id sid tid token lemma pos cana-1091 1 1 communications communication NOUN cana-1091 1 2 on on ADP cana-1091 1 3 applied apply VERB cana-1091 1 4 nonlinear nonlinear ADJ cana-1091 1 5 analysis analysis NOUN cana-1091 1 6 issn issn NOUN cana-1091 1 7 : : PUNCT cana-1091 1 8 1074 1074 NUM cana-1091 1 9 - - PUNCT cana-1091 1 10 133x 133x NUM cana-1091 1 11 vol vol NOUN cana-1091 1 12 31 31 NUM cana-1091 1 13 no no NOUN cana-1091 1 14 . . PUNCT cana-1091 2 1 5s 5s NUM cana-1091 2 2 ( ( PUNCT cana-1091 2 3 2024 2024 NUM cana-1091 2 4 ) ) PUNCT cana-1091 2 5 562 562 NUM cana-1091 2 6 https://internationalpubls.com https://internationalpubls.com X cana-1091 2 7 on on ADP cana-1091 2 8 neutrosophic neutrosophic ADJ cana-1091 2 9 transitivity transitivity NOUN cana-1091 2 10 and and CCONJ cana-1091 2 11 absorbent absorbent ADJ cana-1091 2 12 filters filter NOUN cana-1091 2 13 of of ADP cana-1091 2 14 basic basic ADJ cana-1091 2 15 logic logic NOUN cana-1091 2 16 algebras algebras PROPN cana-1091 2 17 a. a. PROPN cana-1091 2 18 ibrahim1 ibrahim1 PROPN cana-1091 2 19 , , PUNCT cana-1091 2 20 s. s. PROPN cana-1091 2 21 karunya karunya PROPN cana-1091 2 22 helen helen PROPN cana-1091 2 23 gunaseeli2 gunaseeli2 PROPN cana-1091 3 1 1assistant 1assistant NUM cana-1091 3 2 professor professor NOUN cana-1091 3 3 , , PUNCT cana-1091 3 4 p.g p.g PROPN cana-1091 3 5 . . PROPN cana-1091 3 6 and and CCONJ cana-1091 3 7 research research PROPN cana-1091 3 8 department department PROPN cana-1091 3 9 of of ADP cana-1091 3 10 mathematics mathematics PROPN cana-1091 3 11 , , PUNCT cana-1091 3 12 h.h h.h PROPN cana-1091 3 13 . . PROPN cana-1091 4 1 the the DET cana-1091 4 2 rajah rajah NOUN cana-1091 4 3 ’s ’s PART cana-1091 4 4 college college NOUN cana-1091 4 5 , , PUNCT cana-1091 4 6 pudukkottai pudukkottai NOUN cana-1091 4 7 , , PUNCT cana-1091 4 8 affiliated affiliate VERB cana-1091 4 9 to to PART cana-1091 4 10 bharathidasan bharathidasan VERB cana-1091 4 11 university university NOUN cana-1091 4 12 , , PUNCT cana-1091 4 13 trichirappalli trichirappalli PROPN cana-1091 4 14 , , PUNCT cana-1091 4 15 tamilnadu tamilnadu PROPN cana-1091 4 16 , , PUNCT cana-1091 4 17 india india PROPN cana-1091 4 18 . . PUNCT cana-1091 4 19 email email NOUN cana-1091 4 20 : : PUNCT cana-1091 5 1 dribrahimaadhil@gmail.com dribrahimaadhil@gmail.com X cana-1091 5 2 2research 2research NUM cana-1091 5 3 scholar scholar NOUN cana-1091 5 4 , , PUNCT cana-1091 5 5 p.g p.g PROPN cana-1091 5 6 . . PROPN cana-1091 5 7 and and CCONJ cana-1091 5 8 research research PROPN cana-1091 5 9 department department PROPN cana-1091 5 10 of of ADP cana-1091 5 11 mathematics mathematics PROPN cana-1091 5 12 , , PUNCT cana-1091 5 13 h.h h.h PROPN cana-1091 5 14 . . PROPN cana-1091 6 1 the the DET cana-1091 6 2 rajah rajah NOUN cana-1091 6 3 ’s ’s PART cana-1091 6 4 college college NOUN cana-1091 6 5 , , PUNCT cana-1091 6 6 pudukkottai pudukkottai NOUN cana-1091 6 7 , , PUNCT cana-1091 6 8 affiliated affiliate VERB cana-1091 6 9 to to PART cana-1091 6 10 bharathidasan bharathidasan VERB cana-1091 6 11 university university NOUN cana-1091 6 12 , , PUNCT cana-1091 6 13 trichirappalli trichirappalli PROPN cana-1091 6 14 , , PUNCT cana-1091 6 15 tamilnadu tamilnadu PROPN cana-1091 6 16 , , PUNCT cana-1091 6 17 india india PROPN cana-1091 6 18 . . PUNCT cana-1091 6 19 email email NOUN cana-1091 6 20 : : PUNCT cana-1091 7 1 karjes821@gmail.com karjes821@gmail.com X cana-1091 7 2 article article NOUN cana-1091 7 3 history history NOUN cana-1091 7 4 : : PUNCT cana-1091 7 5 received receive VERB cana-1091 7 6 : : PUNCT cana-1091 7 7 16 16 NUM cana-1091 7 8 - - SYM cana-1091 7 9 05 05 NUM cana-1091 7 10 - - PUNCT cana-1091 7 11 2024 2024 NUM cana-1091 7 12 revised revise VERB cana-1091 7 13 : : PUNCT cana-1091 7 14 20 20 NUM cana-1091 7 15 - - SYM cana-1091 7 16 06 06 NUM cana-1091 7 17 - - PUNCT cana-1091 7 18 2024 2024 NUM cana-1091 7 19 accepted accept VERB cana-1091 7 20 : : PUNCT cana-1091 7 21 12 12 NUM cana-1091 7 22 - - PUNCT cana-1091 7 23 07 07 NUM cana-1091 7 24 - - PUNCT cana-1091 7 25 2024 2024 NUM cana-1091 7 26 abstract abstract NOUN cana-1091 7 27 : : PUNCT cana-1091 7 28 the the DET cana-1091 7 29 vital vital ADJ cana-1091 7 30 objective objective NOUN cana-1091 7 31 of of ADP cana-1091 7 32 this this DET cana-1091 7 33 article article NOUN cana-1091 7 34 is be AUX cana-1091 7 35 to to PART cana-1091 7 36 explore explore VERB cana-1091 7 37 the the DET cana-1091 7 38 neutrosophic neutrosophic ADJ cana-1091 7 39 nature nature NOUN cana-1091 7 40 of of ADP cana-1091 7 41 transitive transitive ADJ cana-1091 7 42 and and CCONJ cana-1091 7 43 absorbent absorbent ADJ cana-1091 7 44 filters filter NOUN cana-1091 7 45 in in ADP cana-1091 7 46 basic basic ADJ cana-1091 7 47 logic logic NOUN cana-1091 7 48 ( ( PUNCT cana-1091 7 49 bl bl NOUN cana-1091 7 50 ) ) PUNCT cana-1091 7 51 algebras algebras PROPN cana-1091 7 52 . . PUNCT cana-1091 8 1 we we PRON cana-1091 8 2 establish establish VERB cana-1091 8 3 the the DET cana-1091 8 4 notion notion NOUN cana-1091 8 5 of of ADP cana-1091 8 6 neutrosophic neutrosophic ADJ cana-1091 8 7 transitive transitive NOUN cana-1091 8 8 and and CCONJ cana-1091 8 9 absorbent absorbent ADJ cana-1091 8 10 filters filter NOUN cana-1091 8 11 in in ADP cana-1091 8 12 bl bl NOUN cana-1091 8 13 - - PUNCT cana-1091 8 14 algebras algebras PROPN cana-1091 8 15 with with ADP cana-1091 8 16 suitable suitable ADJ cana-1091 8 17 illustrations illustration NOUN cana-1091 8 18 and and CCONJ cana-1091 8 19 examine examine VERB cana-1091 8 20 a a DET cana-1091 8 21 few few ADJ cana-1091 8 22 of of ADP cana-1091 8 23 their their PRON cana-1091 8 24 properties property NOUN cana-1091 8 25 . . PUNCT cana-1091 9 1 also also ADV cana-1091 9 2 , , PUNCT cana-1091 9 3 we we PRON cana-1091 9 4 prove prove VERB cana-1091 9 5 that that SCONJ cana-1091 9 6 every every DET cana-1091 9 7 neutrosophic neutrosophic ADJ cana-1091 9 8 transitive transitive ADJ cana-1091 9 9 filter filter NOUN cana-1091 9 10 in in ADP cana-1091 9 11 blalgebras blalgebra NOUN cana-1091 9 12 is be AUX cana-1091 9 13 a a DET cana-1091 9 14 neutrosophic neutrosophic ADJ cana-1091 9 15 filter filter NOUN cana-1091 9 16 . . PUNCT cana-1091 10 1 in in ADP cana-1091 10 2 addition addition NOUN cana-1091 10 3 , , PUNCT cana-1091 10 4 we we PRON cana-1091 10 5 confer confer VERB cana-1091 10 6 some some DET cana-1091 10 7 necessary necessary ADJ cana-1091 10 8 and and CCONJ cana-1091 10 9 sufficient sufficient ADJ cana-1091 10 10 conditions condition NOUN cana-1091 10 11 for for ADP cana-1091 10 12 a a DET cana-1091 10 13 neutrosophic neutrosophic ADJ cana-1091 10 14 filter filter NOUN cana-1091 10 15 to to PART cana-1091 10 16 be be AUX cana-1091 10 17 a a DET cana-1091 10 18 transitive transitive ADJ cana-1091 10 19 filter filter NOUN cana-1091 10 20 and and CCONJ cana-1091 10 21 an an DET cana-1091 10 22 extension extension NOUN cana-1091 10 23 property property NOUN cana-1091 10 24 . . PUNCT cana-1091 11 1 further far ADV cana-1091 11 2 , , PUNCT cana-1091 11 3 we we PRON cana-1091 11 4 obtain obtain VERB cana-1091 11 5 ( ( PUNCT cana-1091 11 6 i i NOUN cana-1091 11 7 ) ) PUNCT cana-1091 11 8 every every DET cana-1091 11 9 neutrosophic neutrosophic ADJ cana-1091 11 10 associative associative ADJ cana-1091 11 11 filter filter NOUN cana-1091 11 12 is be AUX cana-1091 11 13 an an DET cana-1091 11 14 absorbent absorbent ADJ cana-1091 11 15 filter filter NOUN cana-1091 11 16 . . PUNCT cana-1091 12 1 ( ( PUNCT cana-1091 12 2 ii ii X cana-1091 12 3 ) ) PUNCT cana-1091 12 4 𝐢 𝐢 PROPN cana-1091 12 5 is be AUX cana-1091 12 6 a a DET cana-1091 12 7 neutrosophic neutrosophic ADJ cana-1091 12 8 positive positive ADJ cana-1091 12 9 implicative implicative ADJ cana-1091 12 10 filter filter NOUN cana-1091 12 11 if if SCONJ cana-1091 13 1 and and CCONJ cana-1091 13 2 only only ADV cana-1091 13 3 if if SCONJ cana-1091 13 4 it it PRON cana-1091 13 5 is be AUX cana-1091 13 6 a a DET cana-1091 13 7 neutrosophic neutrosophic ADJ cana-1091 13 8 absorbent absorbent ADJ cana-1091 13 9 filter filter NOUN cana-1091 13 10 . . PUNCT cana-1091 14 1 ( ( PUNCT cana-1091 14 2 iii iii X cana-1091 14 3 ) ) PUNCT cana-1091 14 4 if if SCONJ cana-1091 14 5 𝐢 𝐢 PROPN cana-1091 14 6 is be AUX cana-1091 14 7 a a DET cana-1091 14 8 neutrosophic neutrosophic ADJ cana-1091 14 9 absorbent absorbent ADJ cana-1091 14 10 filter filter NOUN cana-1091 14 11 , , PUNCT cana-1091 14 12 then then ADV cana-1091 14 13 it it PRON cana-1091 14 14 is be AUX cana-1091 14 15 a a DET cana-1091 14 16 neutrosophic neutrosophic ADJ cana-1091 14 17 fantastic fantastic ADJ cana-1091 14 18 filter filter NOUN cana-1091 14 19 . . PUNCT cana-1091 15 1 in in ADP cana-1091 15 2 the the DET cana-1091 15 3 future future NOUN cana-1091 15 4 , , PUNCT cana-1091 15 5 the the DET cana-1091 15 6 above above ADJ cana-1091 15 7 research research NOUN cana-1091 15 8 can can AUX cana-1091 15 9 be be AUX cana-1091 15 10 extended extend VERB cana-1091 15 11 to to ADP cana-1091 15 12 deductive deductive ADJ cana-1091 15 13 filters filter NOUN cana-1091 15 14 . . PUNCT cana-1091 16 1 moreover moreover ADV cana-1091 16 2 , , PUNCT cana-1091 16 3 these these DET cana-1091 16 4 filters filter NOUN cana-1091 16 5 can can AUX cana-1091 16 6 be be AUX cana-1091 16 7 applied apply VERB cana-1091 16 8 in in ADP cana-1091 16 9 fields field NOUN cana-1091 16 10 such such ADJ cana-1091 16 11 as as ADP cana-1091 16 12 information information NOUN cana-1091 16 13 technology technology NOUN cana-1091 16 14 and and CCONJ cana-1091 16 15 systems system NOUN cana-1091 16 16 . . PUNCT cana-1091 17 1 keywords keyword NOUN cana-1091 17 2 : : PUNCT cana-1091 17 3 bl bl NOUN cana-1091 17 4 - - PUNCT cana-1091 17 5 algebra algebra NOUN cana-1091 17 6 ; ; PUNCT cana-1091 17 7 filter filter NOUN cana-1091 17 8 ; ; PUNCT cana-1091 17 9 neutrosophic neutrosophic ADJ cana-1091 17 10 filter filter NOUN cana-1091 17 11 ; ; PUNCT cana-1091 17 12 neutrosophic neutrosophic ADJ cana-1091 17 13 transitive transitive ADJ cana-1091 17 14 filter filter NOUN cana-1091 17 15 ; ; PUNCT cana-1091 17 16 neutrosophic neutrosophic ADJ cana-1091 17 17 absorbent absorbent ADJ cana-1091 17 18 filter filter NOUN cana-1091 17 19 . . PUNCT cana-1091 18 1 1 1 X cana-1091 18 2 . . X cana-1091 18 3 introduction introduction NOUN cana-1091 18 4 the the DET cana-1091 18 5 greek greek ADJ cana-1091 18 6 term term NOUN cana-1091 18 7 for for ADP cana-1091 18 8 knowledge knowledge NOUN cana-1091 18 9 of of ADP cana-1091 18 10 neutral neutral ADJ cana-1091 18 11 thought thought NOUN cana-1091 18 12 is be AUX cana-1091 18 13 neutrosophy neutrosophy NOUN cana-1091 18 14 . . PUNCT cana-1091 19 1 it it PRON cana-1091 19 2 is be AUX cana-1091 19 3 predicated predicate VERB cana-1091 19 4 on on ADP cana-1091 19 5 an an DET cana-1091 19 6 examination examination NOUN cana-1091 19 7 of of ADP cana-1091 19 8 both both DET cana-1091 19 9 opposing oppose VERB cana-1091 19 10 arguments argument NOUN cana-1091 19 11 and and CCONJ cana-1091 19 12 the the DET cana-1091 19 13 neutralities neutrality NOUN cana-1091 19 14 that that PRON cana-1091 19 15 exist exist VERB cana-1091 19 16 between between ADP cana-1091 19 17 them they PRON cana-1091 19 18 . . PUNCT cana-1091 20 1 since since SCONJ cana-1091 20 2 there there PRON cana-1091 20 3 is be VERB cana-1091 20 4 uncertainty uncertainty NOUN cana-1091 20 5 in in ADP cana-1091 20 6 everything everything PRON cana-1091 20 7 in in ADP cana-1091 20 8 the the DET cana-1091 20 9 world world NOUN cana-1091 20 10 , , PUNCT cana-1091 20 11 the the DET cana-1091 20 12 neutrosophic neutrosophic ADJ cana-1091 20 13 has have AUX cana-1091 20 14 emerged emerge VERB cana-1091 20 15 and and CCONJ cana-1091 20 16 found find VERB cana-1091 20 17 a a DET cana-1091 20 18 home home NOUN cana-1091 20 19 in in ADP cana-1091 20 20 science science NOUN cana-1091 20 21 . . PUNCT cana-1091 21 1 in in ADP cana-1091 21 2 order order NOUN cana-1091 21 3 to to PART cana-1091 21 4 investigate investigate VERB cana-1091 21 5 , , PUNCT cana-1091 21 6 from from ADP cana-1091 21 7 a a DET cana-1091 21 8 semantic semantic ADJ cana-1091 21 9 perspective perspective NOUN cana-1091 21 10 , , PUNCT cana-1091 21 11 the the DET cana-1091 21 12 logical logical ADJ cana-1091 21 13 system system NOUN cana-1091 21 14 whose whose DET cana-1091 21 15 propositional propositional ADJ cana-1091 21 16 value value NOUN cana-1091 21 17 is be AUX cana-1091 21 18 given give VERB cana-1091 21 19 in in ADP cana-1091 21 20 a a DET cana-1091 21 21 lattice lattice NOUN cana-1091 21 22 . . PUNCT cana-1091 22 1 lattice lattice PROPN cana-1091 22 2 implication implication NOUN cana-1091 22 3 algebras algebra NOUN cana-1091 22 4 were be AUX cana-1091 22 5 introduced introduce VERB cana-1091 22 6 and and CCONJ cana-1091 22 7 some some PRON cana-1091 22 8 of of ADP cana-1091 22 9 their their PRON cana-1091 22 10 features feature NOUN cana-1091 22 11 were be AUX cana-1091 22 12 addressed address VERB cana-1091 22 13 by by ADP cana-1091 22 14 y. y. PROPN cana-1091 22 15 xu xu PROPN cana-1091 23 1 [ [ X cana-1091 23 2 1 1 NUM cana-1091 23 3 ] ] PUNCT cana-1091 23 4 . . PUNCT cana-1091 24 1 the the DET cana-1091 24 2 idea idea NOUN cana-1091 24 3 of of ADP cana-1091 24 4 filters filter NOUN cana-1091 24 5 was be AUX cana-1091 24 6 first first ADV cana-1091 24 7 presented