN iajs-2433 95 3 is be AUX iajs-2433 95 4 s s NUM iajs-2433 95 5 closed closed ADJ iajs-2433 95 6 sah sah NOUN iajs-2433 95 7 – – PUNCT iajs-2433 95 8 ideal ideal NOUN iajs-2433 95 9 ∀s ∀s PROPN iajs-2433 95 10 ∈ ∈ PROPN iajs-2433 95 11 𝔜 𝔜 PROPN iajs-2433 95 12 . . PUNCT iajs-2433 96 1 where where SCONJ iajs-2433 96 2 € € SYM iajs-2433 96 3 ς ς PROPN iajs-2433 96 4 ∈ ∈ NOUN iajs-2433 96 5 € € NOUN iajs-2433 96 6 : : SYM iajs-2433 96 7 0 0 NUM iajs-2433 96 8 ∗ ∗ NOUN iajs-2433 96 9 ς ς PROPN iajs-2433 96 10 0 0 PUNCT iajs-2433 96 11 . . PUNCT iajs-2433 97 1 proof proof NOUN iajs-2433 97 2 assume assume VERB iajs-2433 97 3 s s X iajs-2433 97 4 ∈ ∈ PROPN iajs-2433 97 5 𝔜 𝔜 PROPN iajs-2433 97 6 and and CCONJ iajs-2433 97 7 ⊆ ⊆ NUM iajs-2433 97 8 € € NOUN iajs-2433 97 9 . . PUNCT iajs-2433 98 1 then then ADV iajs-2433 98 2 s s VERB iajs-2433 98 3 ∗ ∗ NOUN iajs-2433 98 4 0 0 NUM iajs-2433 98 5 ∗ ∗ NOUN iajs-2433 98 6 ς ς PROPN iajs-2433 98 7 s s NOUN iajs-2433 98 8 ∗ ∗ NOUN iajs-2433 98 9 0 0 NUM iajs-2433 98 10 [ [ PUNCT iajs-2433 98 11 since since SCONJ iajs-2433 98 12 𝔜 𝔜 PROPN iajs-2433 98 13 ⊆ ⊆ NUM iajs-2433 98 14 € € NOUN iajs-2433 98 15 ] ] PUNCT iajs-2433 98 16 = = PUNCT iajs-2433 98 17 s∈ s∈ NOUN iajs-2433 98 18 𝔜 𝔜 PROPN iajs-2433 98 19 ∴ ∴ NOUN iajs-2433 98 20 𝔜 𝔜 PROPN iajs-2433 98 21 is be AUX iajs-2433 98 22 s s NUM iajs-2433 98 23 closed closed ADJ iajs-2433 98 24 sah sah NOUN iajs-2433 98 25 – – PUNCT iajs-2433 98 26 ideal ideal ADJ iajs-2433 98 27 . . PUNCT iajs-2433 99 1 ∎ ∎ PROPN iajs-2433 99 2 4 4 NUM iajs-2433 99 3 . . PUNCT iajs-2433 99 4 conclusion conclusion NOUN iajs-2433 99 5 in in ADP iajs-2433 99 6 this this DET iajs-2433 99 7 paper paper NOUN iajs-2433 99 8 , , PUNCT iajs-2433 99 9 we we PRON iajs-2433 99 10 constructed construct VERB iajs-2433 99 11 the the DET iajs-2433 99 12 idea idea NOUN iajs-2433 99 13 of of ADP iajs-2433 99 14 sah sah NOUN iajs-2433 99 15 – – PUNCT iajs-2433 99 16 ideal ideal ADJ iajs-2433 99 17 , , PUNCT iajs-2433 99 18 closed closed ADJ iajs-2433 99 19 sah sah NOUN iajs-2433 99 20 – – PUNCT iajs-2433 99 21 ideal ideal ADJ iajs-2433 99 22 , , PUNCT iajs-2433 99 23 sclosed sclose VERB iajs-2433 99 24 sah sah NOUN iajs-2433 99 25 – – PUNCT iajs-2433 99 26 ideal ideal ADJ iajs-2433 99 27 , , PUNCT iajs-2433 99 28 𝑆𝐴𝐻 𝑆𝐴𝐻 PROPN iajs-2433 99 29 – – PUNCT iajs-2433 99 30 ideal ideal ADJ iajs-2433 99 31 and and CCONJ iajs-2433 99 32 s𝑆𝐴𝐻 s𝑆𝐴𝐻 PROPN iajs-2433 99 33 – – PUNCT iajs-2433 99 34 ideal ideal NOUN iajs-2433 99 35 of of ADP iajs-2433 99 36 bh bh NOUN iajs-2433 99 37 – – PUNCT iajs-2433 99 38 algebra algebra NOUN iajs-2433 99 39 which which PRON iajs-2433 99 40 are be AUX iajs-2433 99 41 presented present VERB iajs-2433 99 42 with with ADP iajs-2433 99 43 some some PRON iajs-2433 99 44 of of ADP iajs-2433 99 45 their their PRON iajs-2433 99 46 properties property NOUN iajs-2433 99 47 , , PUNCT iajs-2433 99 48 examples example NOUN iajs-2433 99 49 and and CCONJ iajs-2433 99 50 theorems theorem NOUN iajs-2433 99 51 . . PUNCT iajs-2433 100 1 in in ADP iajs-2433 100 2 our our PRON iajs-2433 100 3 future future ADJ iajs-2433 100 4 work work NOUN iajs-2433 100 5 , , PUNCT iajs-2433 100 6 we we PRON iajs-2433 100 7 introduce introduce VERB iajs-2433 100 8 the the DET iajs-2433 100 9 concept concept NOUN iajs-2433 100 10 of of ADP iajs-2433 100 11 fuzzy fuzzy ADJ iajs-2433 100 12 sah sah NOUN iajs-2433 100 13 – – PUNCT iajs-2433 100 14 ideal ideal NOUN iajs-2433 100 15 of of ADP iajs-2433 100 16 bh bh NOUN iajs-2433 100 17 – – PUNCT iajs-2433 100 18 algebra algebra NOUN iajs-2433 100 19 . . PUNCT iajs-2433 101 1 it it PRON iajs-2433 101 2 is be AUX iajs-2433 101 3 our our PRON iajs-2433 101 4 optimism optimism NOUN iajs-2433 101 5 that that SCONJ iajs-2433 101 6 this this DET iajs-2433 101 7 effort effort NOUN iajs-2433 101 8 grows grow VERB iajs-2433 101 9 into into ADP iajs-2433 101 10 other other ADJ iajs-2433 101 11 fundamentals fundamental NOUN iajs-2433 101 12 for for ADP iajs-2433 101 13 further further ADJ iajs-2433 101 14 study study NOUN iajs-2433 101 15 of of ADP iajs-2433 101 16 ideas idea NOUN iajs-2433 101 17 of of ADP iajs-2433 101 18 bh bh NOUN iajs-2433 101 19 - - NOUN iajs-2433 101 20 algebra algebra NOUN iajs-2433 101 21 . . PUNCT iajs-2433 102 1 references reference NOUN iajs-2433 102 2 1 1 NUM iajs-2433 102 3 . . X iajs-2433 103 1 zadah zadah PROPN iajs-2433 103 2 , , PUNCT iajs-2433 103 3 l. l. PROPN iajs-2433 103 4 a. a. PROPN iajs-2433 103 5 fuzzy fuzzy PROPN iajs-2433 103 6 sets set NOUN iajs-2433 103 7 . . PUNCT iajs-2433 104 1 in in ADP iajs-2433 104 2 form form NOUN iajs-2433 104 3 . . PUNCT iajs-2433 105 1 control.1965 control.1965 PROPN iajs-2433 105 2 , , PUNCT iajs-2433 105 3 8 8 NUM iajs-2433 105 4 , , PUNCT iajs-2433 105 5 338 338 NUM iajs-2433 105 6 - - SYM iajs-2433 105 7 353 353 NUM iajs-2433 105 8 . . NOUN iajs-2433 105 9 2 2 NUM iajs-2433 105 10 . . X iajs-2433 105 11 iseki iseki PROPN iajs-2433 105 12 , , PUNCT iajs-2433 105 13 k. k. PROPN iajs-2433 105 14 on on ADP iajs-2433 105 15 bci bci PROPN iajs-2433 105 16 - - PUNCT iajs-2433 105 17 algebras algebra NOUN iajs-2433 105 18 . . PUNCT iajs-2433 106 1 mathematics mathematic NOUN