present VERB cana-1091 24 8 by by ADP cana-1091 24 9 y. y. PROPN cana-1091 24 10 xu xu PROPN cana-1091 24 11 and and CCONJ cana-1091 24 12 k.y k.y PROPN cana-1091 24 13 . . PROPN cana-1091 24 14 qin qin PROPN cana-1091 25 1 [ [ X cana-1091 25 2 2 2 NUM cana-1091 25 3 ] ] PUNCT cana-1091 25 4 . . PUNCT cana-1091 26 1 the the DET cana-1091 26 2 concepts concept NOUN cana-1091 26 3 of of ADP cana-1091 26 4 transitive transitive ADJ cana-1091 26 5 and and CCONJ cana-1091 26 6 absorbent absorbent ADJ cana-1091 26 7 filters filter NOUN cana-1091 26 8 were be AUX cana-1091 26 9 established establish VERB cana-1091 26 10 and and CCONJ cana-1091 26 11 their their PRON cana-1091 26 12 features feature NOUN cana-1091 26 13 were be AUX cana-1091 26 14 examined examine VERB cana-1091 26 15 by by ADP cana-1091 26 16 m. m. NOUN cana-1091 26 17 sambasiva sambasiva PROPN cana-1091 26 18 rao rao PROPN cana-1091 27 1 [ [ X cana-1091 27 2 3 3 X cana-1091 27 3 ] ] PUNCT cana-1091 27 4 in in ADP cana-1091 27 5 lattice lattice PROPN cana-1091 27 6 implication implication NOUN cana-1091 27 7 algebras algebra NOUN cana-1091 27 8 . . PUNCT cana-1091 28 1 the the DET cana-1091 28 2 authors author NOUN cana-1091 28 3 were be AUX cana-1091 28 4 inspired inspire VERB cana-1091 28 5 to to PART cana-1091 28 6 investigate investigate VERB cana-1091 28 7 this this DET cana-1091 28 8 idea idea NOUN cana-1091 28 9 in in ADP cana-1091 28 10 basic basic ADJ cana-1091 28 11 logic logic NOUN cana-1091 28 12 ( ( PUNCT cana-1091 28 13 bl)algebras bl)algebras PROPN cana-1091 28 14 by by ADP cana-1091 28 15 this this PRON cana-1091 28 16 . . PUNCT cana-1091 29 1 recently recently ADV cana-1091 29 2 , , PUNCT cana-1091 29 3 the the DET cana-1091 29 4 authors author NOUN cana-1091 29 5 studied study VERB cana-1091 29 6 the the DET cana-1091 29 7 neutrosophication neutrosophication NOUN cana-1091 29 8 of of ADP cana-1091 29 9 filters filter NOUN cana-1091 29 10 of of ADP cana-1091 29 11 bl bl NOUN cana-1091 29 12 - - PUNCT cana-1091 29 13 algebras algebras X cana-1091 30 1 [ [ X cana-1091 30 2 4 4 NUM cana-1091 30 3 ] ] PUNCT cana-1091 30 4 . . PUNCT cana-1091 31 1 they they PRON cana-1091 31 2 then then ADV cana-1091 31 3 developed develop VERB cana-1091 31 4 the the DET cana-1091 31 5 concept concept NOUN cana-1091 31 6 to to ADP cana-1091 31 7 fantastic fantastic ADJ cana-1091 31 8 , , PUNCT cana-1091 31 9 positive positive ADJ cana-1091 31 10 implicative implicative ADJ cana-1091 31 11 and and CCONJ cana-1091 31 12 associative associative ADJ cana-1091 31 13 filters filter NOUN cana-1091 31 14 [ [ X cana-1091 31 15 5 5 NUM cana-1091 31 16 , , PUNCT cana-1091 31 17 6 6 NUM cana-1091 31 18 ] ] PUNCT cana-1091 31 19 . . PUNCT cana-1091 32 1 our our PRON cana-1091 32 2 major major ADJ cana-1091 32 3 contributions contribution NOUN cana-1091 32 4 : : PUNCT cana-1091 32 5 ➒ ➒ VERB cana-1091 32 6 the the DET cana-1091 32 7 ideas idea NOUN cana-1091 32 8 of of ADP cana-1091 32 9 neutrosophic neutrosophic ADJ cana-1091 32 10 transitive transitive NOUN cana-1091 32 11 and and CCONJ cana-1091 32 12 absorbent absorbent ADJ cana-1091 32 13 filters filter NOUN cana-1091 32 14 are be AUX cana-1091 32 15 applied apply VERB cana-1091 32 16 in in ADP cana-1091 32 17 bl bl NOUN cana-1091 32 18 - - PUNCT cana-1091 32 19 algebras algebras PROPN cana-1091 32 20 . . PUNCT cana-1091 33 1 we we PRON cana-1091 33 2 obtain obtain VERB cana-1091 33 3 a a DET cana-1091 33 4 few few ADJ cana-1091 33 5 equivalent equivalent ADJ cana-1091 33 6 requirements requirement NOUN cana-1091 33 7 for for ADP cana-1091 33 8 a a DET cana-1091 33 9 neutrosophic neutrosophic ADJ cana-1091 33 10 filter filter NOUN cana-1091 33 11 to to PART cana-1091 33 12 be be AUX cana-1091 33 13 neutrosophic neutrosophic ADJ cana-1091 33 14 transitive transitive ADJ cana-1091 33 15 and and CCONJ cana-1091 33 16 absorbent absorbent NOUN cana-1091 33 17 . . PUNCT cana-1091 34 1 finally finally ADV cana-1091 34 2 , , PUNCT cana-1091 34 3 we we PRON cana-1091 34 4 establish establish VERB cana-1091 34 5 the the DET cana-1091 34 6 relationship relationship NOUN cana-1091 34 7 among among ADP cana-1091 34 8 the the DET cana-1091 34 9 neutrosophic neutrosophic ADJ cana-1091 34 10 transitive transitive NOUN cana-1091 34 11 , , PUNCT cana-1091 34 12 absorbent absorbent ADJ cana-1091 34 13 and and CCONJ cana-1091 34 14 associative associative ADJ cana-1091 34 15 filters filter NOUN cana-1091 34 16 in in ADP cana-1091 34 17 bl bl NOUN cana-1091 34 18 - - PUNCT cana-1091 34 19 algebras algebras PROPN cana-1091 34 20 . . PUNCT cana-1091 35 1 mailto:karjes821@gmail.com mailto:karjes821@gmail.com NOUN cana-1091 35 2 communications communication NOUN cana-1091 35 3 on on ADP cana-1091 35 4 applied apply VERB cana-1091 35 5 nonlinear nonlinear ADJ cana-1091 35 6 analysis analysis NOUN cana-1091 35 7 issn issn NOUN cana-1091 35 8 : : PUNCT cana-1091 35 9 1074 1074 NUM cana-1091 35 10 - - PUNCT cana-1091 35 11 133x 133x NUM cana-1091 35 12 vol vol NOUN cana-1091 35 13 31 31 NUM cana-1091 35 14 no no NOUN cana-1091 35 15 . . PUNCT cana-1091 36 1 5s 5s NUM cana-1091 36 2 ( ( PUNCT cana-1091 36 3 2024 2024 NUM cana-1091 36 4 ) ) PUNCT cana-1091 36 5 563 563 NUM cana-1091 36 6 https://internationalpubls.com https://internationalpubls.com X cana-1091 36 7 2 2 NUM cana-1091 36 8 . . PUNCT cana-1091 36 9 preliminaries preliminary NOUN cana-1091 36 10 in in ADP cana-1091 36 11 this this DET cana-1091 36 12 part part NOUN cana-1091 36 13 , , PUNCT cana-1091 36 14 few few ADJ cana-1091 36 15 of of ADP cana-1091 36 16 the the DET cana-1091 36 17 definitions definition NOUN cana-1091 36 18 and and CCONJ cana-1091 36 19 findings finding NOUN cana-1091 36 20 from from ADP cana-1091 36 21 the the DET cana-1091 36 22 literature literature NOUN cana-1091 36 23 are be AUX cana-1091 36 24 referred refer VERB cana-1091 36 25 to to PART cana-1091 36 26 progress progress VERB cana-1091 36 27 the the DET cana-1091 36 28 major major ADJ cana-1091 36 29 conclusions conclusion NOUN cana-1091 36 30 . . PUNCT cana-1091 37 1 definition definition NOUN cana-1091 37 2 2.1[7,8 2.1[7,8 NUM cana-1091 37 3 ] ] X cana-1091 37 4 a a DET cana-1091 37 5 bl bl NOUN cana-1091 37 6 - - PUNCT cana-1091 37 7 algebra algebra NOUN cana-1091 37 8 ( ( PUNCT cana-1091 37 9 𝒒 𝒒 PROPN cana-1091 37 10 , , PUNCT cana-1091 37 11 ∨ ∨ NOUN cana-1091 37 12 , , PUNCT cana-1091 37 13 ∧ ∧ PROPN cana-1091 37 14 , , PUNCT cana-1091 37 15 ∘ ∘ ADJ cana-1091 37 16 , , PUNCT cana-1091 37 17 β†’ β†’ SYM cana-1091 37 18 , , PUNCT cana-1091 37 19 0,1 0,1 NUM cana-1091 37 20 ) ) PUNCT cana-1091 37 21 of of ADP cana-1091 37 22 type type NOUN cana-1091 37 23 ( ( PUNCT cana-1091 37 24 2 2 NUM cana-1091 37 25 , , PUNCT cana-1091 37 26 2 2 NUM cana-1091 37 27 , , PUNCT cana-1091 37 28 2 2 NUM cana-1091 37 29 , , PUNCT cana-1091 37 30 2 2 NUM cana-1091 37 31 , , PUNCT cana-1091 37 32 0 0 NUM cana-1091 37 33 , , PUNCT cana-1091 37 34 0 0 NUM cana-1091 37 35 ) ) PUNCT cana-1091 37 36 such such ADJ cana-1091 37 37 that that SCONJ cana-1091 37 38 the the DET cana-1091 37 39 subsequent subsequent ADJ cana-1091 37 40 requirements requirement NOUN cana-1091 37 41 are be AUX cana-1091 37 42 persuaded persuade VERB cana-1091 37 43 for for ADP cana-1091 37 44 all all DET cana-1091 37 45 𝛼5 𝛼5 NOUN cana-1091 37 46 , , PUNCT cana-1091 37 47 𝛽5 𝛽5 NOUN cana-1091 37 48 , , PUNCT cana-1091 37 49 𝛾5 𝛾5 PROPN cana-1091 37 50 ∈ ∈ PROPN cana-1091 37 51 𝒒 𝒒 PROPN cana-1091 37 52 , , PUNCT cana-1091 37 53 ( ( PUNCT cana-1091 37 54 i i NOUN cana-1091 37 55 ) ) PUNCT cana-1091 37 56 ( ( PUNCT cana-1091 37 57 𝒒 𝒒 PROPN cana-1091 37 58 , , PUNCT cana-1091 37 59 ∨ ∨ NOUN cana-1091 37 60 , , PUNCT cana-1091 37 61 ∧ ∧ PROPN cana-1091 37 62 , , PUNCT cana-1091 37 63 0,1 0,1 NUM cana-1091 37 64 ) ) PUNCT cana-1091 37 65 is be AUX cana-1091 37 66 a a DET cana-1091 37 67 bounded bounded ADJ cana-1091 37 68 lattice lattice NOUN cana-1091 37 69 , , PUNCT cana-1091 37 70 ( ( PUNCT cana-1091 37 71 ii ii NOUN cana-1091 37 72 ) ) PUNCT cana-1091 37 73 ( ( PUNCT cana-1091 37 74 𝒒 𝒒 PROPN cana-1091 37 75 , , PUNCT cana-1091 37 76 ∘ ∘ ADJ cana-1091 37 77 , , PUNCT cana-1091 37 78 1 1 NUM cana-1091 37 79 ) ) PUNCT cana-1091 37 80 is be AUX cana-1091 37 81 a a DET cana-1091 37 82 commutative commutative ADJ cana-1091 37 83 monoid monoid NOUN cana-1091 37 84 , , PUNCT cana-1091 37 85 ( ( PUNCT cana-1091 37 86 iii iii NOUN cana-1091 37 87 ) ) PUNCT cana-1091 37 88 β€² β€² NUM cana-1091 37 89 ∘ ∘ NUM cana-1091 37 90 β€² β€² NUM cana-1091 37 91 , , PUNCT cana-1091 38 1 β€² β€² NUM cana-1091 38 2 β†’ β†’ SYM cana-1091 38 3 β€² β€² NUM cana-1091 38 4 is be AUX cana-1091 38 5 an an DET cana-1091 38 6 adjoint adjoint NOUN cana-1091 38 7 pair pair NOUN cana-1091 38 8 , , PUNCT cana-1091 38 9 that that ADV cana-1091 38 10 is is ADV cana-1091 38 11 , , PUNCT cana-1091 38 12 𝛾5 𝛾5 PROPN cana-1091 38 13 ≀ ≀ NOUN cana-1091 39 1 𝛼5β†’ 𝛼5β†’ X cana-1091 40 1 𝛽5if 𝛽5if PROPN cana-1091 41 1 and and CCONJ cana-1091 41 2 only only ADV cana-1091 41 3 if if SCONJ cana-1091 41 4 𝛼5 𝛼5 ADP cana-1091 41 5 ∘ ∘ NOUN cana-1091 41 6 𝛾5 𝛾5 PROPN cana-1091 41 7 ≀ ≀ NOUN cana-1091 41 8 𝛽5for 𝛽5for ADP cana-1091 41 9 all all DET cana-1091 41 10 𝛼5 𝛼5 NOUN cana-1091 41 11 , , PUNCT cana-1091 41 12 𝛽5 𝛽5 NOUN cana-1091 41 13 , , PUNCT cana-1091 41 14 𝛾5 𝛾5 PROPN cana-1091 41 15 ∈ ∈ PROPN cana-1091 41 16 𝒒 𝒒 PROPN cana-1091 41 17 , , PUNCT cana-1091 41 18 ( ( PUNCT cana-1091 41 19 iv iv X cana-1091 41 20 ) ) PUNCT cana-1091 41 21 𝛼5βˆ§π›½5= 𝛼5βˆ§π›½5= ADJ cana-1091 41 22 𝛼5 𝛼5 NOUN cana-1091 41 23 ∘ ∘ NOUN cana-1091 41 24 ( ( PUNCT cana-1091 41 25 𝛼5→𝛽5 𝛼5→𝛽5 NOUN cana-1091 41 26 ) ) PUNCT cana-1091 41 27 , , PUNCT cana-1091 41 28 ( ( PUNCT cana-1091 41 29 v v NOUN cana-1091 41 30 ) ) PUNCT cana-1091 41 31 ( ( PUNCT cana-1091 41 32 𝛼5→𝛽5 𝛼5→𝛽5 NOUN cana-1091 41 33 ) ) PUNCT cana-1091 41 34 ∨ ∨ NOUN cana-1091 41 35 ( ( PUNCT cana-1091 41 36 𝛽5→𝛼5 𝛽5→𝛼5 ADJ cana-1091 41 37 ) ) PUNCT cana-1091 41 38 = = SYM cana-1091 41 39 1 1 X cana-1091 41 40 . . X cana-1091 42 1 proposition proposition NOUN cana-1091 42 2 2.2[9,10]the 2.2[9,10]the PRON cana-1091 42 3 succeeding succeed VERB cana-1091 42 4 requirements requirement NOUN cana-1091 42 5 are be AUX cana-1091 42 6 persuaded persuade VERB cana-1091 42 7 in in ADP cana-1091 42 8 a a DET cana-1091 42 9 blalgebra blalgebra NOUN cana-1091 42 10 𝒒 𝒒 NOUN cana-1091 42 11 for for ADP cana-1091 42 12 all all DET cana-1091 42 13 𝛼5 𝛼5 NOUN cana-1091 42 14 , , PUNCT cana-1091 42 15 𝛽5 𝛽5 NOUN cana-1091 42 16 , , PUNCT cana-1091 42 17 𝛾5∈ 𝛾5∈ PROPN cana-1091 42 18 𝒒 𝒒 PROPN cana-1091 42 19 , , PUNCT cana-1091 42 20 ( ( PUNCT cana-1091 42 21 i i NOUN cana-1091 42 22 ) ) PUNCT cana-1091 42 23 𝛽5β†’ 𝛽5β†’ X cana-1091 42 24 ( ( PUNCT cana-1091 42 25 𝛼5→𝛾5 𝛼5→𝛾5 NOUN cana-1091 42 26 ) ) PUNCT cana-1091 42 27 = = SYM cana-1091 42 28 𝛼5β†’ 𝛼5β†’ X cana-1091 42 29 ( ( PUNCT cana-1091 42 30 𝛽5→𝛾5 𝛽5→𝛾5 PROPN cana-1091 42 31 ) ) PUNCT cana-1091 42 32 = = PUNCT cana-1091 42 33 ( ( PUNCT cana-1091 42 34 𝛼5 𝛼5 NOUN cana-1091 42 35 ∘ ∘ NOUN cana-1091 42 36 𝛽5 𝛽5 NOUN cana-1091 42 37 ) ) PUNCT cana-1091 42 38 →𝛾5 →𝛾5 X cana-1091 42 39 , , PUNCT cana-1091 42 40 ( ( PUNCT cana-1091 42 41 ii ii NOUN cana-1091 42 42 ) ) PUNCT cana-1091 42 43 1 1 NUM cana-1091 42 44 β†’ β†’ SYM cana-1091 42 45 𝛼5 𝛼5 NOUN cana-1091 42 46 = = NOUN cana-1091 42 47 𝛼5 𝛼5 NOUN cana-1091 42 48 , , PUNCT cana-1091 42 49 ( ( PUNCT cana-1091 42 50 iii iii NOUN cana-1091 42 51 ) ) PUNCT cana-1091 42 52 𝛼5≀ 𝛼5≀ NUM cana-1091 42 53 𝛽5 𝛽5 NOUN cana-1091 42 54 if if SCONJ cana-1091 42 55 and and CCONJ cana-1091 42 56 only only ADV cana-1091 42 57 if if SCONJ cana-1091 42 58 𝛼5 𝛼5 NOUN cana-1091 42 59 β†’ β†’ SYM cana-1091 42 60 𝛽5 𝛽5 NOUN cana-1091 42 61 = = SYM cana-1091 42 62 1 1 NUM cana-1091 42 63 , , PUNCT cana-1091 42 64 ( ( PUNCT cana-1091 42 65 iv iv X cana-1091 42 66 ) ) PUNCT cana-1091 42 67 𝛼5βˆ¨π›½5 𝛼5βˆ¨π›½5 NUM cana-1091 42 68 = = SYM cana-1091 42 69 ( ( PUNCT cana-1091 42 70 ( ( PUNCT cana-1091 42 71 𝛼5→𝛽5 𝛼5→𝛽5 NOUN cana-1091 42 72 ) ) PUNCT cana-1091 42 73 →𝛽5 →𝛽5 ADJ cana-1091 42 74 ) ) PUNCT cana-1091 42 75 ∧ ∧ PROPN cana-1091 42 76 ( ( PUNCT cana-1091 42 77 ( ( PUNCT cana-1091 42 78 𝛽5→𝛼5 𝛽5→𝛼5 ADJ cana-1091 42 79 ) ) PUNCT cana-1091 42 80 →𝛼5 →𝛼5 NUM cana-1091 42 81 ) ) PUNCT cana-1091 42 82 , , PUNCT cana-1091 42 83 ( ( PUNCT cana-1091 42 84 v v NOUN cana-1091 42 85 ) ) PUNCT cana-1091 42 86 𝛼5≀ 𝛼5≀ NUM cana-1091 42 87 𝛽5 𝛽5 NOUN cana-1091 42 88 implies imply VERB cana-1091 42 89 𝛽5 𝛽5 NOUN cana-1091 42 90 β†’ β†’ SYM cana-1091 42 91 𝛾5 𝛾5 ADJ cana-1091 42 92 ≀ ≀ ADJ cana-1091 42 93 𝛼5 𝛼5 NOUN cana-1091 42 94 →𝛾5 →𝛾5 NOUN cana-1091 42 95 , , PUNCT cana-1091 42 96 ( ( PUNCT cana-1091 42 97 vi vi NOUN cana-1091 42 98 ) ) PUNCT cana-1091 42 99 𝛼5≀ 𝛼5≀ NUM cana-1091 42 100 𝛽5 𝛽5 NOUN cana-1091 42 101 implies imply VERB cana-1091 42 102 𝛾5 𝛾5 NOUN cana-1091 42 103 β†’ β†’ SYM cana-1091 42 104 𝛼5≀ 𝛼5≀ NUM cana-1091 42 105 𝛾5 𝛾5 NOUN cana-1091 42 106 →𝛽5 →𝛽5 PROPN cana-1091 42 107 , , PUNCT cana-1091 42 108 ( ( PUNCT cana-1091 42 109 vii vii PROPN cana-1091 42 110 ) ) PUNCT cana-1091 42 111 𝛼5β†’ 𝛼5β†’ NUM cana-1091 42 112 𝛽5 𝛽5 ADJ cana-1091 42 113 ≀ ≀ NOUN cana-1091 42 114 ( ( PUNCT cana-1091 42 115 𝛾5→𝛼5 𝛾5→𝛼5 PROPN cana-1091 42 116 ) ) PUNCT cana-1091 42 117 β†’ β†’ PUNCT cana-1091 42 118 ( ( PUNCT cana-1091 42 119 𝛾5→𝛽5 𝛾5→𝛽5 NOUN cana-1091 42 120 ) ) PUNCT cana-1091 