iajs-2433 106 2 seminar seminar NOUN iajs-2433 106 3 notes.1980 notes.1980 PROPN iajs-2433 106 4 , , PUNCT iajs-2433 106 5 8 8 NUM iajs-2433 106 6 , , PUNCT iajs-2433 106 7 125 125 NUM iajs-2433 106 8 - - SYM iajs-2433 106 9 130 130 NUM iajs-2433 106 10 . . PUNCT iajs-2433 107 1 3 3 X iajs-2433 107 2 . . X iajs-2433 107 3 iseki iseki PROPN iajs-2433 107 4 , , PUNCT iajs-2433 107 5 k. k. PROPN iajs-2433 107 6 ; ; PUNCT iajs-2433 107 7 tanaka tanaka PROPN iajs-2433 107 8 , , PUNCT iajs-2433 107 9 s. s. PROPN iajs-2433 107 10 an an DET iajs-2433 107 11 introduction introduction NOUN iajs-2433 107 12 to to ADP iajs-2433 107 13 theory theory NOUN iajs-2433 107 14 of of ADP iajs-2433 107 15 bck bck PROPN iajs-2433 107 16 – – PUNCT iajs-2433 107 17 algebras algebras PROPN iajs-2433 107 18 . . PUNCT iajs-2433 108 1 math math NOUN iajs-2433 108 2 . . PUNCT iajs-2433 109 1 japonica japonica PROPN iajs-2433 109 2 . . PUNCT iajs-2433 110 1 1978 1978 NUM iajs-2433 110 2 , , PUNCT iajs-2433 110 3 23 23 NUM iajs-2433 110 4 , , PUNCT iajs-2433 110 5 1 1 NUM iajs-2433 110 6 , , PUNCT iajs-2433 110 7 1 1 NUM iajs-2433 110 8 - - SYM iajs-2433 110 9 8 8 NUM iajs-2433 110 10 . . NOUN iajs-2433 111 1 4 4 NUM iajs-2433 111 2 . . X iajs-2433 112 1 jun jun PROPN iajs-2433 112 2 , , PUNCT iajs-2433 112 3 y. y. PROPN iajs-2433 112 4 b. b. PROPN iajs-2433 112 5 ; ; PUNCT iajs-2433 112 6 roh roh PROPN iajs-2433 112 7 , , PUNCT iajs-2433 112 8 e.h e.h PROPN iajs-2433 112 9 . . PROPN iajs-2433 112 10 on on ADP iajs-2433 112 11 bh bh NOUN iajs-2433 112 12 - - NOUN iajs-2433 112 13 algebra algebra NOUN iajs-2433 112 14 . . PUNCT iajs-2433 113 1 scientiae scientiae PROPN iajs-2433 113 2 mathematica.1998 mathematica.1998 PROPN iajs-2433 113 3 , , PUNCT iajs-2433 113 4 1 1 NUM iajs-2433 113 5 , , PUNCT iajs-2433 113 6 1 1 NUM iajs-2433 113 7 , , PUNCT iajs-2433 113 8 347 347 NUM iajs-2433 113 9 - - SYM iajs-2433 113 10 354 354 NUM iajs-2433 113 11 . . PUNCT iajs-2433 113 12     SPACE iajs-2433 114 1 127 127 NUM iajs-2433 114 2     SPACE iajs-2433 114 3 ibn ibn PROPN iajs-2433 114 4 al al PROPN iajs-2433 114 5 - - PUNCT iajs-2433 114 6 haitham haitham PROPN iajs-2433 114 7 jour jour X iajs-2433 114 8 . . PROPN iajs-2433 114 9 for for ADP iajs-2433 114 10 pure pure ADJ iajs-2433 114 11 & & CCONJ iajs-2433 114 12 appl appl PROPN iajs-2433 114 13 . . PUNCT iajs-2433 115 1 sci sci PROPN iajs-2433 115 2 . . PROPN iajs-2433 116 1 33 33 NUM iajs-2433 116 2 ( ( PUNCT iajs-2433 116 3 2 2 NUM iajs-2433 116 4 ) ) PUNCT iajs-2433 116 5 2020 2020 NUM iajs-2433 116 6 5 5 NUM iajs-2433 116 7 . . PUNCT iajs-2433 