42 121 , , PUNCT cana-1091 42 122 ( ( PUNCT cana-1091 42 123 viii viii NOUN cana-1091 42 124 ) ) PUNCT cana-1091 42 125 𝛼5β†’ 𝛼5β†’ NOUN cana-1091 42 126 𝛽5≀ 𝛽5≀ NUM cana-1091 42 127 ( ( PUNCT cana-1091 42 128 𝛽5→𝛾5 𝛽5→𝛾5 PROPN cana-1091 42 129 ) ) PUNCT cana-1091 42 130 β†’ β†’ SYM cana-1091 42 131 ( ( PUNCT cana-1091 42 132 𝛼5→𝛾5 𝛼5→𝛾5 NOUN cana-1091 42 133 ) ) PUNCT cana-1091 42 134 , , PUNCT cana-1091 42 135 ( ( PUNCT cana-1091 42 136 ix ix ADP cana-1091 42 137 ) ) PUNCT cana-1091 42 138 𝛼5≀ 𝛼5≀ NUM cana-1091 42 139 ( ( PUNCT cana-1091 42 140 𝛼5→𝛽5 𝛼5→𝛽5 NOUN cana-1091 42 141 ) ) PUNCT cana-1091 42 142 →𝛽5 →𝛽5 NOUN cana-1091 42 143 , , PUNCT cana-1091 42 144 ( ( PUNCT cana-1091 42 145 x x X cana-1091 42 146 ) ) PUNCT cana-1091 42 147 𝛼5 𝛼5 NOUN cana-1091 42 148 ∘ ∘ NOUN cana-1091 42 149 ( ( PUNCT cana-1091 42 150 𝛼5→𝛽5 𝛼5→𝛽5 NOUN cana-1091 42 151 ) ) PUNCT cana-1091 43 1 = = SYM cana-1091 43 2 𝛼5βˆ§π›½5 𝛼5βˆ§π›½5 PROPN cana-1091 43 3 , , PUNCT cana-1091 43 4 ( ( PUNCT cana-1091 43 5 xi xi NOUN cana-1091 43 6 ) ) PUNCT cana-1091 43 7 𝛼5 𝛼5 NOUN cana-1091 43 8 ∘ ∘ NOUN cana-1091 43 9 𝛽5 𝛽5 NOUN cana-1091 43 10 ≀ ≀ ADJ cana-1091 43 11 𝛼5βˆ§π›½5 𝛼5βˆ§π›½5 PROPN cana-1091 43 12 ( ( PUNCT cana-1091 43 13 xii xii NOUN cana-1091 43 14 ) ) PUNCT cana-1091 43 15 𝛼5β†’ 𝛼5β†’ NUM cana-1091 43 16 𝛽5 𝛽5 ADJ cana-1091 43 17 ≀ ≀ NOUN cana-1091 43 18 ( ( PUNCT cana-1091 43 19 𝛼5 𝛼5 NOUN cana-1091 43 20 ∘ ∘ NOUN cana-1091 43 21 𝛾5 𝛾5 NOUN cana-1091 43 22 ) ) PUNCT cana-1091 43 23 β†’ β†’ PUNCT cana-1091 43 24 ( ( PUNCT cana-1091 43 25 𝛽5 𝛽5 VERB cana-1091 43 26 ∘ ∘ NOUN cana-1091 43 27 𝛾5 𝛾5 NOUN cana-1091 43 28 ) ) PUNCT cana-1091 43 29 , , PUNCT cana-1091 43 30 ( ( PUNCT cana-1091 43 31 xiii xiii X cana-1091 43 32 ) ) PUNCT cana-1091 43 33 𝛼5 𝛼5 NOUN cana-1091 43 34 ∘ ∘ NOUN cana-1091 43 35 ( ( PUNCT cana-1091 43 36 𝛽5→𝛾5 𝛽5→𝛾5 PROPN cana-1091 43 37 ) ) PUNCT cana-1091 43 38 ≀ ≀ NOUN cana-1091 43 39 𝛽5β†’ 𝛽5β†’ NOUN cana-1091 43 40 ( ( PUNCT cana-1091 43 41 𝛼5 𝛼5 ADP cana-1091 43 42 ∘ ∘ NOUN cana-1091 43 43 𝛾5 𝛾5 NOUN cana-1091 43 44 ) ) PUNCT cana-1091 43 45 , , PUNCT cana-1091 43 46 ( ( PUNCT cana-1091 43 47 xiv xiv PROPN cana-1091 43 48 ) ) PUNCT cana-1091 43 49 ( ( PUNCT cana-1091 43 50 𝛼5→𝛽5 𝛼5→𝛽5 NOUN cana-1091 43 51 ) ) PUNCT cana-1091 43 52 ∘ ∘ NOUN cana-1091 43 53 ( ( PUNCT cana-1091 43 54 𝛽5→𝛾5 𝛽5→𝛾5 PROPN cana-1091 43 55 ) ) PUNCT cana-1091 43 56 ≀ ≀ NOUN cana-1091 43 57 𝛼5 𝛼5 NOUN cana-1091 43 58 →𝛾5 →𝛾5 NOUN cana-1091 43 59 , , PUNCT cana-1091 43 60 ( ( PUNCT cana-1091 43 61 xv xv PROPN cana-1091 43 62 ) ) PUNCT cana-1091 43 63 ( ( PUNCT cana-1091 43 64 𝛼5 𝛼5 ADP cana-1091 43 65 ∘ ∘ NOUN cana-1091 43 66 𝛼5 𝛼5 NOUN cana-1091 43 67 βˆ— βˆ— NOUN cana-1091 43 68 ) ) PUNCT cana-1091 43 69 = = SYM cana-1091 44 1 0 0 X cana-1091 44 2 . . PUNCT cana-1091 44 3 definition definition NOUN cana-1091 44 4 2.3[11,12 2.3[11,12 NUM cana-1091 44 5 ] ] PUNCT cana-1091 44 6 a a DET cana-1091 44 7 neutrosophic neutrosophic ADJ cana-1091 44 8 subset subset NOUN cana-1091 44 9 𝐢 𝐢 PROPN cana-1091 44 10 of of ADP cana-1091 44 11 the the DET cana-1091 44 12 universe universe NOUN cana-1091 44 13 π‘ˆ π‘ˆ PROPN cana-1091 44 14 is be AUX cana-1091 44 15 a a DET cana-1091 44 16 triple triple ADJ cana-1091 44 17 ( ( PUNCT cana-1091 44 18 𝑇𝐢 𝑇𝐢 ADJ cana-1091 44 19 , , PUNCT cana-1091 44 20 𝐼𝐢 𝐼𝐢 PROPN cana-1091 44 21 , , PUNCT cana-1091 44 22 𝐹𝐢 𝐹𝐢 PROPN cana-1091 44 23 ) ) PUNCT cana-1091 44 24 where where SCONJ cana-1091 44 25 𝑇𝐢 𝑇𝐢 ADJ cana-1091 44 26 : : PUNCT cana-1091 44 27 π‘ˆβ†’[0,1 π‘ˆβ†’[0,1 NOUN cana-1091 44 28 ] ] PUNCT cana-1091 44 29 , , PUNCT cana-1091 44 30 𝐼𝐢 𝐼𝐢 PROPN cana-1091 44 31 : : PUNCT cana-1091 44 32 π‘ˆβ†’[0,1]and π‘ˆβ†’[0,1]and PUNCT cana-1091 45 1 𝐹𝐢 𝐹𝐢 NUM cana-1091 45 2 : : PUNCT cana-1091 45 3 π‘ˆ π‘ˆ PROPN cana-1091 45 4 β†’ β†’ SYM cana-1091 46 1 [ [ X cana-1091 46 2 0,1 0,1 NUM cana-1091 46 3 ] ] PUNCT cana-1091 46 4 represents represent VERB cana-1091 46 5 truth truth NOUN cana-1091 46 6 membership membership NOUN cana-1091 46 7 , , PUNCT cana-1091 46 8 indeterminacy indeterminacy NOUN cana-1091 46 9 and and CCONJ cana-1091 46 10 false false ADJ cana-1091 46 11 membership membership NOUN cana-1091 46 12 functions function NOUN cana-1091 46 13 respectively respectively ADV cana-1091 46 14 where where SCONJ cana-1091 46 15 0 0 NUM cana-1091 46 16 ≀ ≀ NUM cana-1091 46 17 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 PUNCT cana-1091 46 18 ) ) PUNCT cana-1091 46 19 + + CCONJ cana-1091 47 1 𝐼𝐢(𝛼5 𝐼𝐢(𝛼5 ADP cana-1091 47 2 ) ) PUNCT cana-1091 47 3 + + CCONJ cana-1091 47 4 𝐹𝐢(𝛼5 𝐹𝐢(𝛼5 X cana-1091 47 5 ) ) PUNCT cana-1091 47 6 ≀ ≀ NOUN cana-1091 47 7 3 3 NUM cana-1091 47 8 for for ADP cana-1091 47 9 all all DET cana-1091 47 10 𝛼5 𝛼5 NOUN cana-1091 47 11 ∈ ∈ NOUN cana-1091 47 12 π‘ˆ. π‘ˆ. ADJ cana-1091 47 13 definition definition NOUN cana-1091 47 14 2.4[4 2.4[4 NUM cana-1091 47 15 ] ] X cana-1091 47 16 a a DET cana-1091 47 17 neutrosophic neutrosophic ADJ cana-1091 47 18 set set NOUN cana-1091 47 19 𝐢 𝐢 PROPN cana-1091 47 20 of of ADP cana-1091 47 21 an an DET cana-1091 47 22 algebra𝒒 algebra𝒒 PROPN cana-1091 47 23 is be AUX cana-1091 47 24 called call VERB cana-1091 47 25 a a DET cana-1091 47 26 neutrosophic neutrosophic ADJ cana-1091 47 27 filter filter NOUN cana-1091 47 28 , , PUNCT cana-1091 47 29 if if SCONJ cana-1091 47 30 it it PRON cana-1091 47 31 persuades persuade VERB cana-1091 47 32 the the DET cana-1091 47 33 requirements requirement NOUN cana-1091 47 34 : : PUNCT cana-1091 47 35 ( ( PUNCT cana-1091 47 36 i i NOUN cana-1091 47 37 ) ) PUNCT cana-1091 47 38 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 NOUN cana-1091 47 39 ) ) PUNCT cana-1091 47 40 ≀ ≀ NOUN cana-1091 47 41 𝑇𝐢(1 𝑇𝐢(1 NOUN cana-1091 47 42 ) ) PUNCT cana-1091 47 43 , , PUNCT cana-1091 47 44 𝐼𝐢 𝐼𝐢 PROPN cana-1091 47 45 ( ( PUNCT cana-1091 47 46 𝛼5 𝛼5 NOUN cana-1091 47 47 ) ) PUNCT cana-1091 47 48 β‰₯ β‰₯ NOUN cana-1091 47 49 𝐼𝐢 𝐼𝐢 PROPN cana-1091 47 50 ( ( PUNCT cana-1091 47 51 1 1 NUM cana-1091 47 52 ) ) PUNCT cana-1091 47 53 and and CCONJ cana-1091 47 54 𝐹𝐢(𝛼5 𝐹𝐢(𝛼5 NUM cana-1091 47 55 ) ) PUNCT cana-1091 47 56 β‰₯ β‰₯ NOUN cana-1091 47 57 𝐹𝐢(1 𝐹𝐢(1 NUM cana-1091 47 58 ) ) PUNCT cana-1091 47 59 , , PUNCT cana-1091 47 60 ( ( PUNCT cana-1091 47 61 ii ii NOUN cana-1091 47 62 ) ) PUNCT cana-1091 47 63 min min NOUN cana-1091 47 64 { { PUNCT cana-1091 47 65 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 PROPN cana-1091 47 66 β†’ β†’ SYM cana-1091 47 67 𝛽5 𝛽5 PROPN cana-1091 47 68 ) ) PUNCT cana-1091 47 69 , , PUNCT cana-1091 47 70 𝑇𝐢(𝛼5)}≀ 𝑇𝐢(𝛼5)}≀ PROPN cana-1091 47 71 𝑇𝐢(𝛽5),min{𝐼𝐢 𝑇𝐢(𝛽5),min{𝐼𝐢 X cana-1091 47 72 ( ( PUNCT cana-1091 47 73 𝛼5 𝛼5 NOUN cana-1091 47 74 β†’ β†’ SYM cana-1091 47 75 𝛽5 𝛽5 NUM cana-1091 47 76 ) ) PUNCT cana-1091 47 77 , , PUNCT cana-1091 47 78 𝐼𝐢 𝐼𝐢 PROPN cana-1091 47 79 ( ( PUNCT cana-1091 47 80 𝛼5 𝛼5 NOUN cana-1091 47 81 ) ) PUNCT cana-1091 47 82 } } PUNCT cana-1091 47 83 β‰₯ β‰₯ AUX cana-1091 47 84 𝐼𝐢 𝐼𝐢 PROPN cana-1091 47 85 ( ( PUNCT cana-1091 47 86 𝛽5 𝛽5 NUM cana-1091 47 87 ) ) PUNCT cana-1091 47 88 and and CCONJ cana-1091 47 89 min min NOUN cana-1091 47 90 { { PUNCT cana-1091 47 91 𝐹𝐢 𝐹𝐢 PROPN cana-1091 47 92 ( ( PUNCT cana-1091 47 93 𝛼5 𝛼5 NOUN cana-1091 47 94 β†’ β†’ SYM cana-1091 47 95 𝛽5 𝛽5 NUM cana-1091 47 96 ) ) PUNCT cana-1091 47 97 , , PUNCT cana-1091 47 98 𝐹𝐢(𝛼5)}β‰₯ 𝐹𝐢(𝛼5)}β‰₯ NUM cana-1091 47 99 𝐹𝐢(𝛽5 𝐹𝐢(𝛽5 NUM cana-1091 47 100 ) ) PUNCT cana-1091 47 101 } } PUNCT cana-1091 47 102 for for ADP cana-1091 47 103 all𝛼5 all𝛼5 NOUN cana-1091 47 104 , , PUNCT cana-1091 47 105 𝛽5 𝛽5 VERB cana-1091 47 106 ∈ ∈ NOUN cana-1091 47 107 𝒒. 𝒒. NOUN cana-1091 47 108 proposition proposition NOUN cana-1091 47 109 2.5[4 2.5[4 NUM cana-1091 47 110 ] ] PUNCT cana-1091 47 111 let let VERB cana-1091 47 112 𝐢 𝐢 PROPN cana-1091 47 113 be be AUX cana-1091 47 114 a a DET cana-1091 47 115 neutrosophic neutrosophic ADJ cana-1091 47 116 filter filter NOUN cana-1091 47 117 of of ADP cana-1091 47 118 𝒒 𝒒 PROPN cana-1091 47 119 if if SCONJ cana-1091 48 1 and and CCONJ cana-1091 48 2 only only ADV cana-1091 48 3 if if SCONJ cana-1091 48 4 ( ( PUNCT cana-1091 48 5 i i NOUN cana-1091 48 6 ) ) PUNCT cana-1091 48 7 if if SCONJ cana-1091 48 8 𝛼5 𝛼5 VERB cana-1091 48 9 ≀ ≀ ADV cana-1091 48 10 𝛽5then 𝛽5then ADV cana-1091 48 11 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 PROPN cana-1091 48 12 ) ) PUNCT cana-1091 48 13 ≀ ≀ NOUN cana-1091 48 14 𝑇𝐢(𝛽5 𝑇𝐢(𝛽5 NUM cana-1091 48 15 ) ) PUNCT cana-1091 48 16 , , PUNCT cana-1091 48 17 𝐼𝐢(𝛼5 𝐼𝐢(𝛼5 PROPN cana-1091 48 18 ) ) PUNCT cana-1091 48 19 β‰₯ β‰₯ NOUN cana-1091 48 20 𝐼𝐢(𝛽5 𝐼𝐢(𝛽5 NUM cana-1091 48 21 ) ) PUNCT cana-1091 48 22 and and CCONJ cana-1091 48 23 𝐹𝐢(𝛼5 𝐹𝐢(𝛼5 NUM cana-1091 48 24 ) ) PUNCT cana-1091 48 25 β‰₯ β‰₯ NOUN cana-1091 48 26 𝐹𝐢(𝛽5 𝐹𝐢(𝛽5 NUM cana-1091 48 27 ) ) PUNCT cana-1091 48 28 , , PUNCT cana-1091 48 29 communications communication NOUN cana-1091 48 30 on on ADP cana-1091 48 31 applied apply VERB cana-1091 48 32 nonlinear nonlinear ADJ cana-1091 48 33 analysis analysis NOUN cana-1091 48 34 issn issn NOUN cana-1091 48 35 : : PUNCT cana-1091 48 36 1074 1074 NUM cana-1091 48 37 - - PUNCT cana-1091 48 38 133x 133x NUM cana-1091 48 39 vol vol NOUN cana-1091 48 40 31 31 NUM cana-1091 48 41 no no NOUN cana-1091 48 42 . . PUNCT cana-1091 49 1 5s 5s NUM cana-1091 49 2 ( ( PUNCT cana-1091 49 3 2024 2024 NUM cana-1091 49 4 ) ) PUNCT cana-1091 49 5 564 564 NUM cana-1091 49 6 https://internationalpubls.com https://internationalpubls.com X cana-1091 49 7 ( ( PUNCT cana-1091 49 8 ii ii NOUN cana-1091 49 9 ) ) PUNCT cana-1091 49 10 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 PROPN cana-1091 50 1 ∘ ∘ NUM cana-1091 50 2 𝛽5 𝛽5 ADJ cana-1091 50 3 ) ) PUNCT cana-1091 50 4 β‰₯ β‰₯ PROPN cana-1091 50 5 min min PROPN cana-1091 50 6 { { PUNCT cana-1091 50 7 𝑇𝐢(𝛼5),𝑇𝐢(𝛽5 𝑇𝐢(𝛼5),𝑇𝐢(𝛽5 PROPN cana-1091 50 8 ) ) PUNCT cana-1091 50 9 } } PUNCT cana-1091 50 10 , , PUNCT cana-1091 50 11 𝐼𝐢 𝐼𝐢 PROPN cana-1091 50 12 ( ( PUNCT cana-1091 50 13 𝛼5 𝛼5 NOUN cana-1091 50 14 ∘ ∘ NOUN cana-1091 50 15 𝛽5)≀ 𝛽5)≀ PROPN cana-1091 50 16 min min ADJ cana-1091 50 17 { { PUNCT cana-1091 50 18 𝐼𝐢 𝐼𝐢 PROPN cana-1091 50 19 ( ( PUNCT cana-1091 50 20 𝛼5 𝛼5 NOUN cana-1091 50 21 ) ) PUNCT cana-1091 50 22 , , PUNCT cana-1091 50 23 𝐼𝐢 𝐼𝐢 PROPN cana-1091 50 24 ( ( PUNCT cana-1091 50 25 𝛽5 𝛽5 NOUN cana-1091 50 26 ) ) PUNCT cana-1091 50 27 } } PUNCT cana-1091 50 28 and and CCONJ cana-1091 50 29 𝐹𝐢(𝛼5 𝐹𝐢(𝛼5 NUM cana-1091 50 30 ∘ ∘ NUM cana-1091 50 31 𝛽5 𝛽5 ADJ cana-1091 50 32 ) ) PUNCT cana-1091 50 33 ≀ ≀ NUM cana-1091 50 34 min min NOUN cana-1091 50 35 { { PUNCT cana-1091 50 36 𝐹𝐢(𝛼5 𝐹𝐢(𝛼5 PROPN cana-1091 50 37 ) ) PUNCT cana-1091 50 38 , , PUNCT cana-1091 50 39 𝐹𝐢(𝛽5 𝐹𝐢(𝛽5 NUM cana-1091 50 40 ) ) PUNCT cana-1091 50 41 } } PUNCT cana-1091 50 42 for for ADP cana-1091 50 43 all all DET cana-1091 50 44 𝛼5 𝛼5 NOUN cana-1091 50 45 , , PUNCT cana-1091 50 46 𝛽5 𝛽5 VERB cana-1091 50 47 ∈ ∈ NOUN cana-1091 50 48 𝒒. 𝒒. NOUN cana-1091 50 49 proposition proposition NOUN cana-1091 50 50 2.6[4 2.6[4 NUM cana-1091 50 51 , , PUNCT cana-1091 50 52 5]let 5]let NOUN cana-1091 50 53 𝐢 𝐢 PROPN cana-1091 50 54 be be VERB cana-1091 50 55 a a DET cana-1091 50 56 neutrosophicfilter neutrosophicfilter NOUN cana-1091 50 57 of of ADP cana-1091 50 58 𝒒 𝒒 PROPN cana-1091 50 59 for for ADP cana-1091 50 60 all𝛼5 all𝛼5 NOUN cana-1091 50 61 , , PUNCT cana-1091 50 62 𝛽5 𝛽5 ADJ cana-1091 50 63 , , PUNCT cana-1091 50 64 𝛾5 𝛾5 NOUN cana-1091 50 65 ∈ ∈ PROPN cana-1091 50 66 𝒒 𝒒 PROPN cana-1091 50 67 then then ADV cana-1091 50 68 the the DET cana-1091 50 69 following follow VERB cana-1091 50 70 hold hold NOUN cana-1091 50 71 . . PUNCT cana-1091 51 1 ( ( PUNCT cana-1091 51 2 i i NOUN cana-1091 51 3 ) ) PUNCT cana-1091 51 4 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 PROPN cana-1091 52 1 β†’ β†’ SYM cana-1091 52 2 𝛽5 𝛽5 NUM cana-1091 52 3 ) ) PUNCT cana-1091 52 4 = = SYM cana-1091 52 5 𝑇𝐢(1 𝑇𝐢(1 PROPN cana-1091 52 6 ) ) PUNCT cana-1091 52 7 , , PUNCT cana-1091 52 8 then then ADV cana-1091 52 9 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 X cana-1091 52 10 ) ) PUNCT cana-1091 52 11 ≀ ≀ NOUN cana-1091 52 12 𝑇𝐢(𝛽5 𝑇𝐢(𝛽5 NUM cana-1091 52 13 ) ) PUNCT cana-1091 52 14 , , PUNCT cana-1091 52 15 𝐼𝐢(𝛼5 𝐼𝐢(𝛼5 PROPN cana-1091 52 16 β†’ β†’ SYM cana-1091 52 17 𝛽5 𝛽5 NUM cana-1091 52 18 ) ) PUNCT cana-1091 52 19 = = SYM cana-1091 52 20 𝐼𝐢(1 𝐼𝐢(1 PROPN cana-1091 52 21 ) ) PUNCT cana-1091 52 22 , , PUNCT cana-1091 52 23 then then ADV cana-1091 52 24 𝐼𝐢(𝛼5 𝐼𝐢(𝛼5 ADP cana-1091 52 25 ) ) PUNCT cana-1091 52 26 β‰₯ β‰₯ NOUN cana-1091 52 27 𝐼𝐢(𝛽5 𝐼𝐢(𝛽5 NUM cana-1091 52 28 ) ) PUNCT cana-1091 52 29 , , PUNCT cana-1091 52 30 𝐹𝐢(𝛼5 𝐹𝐢(𝛼5 PROPN cana-1091 52 31 β†’ β†’ SYM cana-1091 52 32 𝛽5 𝛽5 NUM cana-1091 52 33 ) ) PUNCT cana-1091 52 34 = = SYM cana-1091 52 35 𝐹𝐢(1 𝐹𝐢(1 PROPN cana-1091 52 36 ) ) PUNCT cana-1091 52 37 , , PUNCT cana-1091 52 38 then then ADV cana-1091 52 39 𝐹𝐢(𝛼5 𝐹𝐢(𝛼5 NUM cana-1091 52 40 ) ) PUNCT cana-1091 52 41 β‰₯ β‰₯ NOUN cana-1091 52 42 𝐹𝐢(𝛽5 𝐹𝐢(𝛽5 NUM cana-1091 52 43 ) ) PUNCT cana-1091 52 44 ( ( PUNCT cana-1091 52 45 ii ii NOUN cana-1091 52 46 ) ) PUNCT cana-1091 52 47 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 PROPN cana-1091 53 1 ∧ ∧ NOUN cana-1091 53 2 𝛽5 𝛽5 PROPN cana-1091 53 3 ) ) PUNCT cana-1091 53 4 = = SYM cana-1091 53 5 min{𝑇𝐢(𝛼5 min{𝑇𝐢(𝛼5 NOUN cana-1091 53 6 ) ) PUNCT cana-1091 53 7 , , PUNCT cana-1091 53 8 𝑇𝐢(𝛽5 𝑇𝐢(𝛽5 NUM cana-1091 53 9 ) ) PUNCT cana-1091 53 10 } } PUNCT cana-1091 53 11 , , PUNCT cana-1091 53 12 𝐼𝐢(𝛼5 𝐼𝐢(𝛼5 ADP cana-1091 53 13 ∧ ∧ NOUN cana-1091 53 14 𝛽5 𝛽5 PROPN cana-1091 53 15 ) ) PUNCT cana-1091 53 16 = = SYM cana-1091 53 17 min{𝐼𝐢(𝛼5 min{𝐼𝐢(𝛼5 NOUN cana-1091 53 18 ) ) PUNCT cana-1091 53 19 , , PUNCT cana-1091 53 20 𝐼𝐢(𝛽5 𝐼𝐢(𝛽5 NUM cana-1091 53 21 ) ) PUNCT cana-1091 53 22 } } PUNCT cana-1091 53 23 , , PUNCT cana-1091 53 24 𝐹𝐢(𝛼5 𝐹𝐢(𝛼5 PROPN cana-1091 53 25 ∧ ∧ PROPN cana-1091 53 26 𝛽5 𝛽5 NOUN cana-1091 53 27 ) ) PUNCT cana-1091 53 28 = = NOUN cana-1091 53 29 min{𝐹𝐢(𝛼5 min{𝐹𝐢(𝛼5 NOUN cana-1091 53 30 ) ) PUNCT cana-1091 53 31 , , PUNCT cana-1091 53 32 𝐹𝐢(𝛽5 𝐹𝐢(𝛽5 NUM cana-1091 53 33 ) ) PUNCT cana-1091 53 34 } } PUNCT cana-1091 53 35 ( ( PUNCT cana-1091 53 36 iii iii X cana-1091 53 37 ) ) PUNCT cana-1091 53 38 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 PROPN cana-1091 53 39 ∘ ∘ NUM cana-1091 53 40 𝛽5 𝛽5 NOUN cana-1091 53 41 ) ) PUNCT cana-1091 53 42 = = SYM cana-1091 53 43 min{𝑇𝐢(𝛼5 min{𝑇𝐢(𝛼5 NOUN cana-1091 53 44 ) ) PUNCT cana-1091 53 45 , , PUNCT cana-1091 53 46 𝑇𝐢(𝛽5 𝑇𝐢(𝛽5 NUM cana-1091 53 47 ) ) PUNCT cana-1091 53 48 } } PUNCT cana-1091 53 49 , , PUNCT cana-1091 53 50 𝐼𝐢(𝛼5 𝐼𝐢(𝛼5 ADP cana-1091 53 51 ∘ ∘ NUM cana-1091 53 52 𝛽5 𝛽5 NOUN cana-1091 53 53 ) ) PUNCT cana-1091 53 54 = = SYM cana-1091 53 55 min{𝐼𝐢(𝛼5 min{𝐼𝐢(𝛼5 NOUN cana-1091 53 56 ) ) PUNCT cana-1091 53 57 , , PUNCT cana-1091 53 58 𝐼𝐢(𝛽5 𝐼𝐢(𝛽5 NUM cana-1091 53 59 ) ) PUNCT cana-1091 53 60 } } PUNCT cana-1091 53 61 , , PUNCT cana-1091 53 62 𝐹𝐢(𝛼5 𝐹𝐢(𝛼5 PROPN cana-1091 53 63 ∘ ∘ NOUN cana-1091 53 64 𝛽5 𝛽5 NOUN cana-1091 53 65 ) ) PUNCT cana-1091 53 66 = = NOUN cana-1091 53 67 min{𝐹𝐢(𝛼5 min{𝐹𝐢(𝛼5 NOUN cana-1091 53 68 ) ) PUNCT cana-1091 53 69 , , PUNCT cana-1091 53 70 𝐹𝐢(𝛽5 𝐹𝐢(𝛽5 NUM cana-1091 53 71 ) ) PUNCT cana-1091 53 72 } } PUNCT cana-1091 53 73 ( ( PUNCT cana-1091 53 74 iv iv X cana-1091 53 75 ) ) PUNCT cana-1091 53 76 𝑇𝐢(0 𝑇𝐢(0 PROPN cana-1091 53 77 ) ) PUNCT cana-1091 53 78 = = SYM cana-1091 53 79 min{𝑇𝐢(𝛼5 min{𝑇𝐢(𝛼5 NOUN cana-1091 53 80 ) ) PUNCT cana-1091 53 81 , , PUNCT cana-1091 53 82 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 PROPN cana-1091 53 83 βˆ— βˆ— NOUN cana-1091 53 84 ) ) PUNCT cana-1091 53 85 } } PUNCT cana-1091 53 86 , , PUNCT cana-1091 53 87 𝐼𝐢(0 𝐼𝐢(0 PROPN cana-1091 53 88 ) ) PUNCT cana-1091 53 89 = = SYM cana-1091 53 90 min{𝐼𝐢(𝛼5 min{𝐼𝐢(𝛼5 NOUN cana-1091 53 91 ) ) PUNCT cana-1091 53 92 , , PUNCT cana-1091 53 93 𝐼𝐢(𝛼5 𝐼𝐢(𝛼5 ADP cana-1091 53 94 βˆ— βˆ— NOUN cana-1091 53 95 ) ) PUNCT cana-1091 53 96 } } PUNCT cana-1091 53 97 , , PUNCT cana-1091 53 98 𝐹𝐢(0 𝐹𝐢(0 PROPN cana-1091 53 99 ) ) PUNCT cana-1091 53 100 = = SYM cana-1091 53 101 min{𝐹𝐢(𝛼5 min{𝐹𝐢(𝛼5 NOUN cana-1091 53 102 ) ) PUNCT cana-1091 53 103 , , PUNCT cana-1091 53 104 𝐹𝐢(𝛼5 𝐹𝐢(𝛼5 NUM cana-1091 53 105 βˆ— βˆ— NOUN cana-1091 53 106 ) ) PUNCT cana-1091 53 107 } } PUNCT cana-1091 53 108 definition definition NOUN cana-1091 53 109 2.7[5 2.7[5 NUM cana-1091 53 110 ] ] PUNCT cana-1091 53 111 let let VERB cana-1091 53 112 𝐢 𝐢 PROPN cana-1091 53 113 be be AUX cana-1091 53 114 called call VERB cana-1091 53 115 a a DET cana-1091 53 116 neutrosophic neutrosophic ADJ cana-1091 53 117 fantastic fantastic ADJ cana-1091 53 118 filter filter NOUN cana-1091 53 119 of of ADP cana-1091 53 120 𝒒 𝒒 PROPN cana-1091 53 121 , , PUNCT cana-1091 53 122 if if SCONJ cana-1091 53 123 it it PRON cana-1091 53 124 persuades persuade VERB cana-1091 53 125 the the DET cana-1091 53 126 subsequent subsequent ADJ cana-1091 53 127 requirements requirement NOUN cana-1091 53 128 for for ADP cana-1091 53 129 all all DET cana-1091 53 130 𝛼5 𝛼5 NOUN cana-1091 53 131 , , PUNCT cana-1091 53 132 𝛽5 𝛽5 NOUN cana-1091 53 133 , , PUNCT cana-1091 53 134 𝛾5 𝛾5 PROPN cana-1091 53 135 ∈ ∈ PROPN cana-1091 53 136 𝒒 𝒒 PROPN cana-1091 53 137 , , PUNCT cana-1091 53 138 ( ( PUNCT cana-1091 53 139 i i NOUN cana-1091 53 140 ) ) PUNCT cana-1091 54 1 𝑇𝐢(1 𝑇𝐢(1 PROPN cana-1091 54 2 ) ) PUNCT cana-1091 54 3 β‰₯ β‰₯ NOUN cana-1091 54 4 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 NOUN cana-1091 54 5 ) ) PUNCT cana-1091 54 6 , , PUNCT cana-1091 54 7 𝐼𝐢(1 𝐼𝐢(1 PROPN cana-1091 54 8 ) ) PUNCT cana-1091 54 9 ≀ ≀ NOUN cana-1091 55 1 𝐼𝐢(𝛼5 𝐼𝐢(𝛼5 ADP cana-1091 55 2 ) ) PUNCT cana-1091 55 3 , , PUNCT cana-1091 55 4 𝐹𝐢(1 𝐹𝐢(1 PROPN cana-1091 55 5 ) ) PUNCT cana-1091 55 6 ≀ ≀ NOUN cana-1091 56 1 𝐹𝐢(𝛼5 𝐹𝐢(𝛼5 NUM cana-1091 56 2 ) ) PUNCT cana-1091 56 3 . . PUNCT cana-1091 57 1 ( ( PUNCT cana-1091 57 2 ii ii NOUN cana-1091 57 3 ) ) PUNCT cana-1091 57 4 min{𝑇𝐢(𝛼5 min{𝑇𝐢(𝛼5 ADP cana-1091 57 5 β†’ β†’ SYM cana-1091 57 6 𝛽5 𝛽5 NUM cana-1091 57 7 ) ) PUNCT cana-1091 57 8 , , PUNCT cana-1091 57 9 𝑇𝐢(𝛼5)}≀ 𝑇𝐢(𝛼5)}≀ PROPN cana-1091 57 10 𝑇𝐢(𝛽5),min{𝐼𝐢 𝑇𝐢(𝛽5),min{𝐼𝐢 X cana-1091 57 11 ( ( PUNCT cana-1091 57 12 𝛼5 𝛼5 NOUN cana-1091 57 13 β†’ β†’ SYM cana-1091 57 14 𝛽5 𝛽5 NUM cana-1091 57 15 ) ) PUNCT cana-1091 57 16 , , PUNCT cana-1091 57 17 𝐼𝐢 𝐼𝐢 PROPN cana-1091 57 18 ( ( PUNCT cana-1091 57 19 𝛼5 𝛼5 NOUN cana-1091 57 20 ) ) PUNCT cana-1091 57 21 } } PUNCT cana-1091 57 22 β‰₯ β‰₯ AUX cana-1091 57 23 𝐼𝐢 𝐼𝐢 PROPN cana-1091 57 24 ( ( PUNCT cana-1091 57 25 𝛽5 𝛽5 NUM cana-1091 57 26 ) ) PUNCT cana-1091 57 27 and and CCONJ cana-1091 57 28 min min NOUN cana-1091 57 29 { { PUNCT cana-1091 57 30 𝐹𝐢 𝐹𝐢 PROPN cana-1091 57 31 ( ( PUNCT cana-1091 57 32 𝛼5 𝛼5 NOUN cana-1091 57 33 β†’ β†’ SYM cana-1091 57 34 𝛽5 𝛽5 NUM cana-1091 57 35 ) ) PUNCT cana-1091 57 36 , , PUNCT cana-1091 57 37 𝐹𝐢(𝛼5)}β‰₯ 𝐹𝐢(𝛼5)}β‰₯ NUM cana-1091 58 1 𝐹𝐢(𝛽5 𝐹𝐢(𝛽5 NUM cana-1091 58 2 ) ) PUNCT cana-1091 58 3 } } PUNCT cana-1091 58 4 . . PUNCT cana-1091 59 1 ( ( PUNCT cana-1091 59 2 iii iii X cana-1091 59 3 ) ) PUNCT cana-1091 59 4 𝑇𝐢((𝛼5 𝑇𝐢((𝛼5 NOUN cana-1091 59 5 β†’ β†’ SYM cana-1091 59 6 𝛽5 𝛽5 PROPN cana-1091 59 7 ) ) PUNCT cana-1091 59 8 β†’ β†’ SYM cana-1091 59 9 𝛽5 𝛽5 NUM cana-1091 59 10 ) ) PUNCT cana-1091 59 11 β†’ β†’ SYM cana-1091 59 12 𝛼5 𝛼5 NOUN cana-1091 59 13 ) ) PUNCT cana-1091 59 14 β‰₯ β‰₯ NOUN cana-1091 59 15 min{𝑇𝐢(𝛾5 min{𝑇𝐢(𝛾5 X cana-1091 60 1 β†’ β†’ PUNCT cana-1091 60 2 ( ( PUNCT cana-1091 60 3 𝛽5 𝛽5 ADJ cana-1091 60 4 β†’ β†’ SYM cana-1091 60 5 𝛼5 𝛼5 NOUN cana-1091 60 6 ) ) PUNCT cana-1091 60 7 ) ) PUNCT cana-1091 60 8 , , PUNCT cana-1091 60 9 𝑇𝐢(𝛾5 𝑇𝐢(𝛾5 PROPN cana-1091 60 10 ) ) PUNCT cana-1091 60 11 } } PUNCT cana-1091 60 12 , , PUNCT cana-1091 60 13 𝐼𝐢((𝛼5 𝐼𝐢((𝛼5 NOUN cana-1091 60 14 β†’ β†’ SYM cana-1091 60 15 𝛽5 𝛽5 ADJ cana-1091 60 16 ) ) PUNCT cana-1091 60 17 β†’ β†’ SYM cana-1091 60 18 𝛽5 𝛽5 NUM cana-1091 60 19 ) ) PUNCT cana-1091 60 20 β†’ β†’ SYM cana-1091 60 21 𝛼5 𝛼5 NOUN cana-1091 60 22 ) ) PUNCT cana-1091 60 23 ≀ ≀ NUM cana-1091 60 24 min min NOUN cana-1091 60 25 { { PUNCT cana-1091 60 26 𝐼𝐢(𝛾5 𝐼𝐢(𝛾5 X cana-1091 60 27 β†’ β†’ PUNCT cana-1091 60 28 ( ( PUNCT cana-1091 60 29 𝛽5 𝛽5 ADJ cana-1091 60 30 β†’ β†’ SYM cana-1091 60 31 𝛼5 𝛼5 NOUN cana-1091 60 32 ) ) PUNCT cana-1091 60 33 ) ) PUNCT cana-1091 60 34 , , PUNCT cana-1091 60 35 𝐼𝐢(𝛾5 𝐼𝐢(𝛾5 NOUN cana-1091 60 36 ) ) PUNCT cana-1091 60 37 } } PUNCT cana-1091 60 38 , , PUNCT cana-1091 60 39 𝐹𝐢 𝐹𝐢 PROPN cana-1091 60 40 ( ( PUNCT cana-1091 60 41 ( ( PUNCT cana-1091 60 42 𝛼5 𝛼5 NOUN cana-1091 60 43 β†’ β†’ SYM cana-1091 60 44 𝛽5 𝛽5 NUM cana-1091 60 45 ) ) PUNCT cana-1091 60 46 β†’ β†’ SYM cana-1091 60 47 𝛽5 𝛽5 NUM cana-1091 60 48 ) ) PUNCT cana-1091 60 49 β†’ β†’ SYM cana-1091 60 50 𝛼5 𝛼5 NOUN cana-1091 60 51 ) ) PUNCT cana-1091 60 52 ≀ ≀ NUM cana-1091 60 53 min min NOUN cana-1091 60 54 { { PUNCT cana-1091 60 55 𝐹𝐢(𝛾5 𝐹𝐢(𝛾5 PROPN cana-1091 60 56 β†’ β†’ SYM cana-1091 60 57 ( ( PUNCT cana-1091 60 58 𝛽5 𝛽5 ADJ cana-1091 60 59 β†’ β†’ SYM cana-1091 60 60 𝛼5 𝛼5 NOUN cana-1091 60 61 ) ) PUNCT cana-1091 60 62 ) ) PUNCT cana-1091 60 63 , , PUNCT cana-1091 60 64 𝐹𝐢(𝛾5 𝐹𝐢(𝛾5 NUM cana-1091 60 65 ) ) PUNCT cana-1091 60 66 } } PUNCT cana-1091 60 67 . . PUNCT cana-1091 61 1 proposition proposition NOUN cana-1091 61 2 2.8[5 2.8[5 NUM cana-1091 61 3 ] ] PUNCT cana-1091 61 4 let let VERB cana-1091 61 5 𝐢 𝐢 PROPN cana-1091 61 6 be be AUX cana-1091 61 7 a a DET cana-1091 61 8 neutrosophic neutrosophic ADJ cana-1091 61 9 fantastic fantastic ADJ cana-1091 61 10 filter filter NOUN cana-1091 61 11 of of ADP cana-1091 61 12 𝒒 𝒒 PROPN cana-1091 61 13 if if SCONJ cana-1091 62 1 and and CCONJ cana-1091 62 2 only only ADV cana-1091 62 3 if if SCONJ cana-1091 62 4 𝑇𝐢 𝑇𝐢 PROPN cana-1091 62 5 ( ( PUNCT cana-1091 62 6 ( ( PUNCT cana-1091 62 7 ( ( PUNCT cana-1091 62 8 𝛼5 𝛼5 NOUN cana-1091 62 9 β†’ β†’ SYM cana-1091 62 10 𝛽5 𝛽5 NUM cana-1091 62 11 ) ) PUNCT cana-1091 62 12 β†’ β†’ SYM cana-1091 62 13 𝛽5 𝛽5 NUM cana-1091 62 14 ) ) PUNCT cana-1091 62 15 β†’ β†’ SYM cana-1091 62 16 𝛼5 𝛼5 NOUN cana-1091 62 17 ) ) PUNCT cana-1091 62 18 β‰₯ β‰₯ NOUN cana-1091 62 19 𝑇𝐢(𝛽5 𝑇𝐢(𝛽5 NUM cana-1091 62 20 β†’ β†’ SYM cana-1091 62 21 𝛼5 𝛼5 NOUN cana-1091 62 22 ) ) PUNCT cana-1091 62 23 , , PUNCT cana-1091 62 24 𝐼𝐢(((𝛼5 𝐼𝐢(((𝛼5 PROPN cana-1091 62 25 β†’ β†’ SYM cana-1091 62 26 𝛽5 𝛽5 ADJ cana-1091 62 27 ) ) PUNCT cana-1091 62 28 β†’ β†’ SYM cana-1091 62 29 𝛽5 𝛽5 NUM cana-1091 62 30 ) ) PUNCT cana-1091 62 31 β†’ β†’ SYM cana-1091 62 32 𝛼5 𝛼5 NOUN cana-1091 62 33 ) ) PUNCT cana-1091 62 34 ≀ ≀ NOUN cana-1091 63 1 𝐼𝐢(𝛽5 𝐼𝐢(𝛽5 ADP cana-1091 63 2 β†’ β†’ SYM cana-1091 63 3 𝛼5)and𝐹𝐢 𝛼5)and𝐹𝐢 NOUN cana-1091 63 4 ( ( PUNCT cana-1091 63 5 ( ( PUNCT cana-1091 63 6 ( ( PUNCT cana-1091 63 7 𝛼5 𝛼5 NOUN cana-1091 63 8 β†’ β†’ SYM cana-1091 63 9 𝛽5 𝛽5 NUM cana-1091 63 10 ) ) PUNCT cana-1091 63 11 β†’ β†’ SYM cana-1091 63 12 𝛽5 𝛽5 NUM cana-1091 63 13 ) ) PUNCT cana-1091 63 14 β†’ β†’ SYM cana-1091 63 15 𝛼5 𝛼5 NOUN cana-1091 63 16 ) ) PUNCT cana-1091 63 17 ≀ ≀ NOUN cana-1091 63 18 𝐹𝐢(𝛽5 𝐹𝐢(𝛽5 NUM cana-1091 64 1 β†’ β†’ PUNCT cana-1091 64 2 𝛼5)for 𝛼5)for ADP cana-1091 64 3 all all DET cana-1091 64 4 𝛼5 𝛼5 NOUN cana-1091 64 5 , , PUNCT cana-1091 64 6 𝛽5 𝛽5 VERB cana-1091 64 7 ∈ ∈ NOUN cana-1091 64 8 𝒒. 𝒒. NOUN cana-1091 64 9 definition definition NOUN cana-1091 64 10 2.9[6 2.9[6 NUM cana-1091 64 11 ] ] X cana-1091 64 12 let let VERB cana-1091 64 13 𝐢 𝐢 PROPN cana-1091 64 14 be be AUX cana-1091 64 15 a a DET cana-1091 64 16 neutrosophic neutrosophic ADJ cana-1091 64 17 filter filter NOUN cana-1091 64 18 of of ADP cana-1091 64 19 a a DET cana-1091 64 20 bl bl NOUN cana-1091 64 21 - - PUNCT cana-1091 64 22 algebra algebra NOUN cana-1091 64 23 𝒒. 𝒒. PROPN cana-1091 64 24 𝐢 𝐢 PROPN cana-1091 64 25 is be AUX cana-1091 64 26 called call VERB cana-1091 64 27 a a DET cana-1091 64 28 neutrosophic neutrosophic ADJ cana-1091 64 29 positive positive ADJ cana-1091 64 30 implicative implicative ADJ cana-1091 64 31 filter filter NOUN cana-1091 64 32 if if SCONJ cana-1091 64 33 it it PRON cana-1091 64 34 persuades persuade VERB cana-1091 64 35 the the DET cana-1091 64 36 following following NOUN cana-1091 64 37 , , PUNCT cana-1091 64 38 ( ( PUNCT cana-1091 64 39 i i NOUN cana-1091 64 40 ) ) PUNCT cana-1091 65 1 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 NOUN cana-1091 65 2 ) ) PUNCT cana-1091 65 3 ≀ ≀ NOUN cana-1091 66 1 𝑇𝐢(1 𝑇𝐢(1 NOUN cana-1091 66 2 ) ) PUNCT cana-1091 66 3 , , PUNCT cana-1091 66 4 𝐼𝐢 𝐼𝐢 PROPN cana-1091 66 5 ( ( PUNCT cana-1091 66 6 𝛼5 𝛼5 NOUN cana-1091 66 7 ) ) PUNCT cana-1091 66 8 β‰₯ β‰₯ NOUN cana-1091 66 9 𝐼𝐢 𝐼𝐢 PROPN cana-1091 66 10 ( ( PUNCT cana-1091 66 11 1 1 NUM cana-1091 66 12 ) ) PUNCT cana-1091 66 13 and and CCONJ cana-1091 66 14 𝐹𝐢(𝛼5 𝐹𝐢(𝛼5 NUM cana-1091 66 15 ) ) PUNCT cana-1091 66 16 β‰₯ β‰₯ NOUN cana-1091 66 17 𝐹𝐢(1 𝐹𝐢(1 NUM cana-1091 66 18 ) ) PUNCT cana-1091 66 19 , , PUNCT cana-1091 66 20 ( ( PUNCT cana-1091 66 21 ii ii NOUN cana-1091 66 22 ) ) PUNCT cana-1091 66 23 min min PROPN cana-1091 66 24 { { PUNCT cana-1091 66 25 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 PROPN cana-1091 66 26 β†’ β†’ SYM cana-1091 66 27 ( ( PUNCT cana-1091 66 28 ( ( PUNCT cana-1091 66 29 𝛽5 𝛽5 PROPN cana-1091 66 30 β†’ β†’ SYM cana-1091 66 31 𝛾5 𝛾5 NOUN cana-1091 66 32 ) ) PUNCT cana-1091 66 33 β†’ β†’ SYM cana-1091 66 34 𝛽5 𝛽5 NUM cana-1091 66 35 ) ) PUNCT cana-1091 66 36 , , PUNCT cana-1091 66 37 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 PROPN cana-1091 66 38 ) ) PUNCT cana-1091 66 39 } } PUNCT cana-1091 66 40 ≀ ≀ NUM cana-1091 66 41 𝑇𝐢(𝛽5 𝑇𝐢(𝛽5 NUM cana-1091 66 42 ) ) PUNCT cana-1091 66 43 , , PUNCT cana-1091 66 44 min min NOUN cana-1091 66 45 { { PUNCT cana-1091 66 46 𝐼𝐢(𝛼5 𝐼𝐢(𝛼5 X cana-1091 66 47 β†’ β†’ X cana-1091 66 48 ( ( PUNCT cana-1091 66 49 ( ( PUNCT cana-1091 66 50 𝛽5 𝛽5 PROPN cana-1091 66 51 β†’ β†’ SYM cana-1091 66 52 𝛾5 𝛾5 NOUN cana-1091 66 53 ) ) PUNCT cana-1091 66 54 β†’ β†’ SYM cana-1091 66 55 𝛽5 𝛽5 NUM cana-1091 66 56 ) ) PUNCT cana-1091 66 57 , , PUNCT cana-1091 66 58 𝐼𝐢(𝛼5 𝐼𝐢(𝛼5 PROPN cana-1091 66 59 ) ) PUNCT cana-1091 66 60 } } PUNCT cana-1091 66 61 β‰₯ β‰₯ PROPN cana-1091 66 62 𝐼𝐢(𝛽5),min 𝐼𝐢(𝛽5),min X cana-1091 66 63 { { PUNCT cana-1091 66 64 𝐹𝐢(𝛼5 𝐹𝐢(𝛼5 PROPN cana-1091 66 65 β†’ β†’ PUNCT cana-1091 66 66 ( ( PUNCT cana-1091 66 67 ( ( PUNCT cana-1091 66 68 𝛽5 𝛽5 PROPN cana-1091 66 69 β†’ β†’ SYM cana-1091 66 70 𝛾5 𝛾5 NOUN cana-1091 66 71 ) ) PUNCT cana-1091 66 72 β†’ β†’ SYM cana-1091 66 73 𝛽5 𝛽5 NUM cana-1091 66 74 ) ) PUNCT cana-1091 66 75 , , PUNCT cana-1091 66 76 𝐹𝐢(𝛼5 𝐹𝐢(𝛼5 PROPN cana-1091 66 77 ) ) PUNCT cana-1091 66 78 } } PUNCT cana-1091 66 79 β‰₯ β‰₯ NOUN cana-1091 66 80 𝐹𝐢(𝛽5 𝐹𝐢(𝛽5 NUM cana-1091 66 81 ) ) PUNCT cana-1091 66 82 for for ADP cana-1091 66 83 all all DET cana-1091 66 84 𝛼5 𝛼5 NOUN cana-1091 66 85 , , PUNCT cana-1091 66 86 𝛽5 𝛽5 NOUN cana-1091 66 87 , , PUNCT cana-1091 66 88 𝛾5 𝛾5 NOUN cana-1091 66 89 ∈ ∈ PROPN cana-1091 66 90 𝒒. 