117 1 jun jun PROPN iajs-2433 117 2 , , PUNCT iajs-2433 117 3 y.b y.b PROPN iajs-2433 117 4 . . PROPN iajs-2433 117 5 ; ; PUNCT iajs-2433 117 6 kim kim PROPN iajs-2433 117 7 , , PUNCT iajs-2433 117 8 h.s h.s PROPN iajs-2433 117 9 . . PROPN iajs-2433 117 10 ; ; PUNCT iajs-2433 117 11 kondo kondo PROPN iajs-2433 117 12 , , PUNCT iajs-2433 117 13 m. m. NOUN iajs-2433 117 14 on on ADP iajs-2433 117 15 bh bh NOUN iajs-2433 117 16 - - PUNCT iajs-2433 117 17 relations relation NOUN iajs-2433 117 18 in in ADP iajs-2433 117 19 bh bh NOUN iajs-2433 117 20 algebras algebras PROPN iajs-2433 117 21 . . PUNCT iajs-2433 118 1 scientiae scientiae PROPN iajs-2433 118 2 6 6 NUM iajs-2433 118 3 . . PUNCT iajs-2433 119 1 baik baik PROPN iajs-2433 119 2 , , PUNCT iajs-2433 119 3 h.g h.g PROPN iajs-2433 119 4 . . PROPN iajs-2433 120 1 on on ADP iajs-2433 120 2 vague vague ADJ iajs-2433 120 3 bh bh NOUN iajs-2433 120 4 – – PUNCT iajs-2433 120 5 subalgebra subalgebra NOUN iajs-2433 120 6 of of ADP iajs-2433 120 7 bhalgebras bhalgebra NOUN iajs-2433 120 8 . . PUNCT iajs-2433 121 1 international international PROPN iajs-2433 121 2 mathematical mathematical PROPN iajs-2433 121 3 forum.2009 forum.2009 PROPN iajs-2433 121 4 , , PUNCT iajs-2433 121 5 4 4 NUM iajs-2433 121 6 , , PUNCT iajs-2433 121 7 17 17 NUM iajs-2433 121 8 , , PUNCT iajs-2433 121 9 823 823 NUM iajs-2433 121 10 - - SYM iajs-2433 121 11 829 829 NUM iajs-2433 121 12 . . PUNCT iajs-2433 122 1 7 7 X iajs-2433 122 2 . . X iajs-2433 122 3 abass abass PROPN iajs-2433 122 4 , , PUNCT iajs-2433 122 5 h.h h.h PROPN iajs-2433 122 6 . . PROPN iajs-2433 122 7 ; ; PUNCT iajs-2433 122 8 dahham dahham PROPN iajs-2433 122 9 , , PUNCT iajs-2433 122 10 h. h. PROPN iajs-2433 122 11 a. a. NOUN iajs-2433 122 12 on on ADP iajs-2433 122 13 completely completely ADV iajs-2433 122 14 closed close VERB iajs-2433 122 15 ideal ideal ADJ iajs-2433 122 16 with with ADP iajs-2433 122 17 respect respect NOUN iajs-2433 122 18 to to ADP iajs-2433 122 19 an an DET iajs-2433 122 20 element element NOUN iajs-2433 122 21 of of ADP iajs-2433 122 22 a a DET iajs-2433 122 23 bh bh NOUN iajs-2433 122 24 - - NOUN iajs-2433 122 25 algebra algebra NOUN iajs-2433 122 26 . . PUNCT iajs-2433 123 1 journal journal PROPN iajs-2433 123 2 of of ADP iajs-2433 123 3 karbala karbala PROPN iajs-2433 123 4 university.2012 university.2012 PROPN iajs-2433 123 5 , , PUNCT iajs-2433 123 6 10 10 NUM iajs-2433 123 7 , , PUNCT iajs-2433 123 8 3 3 NUM iajs-2433 123 9 , , PUNCT iajs-2433 123 10 302 302 NUM iajs-2433 123 11 - - SYM iajs-2433 123 12 312 312 NUM iajs-2433 123 13 . . PUNCT