𝒒. PROPN cana-1091 66 91 definition definition NOUN cana-1091 66 92 2.10[6 2.10[6 NUM cana-1091 66 93 ] ] PUNCT cana-1091 66 94 let let VERB cana-1091 66 95 𝐢 𝐢 PROPN cana-1091 66 96 be be AUX cana-1091 66 97 a a DET cana-1091 66 98 neutrosophic neutrosophic ADJ cana-1091 66 99 filter filter NOUN cana-1091 66 100 of of ADP cana-1091 66 101 a a DET cana-1091 66 102 bl bl NOUN cana-1091 66 103 - - PUNCT cana-1091 66 104 algebra algebra NOUN cana-1091 66 105 𝒒. 𝒒. PROPN cana-1091 66 106 𝐢 𝐢 PROPN cana-1091 66 107 is be AUX cana-1091 66 108 called call VERB cana-1091 66 109 a a DET cana-1091 66 110 neutrosophic neutrosophic ADJ cana-1091 66 111 associative associative ADJ cana-1091 66 112 filter filter NOUN cana-1091 66 113 if if SCONJ cana-1091 66 114 it it PRON cana-1091 66 115 satisfies satisfy VERB cana-1091 66 116 the the DET cana-1091 66 117 following following NOUN cana-1091 66 118 , , PUNCT cana-1091 66 119 ( ( PUNCT cana-1091 66 120 i i NOUN cana-1091 66 121 ) ) PUNCT cana-1091 67 1 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 NOUN cana-1091 67 2 ) ) PUNCT cana-1091 67 3 ≀ ≀ NOUN cana-1091 68 1 𝑇𝐢(1 𝑇𝐢(1 NOUN cana-1091 68 2 ) ) PUNCT cana-1091 68 3 , , PUNCT cana-1091 68 4 𝐼𝐢 𝐼𝐢 PROPN cana-1091 68 5 ( ( PUNCT cana-1091 68 6 𝛼5 𝛼5 NOUN cana-1091 68 7 ) ) PUNCT cana-1091 68 8 β‰₯ β‰₯ NOUN cana-1091 68 9 𝐼𝐢 𝐼𝐢 PROPN cana-1091 68 10 ( ( PUNCT cana-1091 68 11 1 1 NUM cana-1091 68 12 ) ) PUNCT cana-1091 68 13 and and CCONJ cana-1091 68 14 𝐹𝐢(𝛼5 𝐹𝐢(𝛼5 NUM cana-1091 68 15 ) ) PUNCT cana-1091 68 16 β‰₯ β‰₯ NOUN cana-1091 68 17 𝐹𝐢(1 𝐹𝐢(1 NUM cana-1091 68 18 ) ) PUNCT cana-1091 68 19 , , PUNCT cana-1091 68 20 ( ( PUNCT cana-1091 68 21 ii ii NOUN cana-1091 68 22 ) ) PUNCT cana-1091 68 23 min min PROPN cana-1091 68 24 { { PUNCT cana-1091 68 25 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 PROPN cana-1091 68 26 β†’ β†’ SYM cana-1091 68 27 ( ( PUNCT cana-1091 68 28 𝛽5 𝛽5 PROPN cana-1091 68 29 β†’ β†’ SYM cana-1091 68 30 𝛾5 𝛾5 NOUN cana-1091 68 31 ) ) PUNCT cana-1091 68 32 ) ) PUNCT cana-1091 68 33 , , PUNCT cana-1091 68 34 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 PROPN cana-1091 68 35 β†’ β†’ SYM cana-1091 68 36 𝛽5 𝛽5 NUM cana-1091 68 37 ) ) PUNCT cana-1091 68 38 } } PUNCT cana-1091 68 39 ≀ ≀ NOUN cana-1091 69 1 𝑇𝐢(𝛾5 𝑇𝐢(𝛾5 PROPN cana-1091 69 2 ) ) PUNCT cana-1091 69 3 , , PUNCT cana-1091 69 4 min min NOUN cana-1091 69 5 { { PUNCT cana-1091 69 6 𝐼𝐢(𝛼5 𝐼𝐢(𝛼5 X cana-1091 69 7 β†’ β†’ PUNCT cana-1091 69 8 ( ( PUNCT cana-1091 69 9 𝛽5 𝛽5 ADJ cana-1091 69 10 β†’ β†’ SYM cana-1091 69 11 𝛾5 𝛾5 NOUN cana-1091 69 12 ) ) PUNCT cana-1091 69 13 ) ) PUNCT cana-1091 69 14 , , PUNCT cana-1091 69 15 𝐼𝐢(𝛼5 𝐼𝐢(𝛼5 PROPN cana-1091 69 16 β†’ β†’ SYM cana-1091 69 17 𝛽5 𝛽5 NUM cana-1091 69 18 ) ) PUNCT cana-1091 69 19 } } PUNCT cana-1091 69 20 β‰₯ β‰₯ NOUN cana-1091 69 21 𝐼𝐢(𝛾5 𝐼𝐢(𝛾5 NOUN cana-1091 69 22 ) ) PUNCT cana-1091 69 23 , , PUNCT cana-1091 69 24 min min NOUN cana-1091 69 25 { { PUNCT cana-1091 69 26 𝐹𝐢(𝛼5 𝐹𝐢(𝛼5 PROPN cana-1091 69 27 β†’ β†’ SYM cana-1091 69 28 ( ( PUNCT cana-1091 69 29 𝛽5 𝛽5 ADJ cana-1091 69 30 β†’ β†’ SYM cana-1091 69 31 𝛾5 𝛾5 NOUN cana-1091 69 32 ) ) PUNCT cana-1091 69 33 ) ) PUNCT cana-1091 69 34 , , PUNCT cana-1091 69 35 𝐹𝐢(𝛼5 𝐹𝐢(𝛼5 PROPN cana-1091 69 36 β†’ β†’ SYM cana-1091 69 37 𝛽5 𝛽5 NUM cana-1091 69 38 ) ) PUNCT cana-1091 69 39 } } PUNCT cana-1091 69 40 β‰₯ β‰₯ NOUN cana-1091 69 41 𝐹𝐢(𝛾5 𝐹𝐢(𝛾5 NUM cana-1091 69 42 ) ) PUNCT cana-1091 69 43 for for ADP cana-1091 69 44 all all DET cana-1091 69 45 𝛼5 𝛼5 NOUN cana-1091 69 46 , , PUNCT cana-1091 69 47 𝛽5 𝛽5 NOUN cana-1091 69 48 , , PUNCT cana-1091 69 49 𝛾5 𝛾5 NOUN cana-1091 69 50 ∈ ∈ PROPN cana-1091 69 51 𝒒. 𝒒. PROPN cana-1091 69 52 proposition proposition NOUN cana-1091 69 53 2.11[6 2.11[6 NUM cana-1091 69 54 ] ] X cana-1091 69 55 let let VERB cana-1091 69 56 𝐢 𝐢 PROPN cana-1091 69 57 be be AUX cana-1091 69 58 a a DET cana-1091 69 59 neutrosophic neutrosophic ADJ cana-1091 69 60 filter filter NOUN cana-1091 69 61 of of ADP cana-1091 69 62 𝒒. 𝒒. PROPN cana-1091 69 63 then then ADV cana-1091 69 64 , , PUNCT cana-1091 69 65 𝐢 𝐢 PROPN cana-1091 69 66 is be AUX cana-1091 69 67 a a DET cana-1091 69 68 neutrosophic neutrosophic ADJ cana-1091 69 69 associative associative ADJ cana-1091 69 70 filter filter NOUN cana-1091 70 1 if if SCONJ cana-1091 70 2 and and CCONJ cana-1091 70 3 only only ADV cana-1091 70 4 if if SCONJ cana-1091 70 5 it it PRON cana-1091 70 6 satisfies satisfy VERB cana-1091 70 7 𝑇𝐢((𝛼5 𝑇𝐢((𝛼5 NUM cana-1091 70 8 β†’ β†’ SYM cana-1091 70 9 𝛽5 𝛽5 NUM cana-1091 70 10 ) ) PUNCT cana-1091 70 11 β†’ β†’ SYM cana-1091 70 12 𝛾5 𝛾5 NOUN cana-1091 70 13 ) ) PUNCT cana-1091 70 14 β‰₯ β‰₯ NOUN cana-1091 70 15 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 NUM cana-1091 71 1 β†’ β†’ PUNCT cana-1091 71 2 ( ( PUNCT cana-1091 71 3 𝛽5 𝛽5 PROPN cana-1091 71 4 β†’ β†’ SYM cana-1091 71 5 𝛾5 𝛾5 NOUN cana-1091 71 6 ) ) PUNCT cana-1091 71 7 ) ) PUNCT cana-1091 72 1 , , PUNCT cana-1091 72 2 𝐼𝐢((𝛼5 𝐼𝐢((𝛼5 NOUN cana-1091 72 3 β†’ β†’ SYM cana-1091 72 4 𝛽5 𝛽5 NUM cana-1091 72 5 ) ) PUNCT cana-1091 72 6 β†’ β†’ SYM cana-1091 72 7 𝛾5 𝛾5 NOUN cana-1091 72 8 ) ) PUNCT cana-1091 72 9 ≀ ≀ NOUN cana-1091 73 1 𝐼𝐢(𝛼5 𝐼𝐢(𝛼5 ADP cana-1091 73 2 β†’ β†’ PUNCT cana-1091 73 3 ( ( PUNCT cana-1091 73 4 𝛽5 𝛽5 NOUN cana-1091 73 5 β†’ β†’ SYM cana-1091 73 6 𝛾5)),𝐹𝐢((𝛼5 𝛾5)),𝐹𝐢((𝛼5 ADJ cana-1091 73 7 β†’ β†’ SYM cana-1091 73 8 𝛽5 𝛽5 NUM cana-1091 73 9 ) ) PUNCT cana-1091 73 10 β†’ β†’ SYM cana-1091 73 11 𝛾5 𝛾5 NOUN cana-1091 73 12 ) ) PUNCT cana-1091 73 13 ≀ ≀ NOUN cana-1091 74 1 𝐹𝐢(𝛼5 𝐹𝐢(𝛼5 PROPN cana-1091 74 2 β†’ β†’ SYM cana-1091 74 3 ( ( PUNCT cana-1091 74 4 𝛽5 𝛽5 PROPN cana-1091 74 5 β†’ β†’ SYM cana-1091 74 6 𝛾5 𝛾5 NOUN cana-1091 74 7 ) ) PUNCT cana-1091 74 8 ) ) PUNCT cana-1091 74 9 for for ADP cana-1091 74 10 all all DET cana-1091 74 11 𝛼5 𝛼5 NOUN cana-1091 74 12 , , PUNCT cana-1091 74 13 𝛽5 𝛽5 NOUN cana-1091 74 14 , , PUNCT cana-1091 74 15 𝛾5 𝛾5 NOUN cana-1091 74 16 ∈ ∈ PROPN cana-1091 74 17 𝒒. 𝒒. PROPN cana-1091 74 18 communications communication NOUN cana-1091 74 19 on on ADP cana-1091 74 20 applied apply VERB cana-1091 74 21 nonlinear nonlinear ADJ cana-1091 74 22 analysis analysis NOUN cana-1091 74 23 issn issn NOUN cana-1091 74 24 : : PUNCT cana-1091 74 25 1074 1074 NUM cana-1091 74 26 - - PUNCT cana-1091 74 27 133x 133x NUM cana-1091 74 28 vol vol NOUN cana-1091 74 29 31 31 NUM cana-1091 74 30 no no NOUN cana-1091 74 31 . . PUNCT cana-1091 75 1 5s 5s NUM cana-1091 75 2 ( ( PUNCT cana-1091 75 3 2024 2024 NUM cana-1091 75 4 ) ) PUNCT cana-1091 75 5 565 565 NUM cana-1091 75 6 https://internationalpubls.com https://internationalpubls.com X cana-1091 75 7 proposition proposition NOUN cana-1091 75 8 2.12[6]everyneutrosophic 2.12[6]everyneutrosophic NUM cana-1091 75 9 positive positive ADJ cana-1091 75 10 implicative implicative ADJ cana-1091 75 11 filter filter NOUN cana-1091 75 12 of of ADP cana-1091 75 13 𝒒 𝒒 PROPN cana-1091 75 14 is be AUX cana-1091 75 15 a a DET cana-1091 75 16 neutrosophic neutrosophic ADJ cana-1091 75 17 fantastic fantastic ADJ cana-1091 75 18 filter filter NOUN cana-1091 75 19 . . PUNCT cana-1091 76 1 3 3 X cana-1091 76 2 . . X cana-1091 76 3 neutrosophic neutrosophic ADJ cana-1091 76 4 transitive transitive ADJ cana-1091 76 5 filter filter NOUN cana-1091 76 6 here here ADV cana-1091 76 7 , , PUNCT cana-1091 76 8 we we PRON cana-1091 76 9 put put VERB cana-1091 76 10 forward forward ADV cana-1091 76 11 the the DET cana-1091 76 12 conception conception NOUN cana-1091 76 13 of of ADP cana-1091 76 14 a a DET cana-1091 76 15 neutrosophic neutrosophic ADJ cana-1091 76 16 transitive transitive ADJ cana-1091 76 17 filter filter NOUN cana-1091 76 18 and and CCONJ cana-1091 76 19 confer confer VERB cana-1091 76 20 its its PRON cana-1091 76 21 features feature NOUN cana-1091 76 22 with with ADP cana-1091 76 23 illustrations illustration NOUN cana-1091 76 24 . . PUNCT cana-1091 77 1 definition definition NOUN cana-1091 77 2 3.1 3.1 NUM cana-1091 77 3 let let VERB cana-1091 77 4 𝐢 𝐢 PROPN cana-1091 77 5 be be AUX cana-1091 77 6 called call VERB cana-1091 77 7 a a DET cana-1091 77 8 neutrosophic neutrosophic ADJ cana-1091 77 9 transitive transitive ADJ cana-1091 77 10 filter filter NOUN cana-1091 77 11 of of ADP cana-1091 77 12 𝒒 𝒒 PROPN cana-1091 77 13 , , PUNCT cana-1091 77 14 if if SCONJ cana-1091 77 15 it it PRON cana-1091 77 16 persuades persuade VERB cana-1091 77 17 the the DET cana-1091 77 18 subsequent subsequent ADJ cana-1091 77 19 requirements requirement NOUN cana-1091 77 20 for for ADP cana-1091 77 21 all all DET cana-1091 77 22 𝛼5 𝛼5 NOUN cana-1091 77 23 , , PUNCT cana-1091 77 24 𝛽5 𝛽5 NOUN cana-1091 77 25 , , PUNCT cana-1091 77 26 𝛾5 𝛾5 PROPN cana-1091 77 27 ∈ ∈ PROPN cana-1091 77 28 𝒒 𝒒 PROPN cana-1091 77 29 , , PUNCT cana-1091 77 30 ( ( PUNCT cana-1091 77 31 i i NOUN cana-1091 77 32 ) ) PUNCT cana-1091 78 1 𝑇𝐢(1 𝑇𝐢(1 PROPN cana-1091 78 2 ) ) PUNCT cana-1091 78 3 β‰₯ β‰₯ NOUN cana-1091 78 4 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 NOUN cana-1091 78 5 ) ) PUNCT cana-1091 78 6 , , PUNCT cana-1091 78 7 𝐼𝐢(1 𝐼𝐢(1 PROPN cana-1091 78 8 ) ) PUNCT cana-1091 78 9 ≀ ≀ NOUN cana-1091 79 1 𝐼𝐢(𝛼5 𝐼𝐢(𝛼5 ADP cana-1091 79 2 ) ) PUNCT cana-1091 79 3 , , PUNCT cana-1091 79 4 𝐹𝐢(1 𝐹𝐢(1 PROPN cana-1091 79 5 ) ) PUNCT cana-1091 79 6 ≀ ≀ NOUN cana-1091 80 1 𝐹𝐢(𝛼5 𝐹𝐢(𝛼5 NUM cana-1091 80 2 ) ) PUNCT cana-1091 80 3 . . PUNCT cana-1091 81 1 ( ( PUNCT cana-1091 81 2 ii ii NOUN cana-1091 81 3 ) ) PUNCT cana-1091 81 4 min{𝑇𝐢(𝛼5 min{𝑇𝐢(𝛼5 ADP cana-1091 81 5 β†’ β†’ SYM cana-1091 81 6 𝛽5 𝛽5 NUM cana-1091 81 7 ) ) PUNCT cana-1091 81 8 , , PUNCT cana-1091 81 9 𝑇𝐢(𝛼5)}≀ 𝑇𝐢(𝛼5)}≀ PROPN cana-1091 81 10 𝑇𝐢(𝛽5),min{𝐼𝐢 𝑇𝐢(𝛽5),min{𝐼𝐢 X cana-1091 81 11 ( ( PUNCT cana-1091 81 12 𝛼5 𝛼5 NOUN cana-1091 81 13 β†’ β†’ SYM cana-1091 81 14 𝛽5 𝛽5 NUM cana-1091 81 15 ) ) PUNCT cana-1091 81 16 , , PUNCT cana-1091 81 17 𝐼𝐢 𝐼𝐢 PROPN cana-1091 81 18 ( ( PUNCT cana-1091 81 19 𝛼5 𝛼5 NOUN cana-1091 81 20 ) ) PUNCT cana-1091 81 21 } } PUNCT cana-1091 81 22 β‰₯ β‰₯ AUX cana-1091 81 23 𝐼𝐢 𝐼𝐢 PROPN cana-1091 81 24 ( ( PUNCT cana-1091 81 25 𝛽5 𝛽5 NUM cana-1091 81 26 ) ) PUNCT cana-1091 81 27 and and CCONJ cana-1091 81 28 min min NOUN cana-1091 81 29 { { PUNCT cana-1091 81 30 𝐹𝐢 𝐹𝐢 PROPN cana-1091 81 31 ( ( PUNCT cana-1091 81 32 𝛼5 𝛼5 NOUN cana-1091 81 33 β†’ β†’ SYM cana-1091 81 34 𝛽5 𝛽5 NUM cana-1091 81 35 ) ) PUNCT cana-1091 81 36 , , PUNCT cana-1091 81 37 𝐹𝐢(𝛼5)}β‰₯ 𝐹𝐢(𝛼5)}β‰₯ NUM cana-1091 82 1 𝐹𝐢(𝛽5 𝐹𝐢(𝛽5 NUM cana-1091 82 2 ) ) PUNCT cana-1091 82 3 } } PUNCT cana-1091 82 4 . . PUNCT cana-1091 83 1 ( ( PUNCT cana-1091 83 2 iii iii X cana-1091 83 3 ) ) PUNCT cana-1091 83 4 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 PROPN cana-1091 83 5 β†’ β†’ SYM cana-1091 83 6 𝛾5 𝛾5 NOUN cana-1091 83 7 ) ) PUNCT cana-1091 83 8 β‰₯ β‰₯ NOUN cana-1091 83 9 min{𝑇𝐢(𝛼5 min{𝑇𝐢(𝛼5 ADV cana-1091 83 10 β†’ β†’ SYM cana-1091 83 11 𝛽5 𝛽5 NUM cana-1091 83 12 ) ) PUNCT cana-1091 83 13 , , PUNCT cana-1091 83 14 𝑇𝐢(𝛽5 𝑇𝐢(𝛽5 PROPN cana-1091 83 15 β†’ β†’ SYM cana-1091 83 16 𝛾5 𝛾5 NOUN cana-1091 83 17 ) ) PUNCT cana-1091 83 18 } } PUNCT cana-1091 83 19 , , PUNCT cana-1091 83 20 𝐼𝐢(𝛼5 𝐼𝐢(𝛼5 PROPN cana-1091 83 21 β†’ β†’ SYM cana-1091 83 22 𝛾5 𝛾5 NOUN cana-1091 83 23 ) ) PUNCT cana-1091 83 24 ≀ ≀ NUM cana-1091 83 25 min min NOUN cana-1091 83 26 { { PUNCT cana-1091 83 27 𝐼𝐢(𝛼5 𝐼𝐢(𝛼5 NUM cana-1091 83 28 β†’ β†’ SYM cana-1091 83 29 𝛽5 𝛽5 NUM cana-1091 83 30 ) ) PUNCT cana-1091 83 31 , , PUNCT cana-1091 83 32 𝐼𝐢(𝛽5 𝐼𝐢(𝛽5 PROPN cana-1091 83 33 β†’ β†’ SYM cana-1091 83 34 𝛾5 𝛾5 NOUN cana-1091 83 35 ) ) PUNCT cana-1091 83 36 } } PUNCT cana-1091 83 37 , , PUNCT cana-1091 83 38 𝐹𝐢(𝛼5 𝐹𝐢(𝛼5 PROPN cana-1091 83 39 β†’ β†’ SYM cana-1091 83 40 𝛾5 𝛾5 NOUN cana-1091 83 41 ) ) PUNCT cana-1091 83 42 ≀ ≀ ADJ cana-1091 83 43 min{𝐹𝐢(𝛼5 min{𝐹𝐢(𝛼5 NOUN cana-1091 83 44 β†’ β†’ SYM cana-1091 83 45 𝛽5 𝛽5 NUM cana-1091 83 46 ) ) PUNCT cana-1091 83 47 , , PUNCT cana-1091 83 48 𝐹𝐢(𝛽5 𝐹𝐢(𝛽5 PROPN cana-1091 83 49 β†’ β†’ SYM cana-1091 83 50 𝛾5 𝛾5 NOUN cana-1091 83 51 ) ) PUNCT cana-1091 83 52 } } PUNCT cana-1091 83 53 . . PUNCT cana-1091 84 1 example example NOUN cana-1091 84 2 3.2 3.2 NUM cana-1091 84 3 let let VERB cana-1091 84 4 𝐢 𝐢 PROPN cana-1091 84 5 = = PROPN cana-1091 84 6 { { PUNCT cana-1091 84 7 0 0 NUM cana-1091 84 8 , , PUNCT cana-1091 84 9 πœ–1 πœ–1 PROPN cana-1091 84 10 , , PUNCT cana-1091 84 11 πœ‡1 πœ‡1 PROPN cana-1091 84 12 , , PUNCT cana-1091 84 13 𝜌1 𝜌1 PROPN cana-1091 84 14 , , PUNCT cana-1091 84 15 1}.the 1}.the ADJ cana-1091 84 16 binary binary ADJ cana-1091 84 17 operations operation NOUN cana-1091 84 18 Β° Β° NOUN cana-1091 84 19 and and CCONJ cana-1091 84 20 β†’ β†’ AUX cana-1091 84 21 are be AUX cana-1091 84 22 given give VERB cana-1091 84 23 by by ADP cana-1091 84 24 the the DET cana-1091 84 25 subsequent subsequent ADJ cana-1091 84 26 tables table NOUN cana-1091 84 27 ( ( PUNCT cana-1091 84 28 3.1 3.1 NUM cana-1091 84 29 ) ) PUNCT cana-1091 84 30 and and CCONJ cana-1091 84 31 ( ( PUNCT cana-1091 84 32 3.2 3.2 NUM cana-1091 84 33 ) ) PUNCT cana-1091 84 34 . . PUNCT cana-1091 85 1 table table NOUN cana-1091 85 2 3.1 3.1 NUM cana-1091 85 3 : : PUNCT cana-1091 85 4 β€² β€² NUM cana-1091 85 5 Β° Β° NUM cana-1091 86 1 β€² β€² NUM cana-1091 86 2 operation operation NOUN cana-1091 86 3 table table NOUN cana-1091 86 4 3.2 3.2 NUM cana-1091 86 5 : : PUNCT cana-1091 86 6 β€² β€² NUM cana-1091 86 7 β†’ β†’ SYM cana-1091 86 8 β€²operation β€²operation NOUN cana-1091 86 9 then then ADV cana-1091 86 10 , , PUNCT cana-1091 86 11 ( ( PUNCT cana-1091 86 12 𝒒 𝒒 PROPN cana-1091 86 13 , , PUNCT cana-1091 86 14 ∨ ∨ NOUN cana-1091 86 15 , , PUNCT cana-1091 86 16 ∧ ∧ PROPN cana-1091 86 17 , , PUNCT cana-1091 86 18 ∘ ∘ ADJ cana-1091 86 19 , , PUNCT cana-1091 86 20 β†’ β†’ SYM cana-1091 86 21 , , PUNCT cana-1091 86 22 0 0 NUM cana-1091 86 23 , , PUNCT cana-1091 86 24 1 1 NUM cana-1091 86 25 ) ) PUNCT cana-1091 86 26 is be AUX cana-1091 86 27 a a DET cana-1091 86 28 blalgebra blalgebra NOUN cana-1091 86 29 . . PUNCT cana-1091 87 1 define define VERB cana-1091 87 2 a a DET cana-1091 87 3 neutrosophic neutrosophic ADJ cana-1091 87 4 set set NOUN cana-1091 87 5 𝐢 𝐢 PROPN cana-1091 87 6 of of ADP cana-1091 87 7 𝒒 𝒒 PROPN cana-1091 87 8 as as SCONJ cana-1091 87 9 follows follow VERB cana-1091 87 10 : : PUNCT cana-1091 87 11 𝐢 𝐢 PROPN cana-1091 87 12 = = PRON cana-1091 87 13 { { PUNCT cana-1091 87 14 ( ( PUNCT cana-1091 87 15 0 0 NUM cana-1091 87 16 , , PUNCT cana-1091 87 17 [ [ X cana-1091 87 18 0.5,0.7,0.7 0.5,0.7,0.7 NOUN cana-1091 87 19 ] ] PUNCT cana-1091 87 20 ) ) PUNCT cana-1091 87 21 , , PUNCT cana-1091 87 22 ( ( PUNCT cana-1091 87 23 πœ–1 πœ–1 PROPN cana-1091 87 24 , , PUNCT cana-1091 87 25 [ [ X cana-1091 87 26 0.5,0.7,0.7 0.5,0.7,0.7 NOUN cana-1091 87 27 ] ] PUNCT cana-1091 87 28 ) ) PUNCT cana-1091 87 29 , , PUNCT cana-1091 87 30 ( ( PUNCT cana-1091 87 31 πœ‡1 πœ‡1 ADJ cana-1091 87 32 , , PUNCT cana-1091 87 33 [ [ X cana-1091 87 34 0.5,0.7,0.7 0.5,0.7,0.7 NOUN cana-1091 87 35 ] ] PUNCT cana-1091 87 36 ) ) PUNCT cana-1091 87 37 , , PUNCT cana-1091 87 38 ( ( PUNCT cana-1091 87 39 𝜌1 𝜌1 NOUN cana-1091 87 40 , , PUNCT cana-1091 87 41 [ [ X cana-1091 87 42 0.5,0.7,0.7 0.5,0.7,0.7 NOUN cana-1091 87 43 ] ] PUNCT cana-1091 87 44 ) ) PUNCT cana-1091 87 45 , , PUNCT cana-1091 87 46 ( ( PUNCT cana-1091 87 47 1 1 X cana-1091 87 48 , , PUNCT cana-1091 87 49 [ [ X cana-1091 87 50 0.6,0.7,0.7 0.6,0.7,0.7 NOUN cana-1091 87 51 ] ] PUNCT cana-1091 87 52 ) ) PUNCT cana-1091 87 53 } } PUNCT cana-1091 87 54 . . PUNCT cana-1091 88 1 it it PRON cana-1091 88 2 is be AUX cana-1091 88 3 evident evident ADJ cana-1091 88 4 that that SCONJ cana-1091 88 5 𝐢 𝐢 PROPN cana-1091 88 6 assures assure VERB cana-1091 88 7 the the DET cana-1091 88 8 conditions condition NOUN cana-1091 88 9 ( ( PUNCT cana-1091 88 10 i i NOUN cana-1091 88 11 ) ) PUNCT cana-1091 88 12 and and CCONJ cana-1091 88 13 ( ( PUNCT cana-1091 88 14 ii ii NOUN cana-1091 88 15 ) ) PUNCT cana-1091 88 16 of of ADP cana-1091 88 17 the the DET cana-1091 88 18 definition definition NOUN cana-1091 88 19 3.1 3.1 NUM cana-1091 88 20 and and CCONJ cana-1091 88 21 hence hence ADV cana-1091 88 22 is be AUX cana-1091 88 23 a a DET cana-1091 88 24 neutrosophic neutrosophic ADJ cana-1091 88 25 transitive transitive ADJ cana-1091 88 26 filter filter NOUN cana-1091 88 27 . . PUNCT cana-1091 89 1 proposition proposition NOUN cana-1091 89 2 3.3 3.3 NUM cana-1091 89 3 every every DET cana-1091 89 4 neutrosophic neutrosophic ADJ cana-1091 89 5 transitive transitive ADJ cana-1091 89 6 filter filter NOUN cana-1091 89 7 of of ADP cana-1091 89 8 a a DET cana-1091 89 9 bl bl NOUN cana-1091 89 10 - - PUNCT cana-1091 89 11 algebra algebra NOUN cana-1091 89 12 𝒒 𝒒 PROPN cana-1091 89 13 is be AUX cana-1091 89 14 a a DET cana-1091 89 15 neutrosophic neutrosophic ADJ cana-1091 89 16 filter filter NOUN cana-1091 89 17 with with ADP cana-1091 89 18 respect respect NOUN cana-1091 89 19 to to ADP cana-1091 89 20 1 1 NUM cana-1091 89 21 . . PUNCT cana-1091 90 1 proof proof NOUN cana-1091 90 2 : : PUNCT cana-1091 90 3 let let VERB cana-1091 90 4 𝐢 𝐢 PROPN cana-1091 90 5 be be AUX cana-1091 90 6 a a DET cana-1091 90 7 neutrosophic neutrosophic ADJ cana-1091 90 8 transitive transitive ADJ cana-1091 90 9 filter filter NOUN cana-1091 90 10 of of ADP cana-1091 90 11 a a DET cana-1091 90 12 bl bl VERB cana-1091 90 13 - - PUNCT cana-1091 90 14 algebra𝒒. algebra𝒒. PRON cana-1091 90 15 taking take VERB cana-1091 90 16 𝛼5 𝛼5 NOUN cana-1091 90 17 = = SYM cana-1091 90 18 1 1 NUM cana-1091 90 19 in in ADP cana-1091 90 20 ( ( PUNCT cana-1091 90 21 iii iii NOUN cana-1091 90 22 ) ) PUNCT cana-1091 90 23 of of ADP cana-1091 90 24 the the DET cana-1091 90 25 definition definition NOUN cana-1091 90 26 3.1 3.1 NUM cana-1091 90 27 , , PUNCT cana-1091 90 28 we we PRON cana-1091 90 29 get get VERB cana-1091 90 30 𝑇𝐢(1 𝑇𝐢(1 PROPN cana-1091 90 31 β†’ β†’ SYM cana-1091 90 32 𝛾5 𝛾5 NOUN cana-1091 90 33 ) ) PUNCT cana-1091 90 34 β‰₯ β‰₯ NOUN cana-1091 90 35 min{𝑇𝐢(1 min{𝑇𝐢(1 ADV cana-1091 90 36 β†’ β†’ SYM cana-1091 90 37 𝛽5 𝛽5 NUM cana-1091 90 38 ) ) PUNCT cana-1091 90 39 , , PUNCT cana-1091 90 40 𝑇𝐢(𝛽5 𝑇𝐢(𝛽5 PROPN cana-1091 90 41 β†’ β†’ SYM cana-1091 90 42 𝛾5 𝛾5 NOUN cana-1091 90 43 ) ) PUNCT cana-1091 90 44 } } PUNCT cana-1091 90 45 Β° Β° ADP cana-1091 90 46 0 0 NUM cana-1091 91 1 πœ–1 πœ–1 PROPN cana-1091 91 2 πœ‡1 πœ‡1 PROPN cana-1091 91 3 𝜌1 𝜌1 PROPN cana-1091 91 4 1 1 NUM cana-1091 91 5 0 0 NUM cana-1091 91 6 0 0 NUM cana-1091 91 7 0 0 NUM cana-1091 91 8 0 0 NUM cana-1091 91 9 0 0 NUM cana-1091 91 10 0 0 NUM cana-1091 92 1 πœ–1 πœ–1 NOUN cana-1091 92 2 0 0 NUM cana-1091 92 3 πœ–1 πœ–1 PROPN cana-1091 92 4 𝜌1 𝜌1 PROPN cana-1091 92 5 𝜌1 𝜌1 PROPN cana-1091 92 6 πœ–1 πœ–1 PROPN cana-1091 92 7 πœ‡1 πœ‡1 PROPN cana-1091 92 8 0 0 NUM cana-1091 92 9 𝜌1 𝜌1 PROPN cana-1091 92 10 πœ‡1 πœ‡1 PROPN cana-1091 92 11 𝜌1 𝜌1 PROPN cana-1091 92 12 πœ‡1 πœ‡1 PROPN cana-1091 92 13 𝜌1 𝜌1 NOUN cana-1091 92 14 0 0 NUM cana-1091 92 15 𝜌1 𝜌1 NOUN cana-1091 92 16 𝜌1 𝜌1 NOUN cana-1091 92 17 𝜌1 𝜌1 NOUN cana-1091 92 18 𝜌1 𝜌1 NOUN cana-1091 92 19 1 1 NUM cana-1091 92 20 0 0 NUM cana-1091 92 21 πœ–1 πœ–1 PROPN cana-1091 92 22 πœ‡1 πœ‡1 PROPN cana-1091 92 23 𝜌1 𝜌1 PROPN cana-1091 92 24 1 1 NUM cana-1091 92 25 β†’ β†’ SYM cana-1091 92 26 0 0 NUM cana-1091 92 27 πœ€1 πœ€1 PROPN cana-1091 92 28 πœ‡1 πœ‡1 PROPN cana-1091 92 29 𝜌1 𝜌1 PROPN cana-1091 92 30 1 1 NUM cana-1091 92 31 0 0 NUM cana-1091 92 32 1 1 NUM cana-1091 92 33 1 1 NUM cana-1091 92 34 1 1 NUM cana-1091 92 35 1 1 NUM cana-1091 92 36 1 1 NUM cana-1091 92 37 πœ€1 πœ€1 NOUN cana-1091 92 38 0 0 NUM cana-1091 92 39 1 1 NUM cana-1091 92 40 πœ‡1 πœ‡1 NOUN cana-1091 92 41 πœ‡1 πœ‡1 PROPN cana-1091 92 42 1 1 NUM cana-1091 92 43 πœ‡1 πœ‡1 NOUN cana-1091 92 44 0 0 PUNCT cana-1091 92 45 πœ–1 πœ–1 PROPN cana-1091 92 46 1 1 NUM cana-1091 92 47 πœ–1 πœ–1 PROPN cana-1091 92 48 1 1 NUM cana-1091 92 49 𝜌1 𝜌1 NOUN cana-1091 92 50 0 0 NUM cana-1091 92 51 1 1 NUM cana-1091 92 52 1 1 NUM cana-1091 92 53 1 1 NUM cana-1091 92 54 1 1 NUM cana-1091 92 55 1 1 NUM cana-1091 92 56 0 0 NUM cana-1091 92 57 πœ–1 πœ–1 PROPN cana-1091 92 58 πœ‡1 πœ‡1 PROPN cana-1091 92 59 𝜌1 𝜌1 PROPN cana-1091 92 60 1 1 NUM cana-1091 92 61 communications communication NOUN cana-1091 92 62 on on ADP cana-1091 92 63 applied apply VERB cana-1091 92 64 nonlinear nonlinear ADJ cana-1091 92 65 analysis analysis NOUN cana-1091 92 66 issn issn NOUN cana-1091 92 67 : : PUNCT cana-1091 92 68 1074 1074 NUM cana-1091 92 69 - - PUNCT cana-1091 92 70 133x 133x NUM cana-1091 92 71 vol vol NOUN cana-1091 92 72 31 31 NUM cana-1091 92 73 no no NOUN cana-1091 92 74 . . PUNCT cana-1091 93 1 5s 5s NUM cana-1091 93 2 ( ( PUNCT cana-1091 93 3 2024 2024 NUM cana-1091 93 4 ) ) PUNCT cana-1091 93 5 566 566 NUM cana-1091 93 6 https://internationalpubls.com https://internationalpubls.com X cana-1091 93 7 𝑇𝐢(𝛾5 𝑇𝐢(𝛾5 PROPN cana-1091 93 8 ) ) PUNCT cana-1091 93 9 β‰₯ β‰₯ PROPN cana-1091 93 10 min{𝑇𝐢(𝛽5 min{𝑇𝐢(𝛽5 NOUN cana-1091 93 11 ) ) PUNCT cana-1091 93 12 , , PUNCT cana-1091 93 13 𝑇𝐢(𝛽5 𝑇𝐢(𝛽5 PROPN cana-1091 93 14 β†’ β†’ SYM cana-1091 93 15 𝛾5)}for 𝛾5)}for ADP cana-1091 93 16 all all DET cana-1091 93 17 𝛼5 𝛼5 NOUN cana-1091 93 18 , , PUNCT cana-1091 93 19 𝛽5 𝛽5 VERB cana-1091 93 20 ∈ ∈ NOUN cana-1091 93 21 𝒒. 𝒒. NOUN cana-1091 93 22 similarly similarly ADV cana-1091 93 23 , , PUNCT cana-1091 93 24 we we PRON cana-1091 93 25 can can AUX cana-1091 93 26 prove prove VERB cana-1091 93 27 for for ADP cana-1091 93 28 𝐼𝐢 𝐼𝐢 PROPN cana-1091 93 29 , , PUNCT cana-1091 93 30 𝐹𝐢 𝐹𝐢 PROPN cana-1091 93 31 . . PUNCT cana-1091 94 1 the the DET cana-1091 94 2 converse converse NOUN cana-1091 94 3 part part NOUN cana-1091 94 4 may may AUX cana-1091 94 5 not not PART cana-1091 94 6 be be AUX cana-1091 94 7 true true ADJ cana-1091 94 8 . . PUNCT cana-1091 95 1 this this PRON cana-1091 95 2 can can AUX cana-1091 95 3 be be AUX cana-1091 95 4 proved prove VERB cana-1091 95 5 by by ADP cana-1091 95 6 an an DET cana-1091 95 7 example example NOUN cana-1091 95 8 . . PUNCT cana-1091 96 1 example example NOUN cana-1091 96 2 3.4 3.4 NUM cana-1091 96 3 let let VERB cana-1091 96 4 𝐷 𝐷 NOUN cana-1091 96 5 = = SYM cana-1091 96 6 { { PUNCT cana-1091 96 7 0 0 NUM cana-1091 96 8 , , PUNCT cana-1091 96 9 πœ–1 πœ–1 PROPN cana-1091 96 10 , , PUNCT cana-1091 96 11 πœ‡1 πœ‡1 PROPN cana-1091 96 12 , , PUNCT cana-1091 96 13 𝜌1 𝜌1 NOUN cana-1091 96 14 , , PUNCT cana-1091 96 15 1 1 NUM cana-1091 96 16 } } PUNCT cana-1091 96 17 . . PUNCT cana-1091 97 1 the the DET cana-1091 97 2 bi bi ADJ cana-1091 97 3 - - ADJ cana-1091 97 4 fold fold ADJ cana-1091 97 5 operations operation NOUN cana-1091 97 6 are be AUX cana-1091 97 7 specified specify VERB cana-1091 97 8 by by ADP cana-1091 97 9 the the DET cana-1091 97 10 tables table NOUN cana-1091 97 11 ( ( PUNCT cana-1091 97 12 3.1 3.1 NUM cana-1091 97 13 ) ) PUNCT cana-1091 97 14 and and CCONJ cana-1091 97 15 ( ( PUNCT cana-1091 97 16 3.2 3.2 NUM cana-1091 97 17 ) ) PUNCT cana-1091 97 18 . . PUNCT cana-1091 98 1 let let VERB cana-1091 98 2 𝐷 𝐷 NOUN cana-1091 98 3 = = PRON cana-1091 98 4 { { PUNCT cana-1091 98 5 ( ( PUNCT cana-1091 98 6 0 0 NUM cana-1091 98 7 , , PUNCT cana-1091 98 8 [ [ X cana-1091 98 9 0.3,0.7,0.7 0.3,0.7,0.7 NOUN cana-1091 98 10 ] ] X cana-1091 98 11 ) ) PUNCT cana-1091 98 12 , , PUNCT cana-1091 98 13 ( ( PUNCT cana-1091 98 14 πœ–1 πœ–1 PROPN cana-1091 98 15 , , PUNCT cana-1091 98 16 [ [ X cana-1091 98 17 0.5,0.7,0.7 0.5,0.7,0.7 NOUN cana-1091 98 18 ] ] PUNCT cana-1091 98 19 ) ) PUNCT cana-1091 98 20 , , PUNCT cana-1091 98 21 ( ( PUNCT cana-1091 98 22 πœ‡1 πœ‡1 ADJ cana-1091 98 23 , , PUNCT cana-1091 98 24 [ [ X cana-1091 98 25 0.3,0.7,0.7 0.3,0.7,0.7 NOUN cana-1091 98 26 ] ] X cana-1091 98 27 ) ) PUNCT cana-1091 98 28 , , PUNCT cana-1091 98 29 ( ( PUNCT cana-1091 98 30 𝜌1 𝜌1 NOUN cana-1091 98 31 , , PUNCT cana-1091 98 32 [ [ X cana-1091 98 33 0.3,0.7,0.7 0.3,0.7,0.7 NOUN cana-1091 98 34 ] ] X cana-1091 98 35 ) ) PUNCT cana-1091 98 36 , , PUNCT cana-1091 98 37 ( ( PUNCT cana-1091 98 38 1 1 X cana-1091 98 39 , , PUNCT cana-1091 98 40 [ [ X cana-1091 98 41 0.4,0.6,0.6 0.4,0.6,0.6 NUM cana-1091 98 42 ] ] PUNCT cana-1091 98 43 ) ) PUNCT cana-1091 98 44 } } PUNCT cana-1091 98 45 . . PUNCT cana-1091 99 1 here here ADV cana-1091 99 2 , , PUNCT cana-1091 99 3 𝐷 𝐷 PROPN cana-1091 99 4 is be AUX cana-1091 99 5 not not PART cana-1091 99 6 a a DET cana-1091 99 7 neutrosophic neutrosophic ADJ cana-1091 99 8 transitive transitive ADJ cana-1091 99 9 filter filter NOUN cana-1091 99 10 . . PUNCT cana-1091 100 1 since,𝑇𝐷(1 since,𝑇𝐷(1 VERB cana-1091 100 2 ) ) PUNCT cana-1091 100 3 = = PUNCT cana-1091 101 1 0.4 0.4 NUM cana-1091 101 2 ≱ ≱ PROPN cana-1091 101 3 0.5 0.5 NUM cana-1091 101 4 = = SYM cana-1091 101 5 𝑇𝐷(πœ€1 𝑇𝐷(πœ€1 PROPN cana-1091 101 6 ) ) PUNCT cana-1091 101 7 . . PUNCT cana-1091 102 1 proposition proposition NOUN cana-1091 102 2 3.5 3.5 NUM cana-1091 102 3 let let VERB cana-1091 102 4 𝐢 𝐢 PROPN cana-1091 102 5 be be AUX cana-1091 102 6 a a DET cana-1091 102 7 neutrosophic neutrosophic ADJ cana-1091 102 8 filter filter NOUN cana-1091 102 9 of of ADP cana-1091 102 10 a a DET cana-1091 102 11 bl bl NOUN cana-1091 102 12 - - PUNCT cana-1091 102 13 algebra algebra NOUN cana-1091 102 14 𝒒. 𝒒. PROPN cana-1091 102 15 then then ADV cana-1091 102 16 , , PUNCT cana-1091 102 17 𝐢 𝐢 PROPN cana-1091 102 18 is be AUX cana-1091 102 19 a a DET cana-1091 102 20 neutrosophic neutrosophic ADJ cana-1091 102 21 transitive transitive ADJ cana-1091 102 22 filter filter NOUN cana-1091 102 23 of of ADP cana-1091 102 24 𝒒 𝒒 PROPN cana-1091 102 25 if if SCONJ cana-1091 103 1 and and CCONJ cana-1091 103 2 only only ADV cana-1091 103 3 if if SCONJ cana-1091 103 4 it it PRON cana-1091 103 5 satisfies satisfy VERB cana-1091 103 6 the the DET cana-1091 103 7 following follow VERB cana-1091 103 8 conditions condition NOUN cana-1091 103 9 for for ADP cana-1091 103 10 all all DET cana-1091 103 11 𝛼5 𝛼5 NOUN cana-1091 103 12 , , PUNCT cana-1091 103 13 𝛽5 𝛽5 NOUN cana-1091 103 14 , , PUNCT cana-1091 103 15 𝛾5 𝛾5 PROPN cana-1091 103 16 ∈ ∈ PROPN cana-1091 103 17 𝒒 𝒒 PROPN cana-1091 103 18 , , PUNCT cana-1091 103 19 𝑇𝐢((𝛼5 𝑇𝐢((𝛼5 NUM cana-1091 103 20 β†’ β†’ SYM cana-1091 103 21 𝛽5 𝛽5 NUM cana-1091 103 22 ) ) PUNCT cana-1091 103 23 β†’ β†’ SYM cana-1091 103 24 ( ( PUNCT cana-1091 103 25 𝛼5 𝛼5 NOUN cana-1091 103 26 β†’ β†’ SYM cana-1091 103 27 𝛾5 𝛾5 NOUN cana-1091 103 28 ) ) PUNCT cana-1091 103 29 ) ) PUNCT cana-1091 104 1 β‰₯ β‰₯ NOUN cana-1091 104 2 min{𝑇𝐢((𝛼5 min{𝑇𝐢((𝛼5 NOUN cana-1091 104 3 β†’ β†’ SYM cana-1091 104 4 𝛽5 𝛽5 PROPN cana-1091 104 5 ) ) PUNCT cana-1091 104 6 β†’ β†’ SYM cana-1091 104 7 ( ( PUNCT cana-1091 104 8 𝛽5 𝛽5 PROPN cana-1091 104 9 β†’ β†’ SYM cana-1091 104 10 𝛾5 𝛾5 NOUN cana-1091 104 11 ) ) PUNCT cana-1091 104 12 , , PUNCT cana-1091 104 13 𝑇𝐢((𝛼5 𝑇𝐢((𝛼5 NUM cana-1091 104 14 β†’ β†’ SYM cana-1091 104 15 𝛽5 𝛽5 NUM cana-1091 104 16 ) ) PUNCT cana-1091 104 17 } } PUNCT cana-1091 104 18 , , PUNCT cana-1091 104 19 𝐼𝐢((𝛼5 𝐼𝐢((𝛼5 NOUN cana-1091 104 20 β†’ β†’ SYM cana-1091 104 21 𝛽5 𝛽5 NUM cana-1091 104 22 ) ) PUNCT cana-1091 104 23 β†’ β†’ SYM cana-1091 104 24 ( ( PUNCT cana-1091 104 25 𝛼5 𝛼5 NOUN cana-1091 104 26 β†’ β†’ SYM cana-1091 104 27 𝛾5 𝛾5 NOUN cana-1091 104 28 ) ) PUNCT cana-1091 104 29 ) ) PUNCT cana-1091 105 1 ≀ ≀ NUM cana-1091 105 2 min{𝐼𝐢((𝛼5 min{𝐼𝐢((𝛼5 NOUN cana-1091 105 3 β†’ β†’ SYM cana-1091 105 4 𝛽5 𝛽5 PROPN cana-1091 105 5 ) ) PUNCT cana-1091 105 6 β†’ β†’ SYM cana-1091 105 7 ( ( PUNCT cana-1091 105 8 𝛽5 𝛽5 PROPN cana-1091 105 9 β†’ β†’ SYM cana-1091 105 10 𝛾5 𝛾5 NOUN cana-1091 105 11 ) ) PUNCT cana-1091 105 12 , , PUNCT cana-1091 105 13 𝐼𝐢((𝛼5 𝐼𝐢((𝛼5 NOUN cana-1091 105 14 β†’ β†’ SYM cana-1091 105 15 𝛽5 𝛽5 NUM cana-1091 105 16 ) ) PUNCT cana-1091 105 17 } } PUNCT cana-1091 105 18 𝐹𝐢((𝛼5 𝐹𝐢((𝛼5 NUM cana-1091 105 19 β†’ β†’ SYM cana-1091 105 20 𝛽5 𝛽5 NUM cana-1091 105 21 ) ) PUNCT cana-1091 105 22 β†’ β†’ SYM cana-1091 105 23 ( ( PUNCT cana-1091 105 24 𝛼5 𝛼5 NOUN cana-1091 105 25 β†’ β†’ SYM cana-1091 105 26 𝛾5 𝛾5 NOUN cana-1091 105 27 ) ) PUNCT cana-1091 105 28 ) ) PUNCT cana-1091 106 1 ≀ ≀ NUM cana-1091 106 2 min{𝐹𝐢((𝛼5 min{𝐹𝐢((𝛼5 NOUN cana-1091 106 3 β†’ β†’ SYM cana-1091 106 4 𝛽5 𝛽5 NUM cana-1091 106 5 ) ) PUNCT cana-1091 106 6 β†’ β†’ SYM cana-1091 106 7 ( ( PUNCT cana-1091 106 8 𝛽5 𝛽5 PROPN cana-1091 106 9 β†’ β†’ SYM cana-1091 106 10 𝛾5 𝛾5 NOUN cana-1091 106 11 ) ) PUNCT cana-1091 106 12 , , PUNCT cana-1091 106 13 𝐹𝐢((𝛼5 𝐹𝐢((𝛼5 NUM cana-1091 106 14 β†’ β†’ SYM cana-1091 106 15 𝛽5 𝛽5 NUM cana-1091 106 16 ) ) PUNCT cana-1091 106 17 } } PUNCT cana-1091 106 18 for for ADP cana-1091 106 19 all all DET cana-1091 106 20 𝛼5 𝛼5 NOUN cana-1091 106 21 , , PUNCT cana-1091 106 22 𝛽5 𝛽5 NOUN cana-1091 106 23 , , PUNCT cana-1091 106 24 𝛾5 𝛾5 NOUN cana-1091 106 25 ∈ ∈ PROPN cana-1091 106 26 𝒒. 𝒒. NOUN cana-1091 106 27 proof proof NOUN cana-1091 106 28 : : PUNCT cana-1091 106 29 let let VERB cana-1091 106 30 𝐢 𝐢 PROPN cana-1091 106 31 be be AUX cana-1091 106 32 a a DET cana-1091 106 33 neutrosophic neutrosophic ADJ cana-1091 106 34 filter filter NOUN cana-1091 106 35 of of ADP cana-1091 106 36 a a DET cana-1091 106 37 bl bl NOUN cana-1091 106 38 - - PUNCT cana-1091 106 39 algebra algebra NOUN cana-1091 106 40 𝒒. 𝒒. PROPN cana-1091 106 41 assume assume VERB cana-1091 106 42 that that SCONJ cana-1091 106 43 𝐢 𝐢 PROPN cana-1091 106 44 is be AUX cana-1091 106 45 a a DET cana-1091 106 46 neutrosophic neutrosophic ADJ cana-1091 106 47 transitive transitive ADJ cana-1091 106 48 filter filter NOUN cana-1091 106 49 of of ADP cana-1091 106 50 𝒒. 𝒒. PROPN cana-1091 106 51 𝑇𝐢((𝛼5 𝑇𝐢((𝛼5 PROPN cana-1091 106 52 β†’ β†’ SYM cana-1091 106 53 𝛽5 𝛽5 NUM cana-1091 106 54 ) ) PUNCT cana-1091 106 55 β†’ β†’ SYM cana-1091 106 56 ( ( PUNCT cana-1091 106 57 𝛼5 𝛼5 NOUN cana-1091 106 58 β†’ β†’ SYM cana-1091 106 59 𝛾5 𝛾5 NOUN cana-1091 106 60 ) ) PUNCT cana-1091 106 61 ) ) PUNCT cana-1091 107 1 β‰₯ β‰₯ NOUN cana-1091 107 2 min{𝑇𝐢((𝛼5 min{𝑇𝐢((𝛼5 NOUN cana-1091 107 3 β†’ β†’ SYM cana-1091 107 4 𝛽5 𝛽5 PROPN cana-1091 107 5 ) ) PUNCT cana-1091 107 6 β†’ β†’ SYM cana-1091 107 7 ( ( PUNCT cana-1091 107 8 𝛽5 𝛽5 PROPN cana-1091 107 9 β†’ β†’ SYM cana-1091 107 10 𝛾5 𝛾5 NOUN cana-1091 107 11 ) ) PUNCT cana-1091 107 12 ) ) PUNCT cana-1091 107 13 , , PUNCT cana-1091 107 14 𝑇𝐢((𝛽5 𝑇𝐢((𝛽5 NUM cana-1091 107 15 β†’ β†’ SYM cana-1091 107 16 𝛾5 𝛾5 NOUN cana-1091 107 17 ) ) PUNCT cana-1091 107 18 β†’ β†’ SYM cana-1091 107 19 ( ( PUNCT cana-1091 107 20 𝛼5 𝛼5 NOUN cana-1091 107 21 β†’ β†’ SYM cana-1091 107 22 𝛾5 𝛾5 NOUN cana-1091 107 23 ) ) PUNCT cana-1091 107 24 ) ) PUNCT cana-1091 107 25 } } PUNCT cana-1091 107 26 , , PUNCT cana-1091 107 27 β‰₯ β‰₯ X cana-1091 107 28 min{𝑇𝐢((𝛼5 min{𝑇𝐢((𝛼5 NOUN cana-1091 107 29 β†’ β†’ SYM cana-1091 107 30 𝛽5 𝛽5 PROPN cana-1091 107 31 ) ) PUNCT cana-1091 107 32 β†’ β†’ SYM cana-1091 107 33 ( ( PUNCT cana-1091 107 34 𝛽5 𝛽5 PROPN cana-1091 107 35 β†’ β†’ SYM cana-1091 107 36 𝛾5 𝛾5 NOUN cana-1091 107 37 ) ) PUNCT cana-1091 107 38 ) ) PUNCT cana-1091 107 39 , , PUNCT cana-1091 108 1 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 PROPN cana-1091 108 2 β†’ β†’ SYM cana-1091 108 3 𝛽5 𝛽5 NUM cana-1091 108 4 ) ) PUNCT cana-1091 108 5 } } PUNCT cana-1091 108 6 , , PUNCT cana-1091 108 7 similarly similarly ADV cana-1091 108 8 , , PUNCT cana-1091 108 9 we we PRON cana-1091 108 10 can can AUX cana-1091 108 11 prove prove VERB cana-1091 108 12 for for ADP cana-1091 108 13 𝐼𝐢 𝐼𝐢 PROPN cana-1091 108 14 , , PUNCT cana-1091 108 15 𝐹𝐢 𝐹𝐢 PROPN cana-1091 108 16 . . PUNCT cana-1091 109 1 conversely conversely ADV cana-1091 109 2 , , PUNCT cana-1091 109 3 consider consider VERB cana-1091 109 4 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 PRON cana-1091 109 5 β†’ β†’ SYM cana-1091 109 6 𝛾5 𝛾5 NOUN cana-1091 109 7 ) ) PUNCT cana-1091 109 8 β‰₯ β‰₯ NOUN cana-1091 109 9 𝑇𝐢((𝛼5 𝑇𝐢((𝛼5 NUM cana-1091 109 10 β†’ β†’ SYM cana-1091 109 11 𝛽5 𝛽5 NUM cana-1091 109 12 ) ) PUNCT cana-1091 109 13 β†’ β†’ SYM cana-1091 109 14 ( ( PUNCT cana-1091 109 15 𝛼5 𝛼5 NOUN cana-1091 109 16 β†’ β†’ SYM cana-1091 109 17 𝛾5 𝛾5 NOUN cana-1091 109 18 ) ) PUNCT cana-1091 109 19 ) ) PUNCT cana-1091 109 20 β‰₯ β‰₯ PROPN cana-1091 109 21 min min PROPN cana-1091 109 22 { { PUNCT cana-1091 109 23 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 PROPN cana-1091 109 24 β†’ β†’ SYM cana-1091 109 25 𝛽5 𝛽5 PROPN cana-1091 109 26 ) ) PUNCT cana-1091 109 27 , , PUNCT cana-1091 109 28 𝑇𝐢((𝛼5 𝑇𝐢((𝛼5 NUM cana-1091 109 29 β†’ β†’ SYM cana-1091 109 30 𝛽5 𝛽5 NUM cana-1091 109 31 ) ) PUNCT cana-1091 109 32 β†’ β†’ SYM cana-1091 109 33 ( ( PUNCT cana-1091 109 34 𝛽5 𝛽5 PROPN cana-1091 109 35 β†’ β†’ SYM cana-1091 109 36 𝛾5 𝛾5 NOUN cana-1091 109 37 ) ) PUNCT cana-1091 109 38 ) ) PUNCT cana-1091 109 39 } } PUNCT cana-1091 109 40 β‰₯ β‰₯ NOUN cana-1091 109 41 min min NOUN cana-1091 109 42 { { PUNCT cana-1091 109 43 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 PROPN cana-1091 109 44 β†’ β†’ SYM cana-1091 109 45 𝛽5 𝛽5 NUM cana-1091 109 46 ) ) PUNCT cana-1091 109 47 , , PUNCT cana-1091 109 48 𝑇𝐢(𝛽5 𝑇𝐢(𝛽5 PROPN cana-1091 109 49 β†’ β†’ SYM cana-1091 109 50 𝛾5)}for 𝛾5)}for ADP cana-1091 109 51 all all DET cana-1091 109 52 𝛼5 𝛼5 NOUN cana-1091 109 53 , , PUNCT cana-1091 109 54 𝛽5 𝛽5 NOUN cana-1091 109 55 , , PUNCT cana-1091 109 56 𝛾5 𝛾5 PROPN cana-1091 109 57 ∈ ∈ PROPN cana-1091 109 58 𝒒. 𝒒. PROPN cana-1091 109 59 similarly similarly ADV cana-1091 109 60 , , PUNCT cana-1091 109 61 we we PRON cana-1091 109 62 can can AUX cana-1091 109 63 prove prove VERB cana-1091 109 64 for for ADP cana-1091 109 65 𝐼𝐢 𝐼𝐢 PROPN cana-1091 109 66 , , PUNCT cana-1091 109 67 𝐹𝐢 𝐹𝐢 PROPN cana-1091 109 68 . . PUNCT cana-1091 110 1 hence hence ADV cana-1091 110 2 , , PUNCT cana-1091 110 3 𝐢 𝐢 PROPN cana-1091 110 4 is be AUX cana-1091 110 5 a a DET cana-1091 110 6 neutrosophic neutrosophic ADJ cana-1091 110 7 transitive transitive ADJ cana-1091 110 8 filter filter NOUN cana-1091 110 9 of𝒒. of𝒒. ADP cana-1091 110 10 proposition proposition NOUN cana-1091 110 11 3.6 3.6 NUM cana-1091 110 12 let let VERB cana-1091 110 13 𝐢 𝐢 PROPN cana-1091 110 14 be be AUX cana-1091 110 15 a a DET cana-1091 110 16 neutrosophic neutrosophic ADJ cana-1091 110 17 filterof𝒒 filterof𝒒 PROPN cana-1091 110 18 . . PUNCT cana-1091 111 1 then then ADV cana-1091 111 2 , , PUNCT cana-1091 111 3 𝐢 𝐢 PROPN cana-1091 111 4 is be AUX cana-1091 111 5 a a DET cana-1091 111 6 neutrosophic neutrosophic ADJ cana-1091 111 7 transitive transitive ADJ cana-1091 111 8 filter filter NOUN cana-1091 111 9 of of ADP cana-1091 111 10 𝒒 𝒒 PROPN cana-1091 111 11 if if SCONJ cana-1091 111 12 it it PRON cana-1091 111 13 satisfies satisfy VERB cana-1091 111 14 𝑇𝐢((𝛼5 𝑇𝐢((𝛼5 NUM cana-1091 111 15 β†’ β†’ SYM cana-1091 111 16 𝛽5 𝛽5 NUM cana-1091 111 17 ) ) PUNCT cana-1091 111 18 β†’ β†’ SYM cana-1091 111 19 𝛾5 𝛾5 NOUN cana-1091 111 20 ) ) PUNCT cana-1091 111 21 β‰₯ β‰₯ NOUN cana-1091 111 22 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 NUM cana-1091 111 23 β†’ β†’ PUNCT cana-1091 111 24 ( ( PUNCT cana-1091 111 25 𝛽5 𝛽5 PROPN cana-1091 111 26 β†’ β†’ SYM cana-1091 111 27 𝛾5 𝛾5 NOUN cana-1091 111 28 ) ) PUNCT cana-1091 111 29 ) ) PUNCT cana-1091 111 30 , , PUNCT cana-1091 111 31 𝐼𝐢((𝛼5 𝐼𝐢((𝛼5 NOUN cana-1091 111 32 β†’ β†’ SYM cana-1091 111 33 𝛽5 𝛽5 NUM cana-1091 111 34 ) ) PUNCT cana-1091 111 35 β†’ β†’ SYM cana-1091 111 36 𝛾5 𝛾5 NOUN cana-1091 111 37 ) ) PUNCT cana-1091 111 38 ≀ ≀ NOUN cana-1091 112 1 𝐼𝐢(𝛼5 𝐼𝐢(𝛼5 ADP cana-1091 112 2 β†’ β†’ PUNCT cana-1091 112 3 ( ( PUNCT cana-1091 112 4 𝛽5 𝛽5 NOUN cana-1091 112 5 β†’ β†’ SYM cana-1091 112 6 𝛾5)),𝐹𝐢((𝛼5 𝛾5)),𝐹𝐢((𝛼5 ADJ cana-1091 112 7 β†’ β†’ SYM cana-1091 112 8 𝛽5 𝛽5 NUM cana-1091 112 9 ) ) PUNCT cana-1091 112 10 β†’ β†’ SYM cana-1091 112 11 𝛾5 𝛾5 NOUN cana-1091 112 12 ) ) PUNCT cana-1091 112 13 ≀ ≀ NOUN cana-1091 113 1 𝐹𝐢(𝛼5 𝐹𝐢(𝛼5 PROPN cana-1091 113 2 β†’ β†’ SYM cana-1091 113 3 ( ( PUNCT cana-1091 113 4 𝛽5 𝛽5 PROPN cana-1091 113 5 β†’ β†’ SYM cana-1091 113 6 𝛾5 𝛾5 NOUN cana-1091 113 7 ) ) PUNCT cana-1091 113 8 ) ) PUNCT cana-1091 113 9 for for ADP cana-1091 113 10 all all DET cana-1091 113 11 𝛼5 𝛼5 NOUN cana-1091 113 12 , , PUNCT cana-1091 113 13 𝛽5 𝛽5 NOUN cana-1091 113 14 , , PUNCT cana-1091 113 15 𝛾5 𝛾5 PROPN cana-1091 113 16 ∈ ∈ PROPN cana-1091 113 17 𝒒. 𝒒. PROPN cana-1091 113 18 every every DET cana-1091 113 19 neutrosophic neutrosophic ADJ cana-1091 113 20 transitive transitive ADJ cana-1091 113 21 filter filter NOUN cana-1091 113 22 of of ADP cana-1091 113 23 a a DET cana-1091 113 24 bl bl NOUN cana-1091 113 25 - - PUNCT cana-1091 113 26 algebra𝒒 algebra𝒒 PROPN cana-1091 113 27 is be AUX cana-1091 113 28 an an DET cana-1091 113 29 associative associative ADJ cana-1091 113 30 filter filter NOUN cana-1091 113 31 . . PUNCT cana-1091 114 1 proof proof NOUN cana-1091 114 2 : : PUNCT cana-1091 114 3 let𝐢 let𝐢 NOUN cana-1091 114 4 be be AUX cana-1091 114 5 a a DET cana-1091 114 6 neutrosophicfilter neutrosophicfilter NOUN cana-1091 114 7 of of ADP cana-1091 114 8 𝒒 𝒒 NOUN cana-1091 114 9 satisfying satisfy VERB cana-1091 114 10 the the DET cana-1091 114 11 given give VERB cana-1091 114 12 condition condition NOUN cana-1091 114 13 . . PUNCT cana-1091 115 1 then then ADV cana-1091 115 2 , , PUNCT cana-1091 115 3 min{𝑇𝐢(𝛼5 min{𝑇𝐢(𝛼5 NOUN cana-1091 115 4 β†’ β†’ SYM cana-1091 115 5 𝛽5 𝛽5 NUM cana-1091 115 6 ) ) PUNCT cana-1091 115 7 , , PUNCT cana-1091 115 8 𝑇𝐢(𝛽5 𝑇𝐢(𝛽5 PROPN cana-1091 115 9 β†’ β†’ SYM cana-1091 115 10 𝛾5 𝛾5 NOUN cana-1091 115 11 ) ) PUNCT cana-1091 115 12 } } PUNCT cana-1091 115 13 ≀ ≀ NUM cana-1091 115 14 min min NOUN cana-1091 115 15 { { PUNCT cana-1091 115 16 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 PROPN cana-1091 115 17 β†’ β†’ SYM cana-1091 115 18 𝛽5 𝛽5 NUM cana-1091 115 19 ) ) PUNCT cana-1091 115 20 , , PUNCT cana-1091 115 21 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 PROPN cana-1091 115 22 β†’ β†’ PUNCT cana-1091 115 23 ( ( PUNCT cana-1091 115 24 𝛽5 𝛽5 PROPN cana-1091 115 25 β†’ β†’ SYM cana-1091 115 26 𝛾5 𝛾5 NOUN cana-1091 115 27 ) ) PUNCT cana-1091 115 28 ) ) PUNCT cana-1091 115 29 } } PUNCT cana-1091 115 30 ≀ ≀ NUM cana-1091 115 31 min min NOUN cana-1091 115 32 { { PUNCT cana-1091 115 33 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 PROPN cana-1091 115 34 β†’ β†’ SYM cana-1091 115 35 𝛽5 𝛽5 PROPN cana-1091 115 36 ) ) PUNCT cana-1091 115 37 , , PUNCT cana-1091 115 38 𝑇𝐢((𝛼5 𝑇𝐢((𝛼5 NUM cana-1091 115 39 β†’ β†’ SYM cana-1091 115 40 𝛽5 𝛽5 NUM cana-1091 115 41 ) ) PUNCT cana-1091 115 42 β†’ β†’ SYM cana-1091 115 43 𝛾5 𝛾5 NOUN cana-1091 115 44 ) ) PUNCT cana-1091 115 45 } } PUNCT cana-1091 115 46 ≀ ≀ NOUN cana-1091 115 47 𝑇𝐢(𝛾5 𝑇𝐢(𝛾5 PRON cana-1091 115 48 ) ) PUNCT cana-1091 115 49 ≀ ≀ PUNCT cana-1091 115 50 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 PUNCT cana-1091 115 51 β†’ β†’ SYM cana-1091 115 52 𝛾5 𝛾5 NOUN cana-1091 115 53 ) ) PUNCT cana-1091 115 54 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 PUNCT cana-1091 115 55 β†’ β†’ SYM cana-1091 115 56 𝛾5 𝛾5 NOUN cana-1091 115 57 ) ) PUNCT cana-1091 115 58 β‰₯ β‰₯ NOUN cana-1091 115 59 min{𝑇𝐢(𝛼5 min{𝑇𝐢(𝛼5 ADV cana-1091 115 60 β†’ β†’ SYM cana-1091 115 61 𝛽5 𝛽5 NUM cana-1091 115 62 ) ) PUNCT cana-1091 115 63 , , PUNCT cana-1091 115 64 𝑇𝐢(𝛽5 𝑇𝐢(𝛽5 PROPN cana-1091 115 65 β†’ β†’ SYM cana-1091 115 66 𝛾5 𝛾5 NOUN cana-1091 115 67 ) ) PUNCT cana-1091 115 68 } } PUNCT cana-1091 115 69 similarly similarly ADV cana-1091 115 70 , , PUNCT cana-1091 115 71 we we PRON cana-1091 115 72 can can AUX cana-1091 115 73 prove prove VERB cana-1091 115 74 for for ADP cana-1091 115 75 𝐼𝐢 𝐼𝐢 PROPN cana-1091 115 76 , , PUNCT cana-1091 115 77 𝐹𝐢 𝐹𝐢 PROPN cana-1091 115 78 . . PUNCT cana-1091 116 1 hence hence ADV cana-1091 116 2 , , PUNCT cana-1091 116 3 𝐢 𝐢 PROPN cana-1091 116 4 is be AUX cana-1091 116 5 a a DET cana-1091 116 6 neutrosophic neutrosophic ADJ cana-1091 116 7 transitive transitive ADJ cana-1091 116 8 filter filter NOUN cana-1091 116 9 of𝒒. of𝒒. ADP cana-1091 116 10 communications communication NOUN cana-1091 116 11 on on ADP cana-1091 116 12 applied apply VERB cana-1091 116 13 nonlinear nonlinear ADJ cana-1091 116 14 analysis analysis NOUN cana-1091 116 15 issn issn NOUN cana-1091 116 16 : : PUNCT cana-1091 116 17 1074 1074 NUM cana-1091 116 18 - - PUNCT cana-1091 116 19 133x 133x NUM cana-1091 116 20 vol vol NOUN cana-1091 116 21 31 31 NUM cana-1091 116 22 no no NOUN cana-1091 116 23 . . PUNCT cana-1091 117 1 5s 5s NUM cana-1091 117 2 ( ( PUNCT cana-1091 117 3 2024 2024 NUM cana-1091 117 4 ) ) PUNCT cana-1091 117 5 567 567 NUM cana-1091 117 6 https://internationalpubls.com https://internationalpubls.com X cana-1091 117 7 corollary corollary NOUN cana-1091 117 8 3.7 3.7 NUM cana-1091 117 9 let let VERB cana-1091 117 10 𝐢 𝐢 PROPN cana-1091 117 11 be be AUX cana-1091 117 12 a a DET cana-1091 117 13 neutrosophic neutrosophic ADJ cana-1091 117 14 filter filter NOUN cana-1091 117 15 of of ADP cana-1091 117 16 a a DET cana-1091 117 17 bl bl NOUN cana-1091 117 18 - - PUNCT cana-1091 117 19 algebra algebra NOUN cana-1091 117 20 𝒒. 𝒒. PROPN cana-1091 117 21 if if SCONJ cana-1091 117 22 𝐢 𝐢 PROPN cana-1091 117 23 is be AUX cana-1091 117 24 a a DET cana-1091 117 25 neutrosophic neutrosophic ADJ cana-1091 117 26 associative associative ADJ cana-1091 117 27 filter filter NOUN cana-1091 117 28 , , PUNCT cana-1091 117 29 then then ADV cana-1091 117 30 it it PRON cana-1091 117 31 is be AUX cana-1091 117 32 a a DET cana-1091 117 33 neutrosophic neutrosophic ADJ cana-1091 117 34 transitive transitive ADJ cana-1091 117 35 filter filter NOUN cana-1091 117 36 . . PUNCT cana-1091 118 1 proof proof NOUN cana-1091 118 2 : : PUNCT cana-1091 118 3 by by ADP cana-1091 118 4 the the DET cana-1091 118 5 proposition proposition NOUN cana-1091 118 6 3.6 3.6 NUM cana-1091 118 7 and and CCONJ cana-1091 118 8 2.11 2.11 NUM cana-1091 118 9 , , PUNCT cana-1091 118 10 the the DET cana-1091 118 11 proof proof NOUN cana-1091 118 12 is be AUX cana-1091 118 13 obvious obvious ADJ cana-1091 118 14 . . PUNCT cana-1091 119 1 proposition proposition NOUN cana-1091 119 2 3.8 3.8 NUM cana-1091 119 3 each each DET cana-1091 119 4 neutrosophic neutrosophic ADJ cana-1091 119 5 filter filter NOUN cana-1091 119 6 𝐢 𝐢 PROPN cana-1091 119 7 of of ADP cana-1091 119 8 a a DET cana-1091 119 9 bl bl NOUN cana-1091 119 10 - - PUNCT cana-1091 119 11 algebra algebra NOUN cana-1091 119 12 𝒒 𝒒 PROPN cana-1091 119 13 is be AUX cana-1091 119 14 a a DET cana-1091 119 15 neutrosophic neutrosophic ADJ cana-1091 119 16 transitive transitive ADJ cana-1091 119 17 filter filter NOUN cana-1091 119 18 if if SCONJ cana-1091 119 19 it it PRON cana-1091 119 20 satisfies𝑇𝐢(𝛾5 satisfies𝑇𝐢(𝛾5 VERB cana-1091 119 21 ) ) PUNCT cana-1091 119 22 β‰₯ β‰₯ NOUN cana-1091 119 23 min{𝑇𝐢(𝛼5 min{𝑇𝐢(𝛼5 ADV cana-1091 119 24 β†’ β†’ SYM cana-1091 119 25 𝛽5 𝛽5 NUM cana-1091 119 26 ) ) PUNCT cana-1091 119 27 , , PUNCT cana-1091 119 28 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 PROPN cana-1091 120 1 β†’ β†’ PUNCT cana-1091 120 2 ( ( PUNCT cana-1091 120 3 𝛽5 𝛽5 PROPN cana-1091 120 4 β†’ β†’ SYM cana-1091 120 5 𝛾5 𝛾5 NOUN cana-1091 120 6 ) ) PUNCT cana-1091 120 7 ) ) PUNCT cana-1091 120 8 } } PUNCT cana-1091 121 1 𝐼𝐢(𝛾5 𝐼𝐢(𝛾5 X cana-1091 121 2 ) ) PUNCT cana-1091 121 3 ≀ ≀ ADV cana-1091 121 4 min{𝐼𝐢(𝛼5 min{𝐼𝐢(𝛼5 INTJ cana-1091 121 5 β†’ β†’ SYM cana-1091 121 6 𝛽5 𝛽5 NUM cana-1091 121 7 ) ) PUNCT cana-1091 121 8 , , PUNCT cana-1091 121 9 𝐼𝐢(𝛼5 𝐼𝐢(𝛼5 PROPN cana-1091 121 10 β†’ β†’ PUNCT cana-1091 121 11 ( ( PUNCT cana-1091 121 12 𝛽5 𝛽5 ADJ cana-1091 121 13 β†’ β†’ SYM cana-1091 121 14 𝛾5 𝛾5 NOUN cana-1091 121 15 ) ) PUNCT cana-1091 121 16 ) ) PUNCT cana-1091 121 17 } } PUNCT cana-1091 121 18 𝐹𝐢(𝛾5 𝐹𝐢(𝛾5 X cana-1091 121 19 ) ) PUNCT cana-1091 121 20 ≀ ≀ ADJ cana-1091 121 21 min{𝐹𝐢(𝛼5 min{𝐹𝐢(𝛼5 NOUN cana-1091 121 22 β†’ β†’ SYM cana-1091 121 23 𝛽5 𝛽5 NUM cana-1091 121 24 ) ) PUNCT cana-1091 121 25 , , PUNCT cana-1091 121 26 𝐹𝐢(𝛼5 𝐹𝐢(𝛼5 PROPN cana-1091 121 27 β†’ β†’ SYM cana-1091 121 28 ( ( PUNCT cana-1091 121 29 𝛽5 𝛽5 NOUN cana-1091 121 30 β†’ β†’ SYM cana-1091 121 31 𝛾5))}for 𝛾5))}for ADP cana-1091 121 32 all all DET cana-1091 121 33 𝛼5 𝛼5 NOUN cana-1091 121 34 , , PUNCT cana-1091 121 35 𝛽5 𝛽5 NOUN cana-1091 121 36 , , PUNCT cana-1091 121 37 𝛾5 𝛾5 NOUN cana-1091 121 38 ∈ ∈ PROPN cana-1091 121 39 𝒒. 𝒒. NOUN cana-1091 121 40 proof proof NOUN cana-1091 121 41 : : PUNCT cana-1091 121 42 let𝐢 let𝐢 NOUN cana-1091 121 43 be be AUX cana-1091 121 44 a a DET cana-1091 121 45 neutrosophic neutrosophic ADJ cana-1091 121 46 filter filter NOUN cana-1091 121 47 of of ADP cana-1091 121 48 a a DET cana-1091 121 49 bl bl NOUN cana-1091 121 50 - - PUNCT cana-1091 121 51 algebra algebra NOUN cana-1091 121 52 𝒒. 𝒒. PROPN cana-1091 121 53 assume assume VERB cana-1091 121 54 that that SCONJ cana-1091 121 55 𝑇𝐢(𝛾5 𝑇𝐢(𝛾5 PROPN cana-1091 121 56 ) ) PUNCT cana-1091 121 57 β‰₯ β‰₯ NOUN cana-1091 121 58 min{𝑇𝐢(𝛼5 min{𝑇𝐢(𝛼5 ADV cana-1091 121 59 β†’ β†’ SYM cana-1091 121 60 𝛽5 𝛽5 NUM cana-1091 121 61 ) ) PUNCT cana-1091 121 62 , , PUNCT cana-1091 122 1 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 PROPN cana-1091 122 2 β†’ β†’ PUNCT cana-1091 122 3 ( ( PUNCT cana-1091 122 4 𝛽5 𝛽5 PROPN cana-1091 122 5 β†’ β†’ SYM cana-1091 122 6 𝛾5 𝛾5 NOUN cana-1091 122 7 ) ) PUNCT cana-1091 122 8 ) ) PUNCT cana-1091 122 9 } } PUNCT cana-1091 122 10 . . PUNCT cana-1091 123 1 consider consider VERB cana-1091 123 2 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 PRON cana-1091 123 3 β†’ β†’ SYM cana-1091 123 4 𝛾5 𝛾5 NOUN cana-1091 123 5 ) ) PUNCT cana-1091 123 6 β‰₯ β‰₯ NOUN cana-1091 123 7 𝑇𝐢(𝛾5 𝑇𝐢(𝛾5 NOUN cana-1091 123 8 ) ) PUNCT cana-1091 123 9 β‰₯ β‰₯ PROPN cana-1091 123 10 min min PROPN cana-1091 123 11 { { PUNCT cana-1091 123 12 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 PROPN cana-1091 123 13 β†’ β†’ SYM cana-1091 123 14 𝛽5 𝛽5 PROPN cana-1091 123 15 ) ) PUNCT cana-1091 123 16 , , PUNCT cana-1091 123 17 𝑇𝐢((𝛼5 𝑇𝐢((𝛼5 NUM cana-1091 123 18 β†’ β†’ SYM cana-1091 123 19 𝛽5 𝛽5 NUM cana-1091 123 20 ) ) PUNCT cana-1091 123 21 β†’ β†’ SYM cana-1091 123 22 𝛾5 𝛾5 NOUN cana-1091 123 23 ) ) PUNCT cana-1091 123 24 } } PUNCT cana-1091 123 25 β‰₯ β‰₯ PROPN cana-1091 123 26 min min X cana-1091 123 27 { { PUNCT cana-1091 123 28 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 PROPN cana-1091 123 29 β†’ β†’ SYM cana-1091 123 30 𝛽5 𝛽5 NUM cana-1091 123 31 ) ) PUNCT cana-1091 123 32 , , PUNCT cana-1091 123 33 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 PROPN cana-1091 123 34 β†’ β†’ PUNCT cana-1091 123 35 ( ( PUNCT cana-1091 123 36 𝛽5 𝛽5 PROPN cana-1091 123 37 β†’ β†’ SYM cana-1091 123 38 𝛾5 𝛾5 NOUN cana-1091 123 39 ) ) PUNCT cana-1091 123 40 ) ) PUNCT cana-1091 123 41 } } PUNCT cana-1091 123 42 β‰₯ β‰₯ X cana-1091 123 43 min{𝑇𝐢(𝛼5 min{𝑇𝐢(𝛼5 ADV cana-1091 123 44 β†’ β†’ SYM cana-1091 123 45 𝛽5 𝛽5 NUM cana-1091 123 46 ) ) PUNCT cana-1091 123 47 , , PUNCT cana-1091 123 48 𝑇𝐢(𝛽5 𝑇𝐢(𝛽5 PROPN cana-1091 123 49 β†’ β†’ SYM cana-1091 123 50 𝛾5 𝛾5 NOUN cana-1091 123 51 ) ) PUNCT cana-1091 123 52 } } PUNCT cana-1091 123 53 similarly similarly ADV cana-1091 123 54 , , PUNCT cana-1091 123 55 we we PRON cana-1091 123 56 can can AUX cana-1091 123 57 prove prove VERB cana-1091 123 58 for for ADP cana-1091 123 59 𝐼𝐢 𝐼𝐢 PROPN cana-1091 123 60 , , PUNCT cana-1091 123 61 𝐹𝐢 𝐹𝐢 PROPN cana-1091 123 62 . . PUNCT cana-1091 124 1 hence hence ADV cana-1091 124 2 , , PUNCT cana-1091 124 3 𝐢 𝐢 PROPN cana-1091 124 4 is be AUX cana-1091 124 5 a a DET cana-1091 124 6 neutrosophic neutrosophic ADJ cana-1091 124 7 transitive transitive ADJ cana-1091 124 8 filter filter NOUN cana-1091 124 9 of𝒒. of𝒒. ADP cana-1091 124 10 proposition proposition NOUN cana-1091 124 11 3.9 3.9 NUM cana-1091 124 12 let let VERB cana-1091 124 13 𝐢 𝐢 PROPN cana-1091 124 14 and and CCONJ cana-1091 124 15 𝐷 𝐷 PROPN cana-1091 124 16 be be VERB cana-1091 124 17 two two NUM cana-1091 124 18 neutrosophic neutrosophic ADJ cana-1091 124 19 filters filter NOUN cana-1091 124 20 of of ADP cana-1091 124 21 𝒒 𝒒 PROPN cana-1091 124 22 and and CCONJ cana-1091 124 23 𝐢 𝐢 PROPN cana-1091 124 24 be be VERB cana-1091 124 25 a a DET cana-1091 124 26 neutrosophic neutrosophic ADJ cana-1091 124 27 transitive transitive ADJ cana-1091 124 28 filter filter NOUN cana-1091 124 29 of of ADP cana-1091 124 30 𝒒 𝒒 PROPN cana-1091 124 31 . . PUNCT cana-1091 125 1 if if SCONJ cana-1091 125 2 𝐢 𝐢 PROPN cana-1091 125 3 βŠ† βŠ† NUM cana-1091 125 4 𝐷 𝐷 PROPN cana-1091 125 5 , , PUNCT cana-1091 125 6 then then ADV cana-1091 125 7 𝑇𝐢(1 𝑇𝐢(1 PROPN cana-1091 125 8 ) ) PUNCT cana-1091 125 9 = = SYM cana-1091 125 10 𝑇𝐷(1 𝑇𝐷(1 PROPN cana-1091 125 11 ) ) PUNCT cana-1091 125 12 , , PUNCT cana-1091 125 13 𝐼𝐢(1 𝐼𝐢(1 PROPN cana-1091 125 14 ) ) PUNCT cana-1091 125 15 = = SYM cana-1091 125 16 𝐼𝐷(1 𝐼𝐷(1 PROPN cana-1091 125 17 ) ) PUNCT cana-1091 125 18 , , PUNCT cana-1091 125 19 𝐹𝐢(1 𝐹𝐢(1 PROPN cana-1091 125 20 ) ) PUNCT cana-1091 125 21 = = SYM cana-1091 125 22 𝐹𝐷(1 𝐹𝐷(1 PROPN cana-1091 125 23 ) ) PUNCT cana-1091 125 24 and and CCONJ cana-1091 125 25 𝐷 𝐷 PROPN cana-1091 125 26 is be AUX cana-1091 125 27 also also ADV cana-1091 125 28 a a DET cana-1091 125 29 neutrosophic neutrosophic ADJ cana-1091 125 30 transitive transitive ADJ cana-1091 125 31 filter filter NOUN cana-1091 125 32 . . PUNCT cana-1091 126 1 proof proof NOUN cana-1091 126 2 : : PUNCT cana-1091 126 3 let let VERB cana-1091 126 4 𝐢 𝐢 PROPN cana-1091 126 5 be be AUX cana-1091 126 6 a a DET cana-1091 126 7 neutrosophic neutrosophic ADJ cana-1091 126 8 transitive transitive ADJ cana-1091 126 9 filter filter NOUN cana-1091 126 10 of of ADP cana-1091 126 11 𝒒. 𝒒. PROPN cana-1091 126 12 𝑇𝐷((𝛽5 𝑇𝐷((𝛽5 PROPN cana-1091 126 13 β†’ β†’ SYM cana-1091 126 14 𝛾5 𝛾5 NOUN cana-1091 126 15 ) ) PUNCT cana-1091 126 16 β†’ β†’ SYM cana-1091 126 17 ( ( PUNCT cana-1091 126 18 ( ( PUNCT cana-1091 126 19 𝛼5 𝛼5 NOUN cana-1091 126 20 β†’ β†’ SYM cana-1091 126 21 𝛽5 𝛽5 NUM cana-1091 126 22 ) ) PUNCT cana-1091 126 23 β†’ β†’ SYM cana-1091 126 24 ( ( PUNCT cana-1091 126 25 𝛼5 𝛼5 NOUN cana-1091 126 26 β†’ β†’ SYM cana-1091 126 27 𝛾5 𝛾5 NOUN cana-1091 126 28 ) ) PUNCT cana-1091 126 29 ) ) PUNCT cana-1091 126 30 ) ) PUNCT cana-1091 127 1 β‰₯ β‰₯ NOUN cana-1091 128 1 𝑇𝐢((𝛽5 𝑇𝐢((𝛽5 NUM cana-1091 128 2 β†’ β†’ SYM cana-1091 128 3 𝛾5 𝛾5 NOUN cana-1091 128 4 ) ) PUNCT cana-1091 128 5 β†’ β†’ SYM cana-1091 128 6 ( ( PUNCT cana-1091 128 7 ( ( PUNCT cana-1091 128 8 𝛼5 𝛼5 NOUN cana-1091 128 9 β†’ β†’ SYM cana-1091 128 10 𝛽5 𝛽5 NUM cana-1091 128 11 ) ) PUNCT cana-1091 128 12 β†’ β†’ SYM cana-1091 128 13 ( ( PUNCT cana-1091 128 14 𝛼5 𝛼5 NOUN cana-1091 128 15 β†’ β†’ SYM cana-1091 128 16 𝛾5 𝛾5 NOUN cana-1091 128 17 ) ) PUNCT cana-1091 128 18 ) ) PUNCT cana-1091 128 19 ) ) PUNCT cana-1091 129 1 = = PUNCT cana-1091 129 2 𝑇𝐢((𝛼5 𝑇𝐢((𝛼5 NUM cana-1091 129 3 β†’ β†’ SYM cana-1091 129 4 𝛽5 𝛽5 NUM cana-1091 129 5 ) ) PUNCT cana-1091 129 6 β†’ β†’ SYM cana-1091 129 7 ( ( PUNCT cana-1091 129 8 ( ( PUNCT cana-1091 129 9 𝛽5 𝛽5 PROPN cana-1091 129 10 β†’ β†’ SYM cana-1091 129 11 𝛾5 𝛾5 NOUN cana-1091 129 12 ) ) PUNCT cana-1091 129 13 β†’ β†’ SYM cana-1091 129 14 ( ( PUNCT cana-1091 129 15 𝛼5 𝛼5 NOUN cana-1091 129 16 β†’ β†’ SYM cana-1091 129 17 𝛾5 𝛾5 NOUN cana-1091 129 18 ) ) PUNCT cana-1091 129 19 ) ) PUNCT cana-1091 129 20 ) ) PUNCT cana-1091 130 1 = = PUNCT cana-1091 130 2 𝑇𝐢((𝛼5 𝑇𝐢((𝛼5 NUM cana-1091 130 3 β†’ β†’ SYM cana-1091 130 4 𝛽5 𝛽5 NUM cana-1091 130 5 ) ) PUNCT cana-1091 130 6 β†’ β†’ SYM cana-1091 130 7 ( ( PUNCT cana-1091 130 8 𝛼5 𝛼5 NOUN cana-1091 130 9 β†’ β†’ SYM cana-1091 130 10 ( ( PUNCT cana-1091 130 11 𝛽5 𝛽5 PROPN cana-1091 130 12 β†’ β†’ SYM cana-1091 130 13 𝛾5 𝛾5 NOUN cana-1091 130 14 ) ) PUNCT cana-1091 130 15 β†’ β†’ SYM cana-1091 130 16 𝛾5 𝛾5 NOUN cana-1091 130 17 ) ) PUNCT cana-1091 130 18 ) ) PUNCT cana-1091 131 1 = = PUNCT cana-1091 131 2 𝑇𝐢((𝛼5 𝑇𝐢((𝛼5 NUM cana-1091 131 3 β†’ β†’ SYM cana-1091 131 4 𝛽5 𝛽5 NUM cana-1091 131 5 ) ) PUNCT cana-1091 131 6 β†’ β†’ SYM cana-1091 131 7 ( ( PUNCT cana-1091 131 8 ( ( PUNCT cana-1091 131 9 𝛾5 𝛾5 NOUN cana-1091 131 10 β†’ β†’ SYM cana-1091 131 11 𝛽5 𝛽5 PROPN cana-1091 131 12 ) ) PUNCT cana-1091 131 13 β†’ β†’ SYM cana-1091 131 14 ( ( PUNCT cana-1091 131 15 𝛼5 𝛼5 NOUN cana-1091 131 16 β†’ β†’ SYM cana-1091 131 17 𝛽5 𝛽5 NUM cana-1091 131 18 ) ) PUNCT cana-1091 131 19 ) ) PUNCT cana-1091 131 20 ) ) PUNCT cana-1091 132 1 = = PUNCT cana-1091 132 2 𝑇𝐢((𝛾5 𝑇𝐢((𝛾5 ADJ cana-1091 132 3 β†’ β†’ SYM cana-1091 132 4 𝛽5 𝛽5 NUM cana-1091 132 5 ) ) PUNCT cana-1091 132 6 β†’ β†’ SYM cana-1091 132 7 ( ( PUNCT cana-1091 132 8 ( ( PUNCT cana-1091 132 9 𝛼5 𝛼5 NOUN cana-1091 132 10 β†’ β†’ SYM cana-1091 132 11 𝛽5 𝛽5 NUM cana-1091 132 12 ) ) PUNCT cana-1091 132 13 β†’ β†’ SYM cana-1091 132 14 ( ( PUNCT cana-1091 132 15 𝛼5 𝛼5 NOUN cana-1091 132 16 β†’ β†’ SYM cana-1091 132 17 𝛽5 𝛽5 NUM cana-1091 132 18 ) ) PUNCT cana-1091 132 19 ) ) PUNCT cana-1091 132 20 ) ) PUNCT cana-1091 133 1 = = PUNCT cana-1091 133 2 𝑇𝐢((𝛾5 𝑇𝐢((𝛾5 ADJ cana-1091 133 3 β†’ β†’ SYM cana-1091 133 4 𝛽5 𝛽5 NUM cana-1091 133 5 ) ) PUNCT cana-1091 133 6 β†’ β†’ SYM cana-1091 133 7 1 1 X cana-1091 133 8 ) ) PUNCT cana-1091 133 9 = = SYM cana-1091 133 10 𝑇𝐢(1 𝑇𝐢(1 PROPN cana-1091 133 11 ) ) PUNCT cana-1091 133 12 = = SYM cana-1091 133 13 𝑇𝐷(1 𝑇𝐷(1 PROPN cana-1091 133 14 ) ) PUNCT cana-1091 133 15 since since SCONJ cana-1091 133 16 𝐷 𝐷 NOUN cana-1091 133 17 is be AUX cana-1091 133 18 a a DET cana-1091 133 19 neutrosophic neutrosophic ADJ cana-1091 133 20 filter filter NOUN cana-1091 133 21 , , PUNCT cana-1091 133 22 𝑇𝐷((𝛼5 𝑇𝐷((𝛼5 NUM cana-1091 133 23 β†’ β†’ SYM cana-1091 133 24 𝛽5 𝛽5 NUM cana-1091 133 25 ) ) PUNCT cana-1091 133 26 β†’ β†’ SYM cana-1091 133 27 ( ( PUNCT cana-1091 133 28 𝛼5 𝛼5 NOUN cana-1091 133 29 β†’ β†’ SYM cana-1091 133 30 𝛾5 𝛾5 NOUN cana-1091 133 31 ) ) PUNCT cana-1091 133 32 ) ) PUNCT cana-1091 134 1 β‰₯ β‰₯ PROPN cana-1091 134 2 min{𝑇𝐷(𝛽5 min{𝑇𝐷(𝛽5 PROPN cana-1091 134 3 β†’ β†’ SYM cana-1091 134 4 𝛾5 𝛾5 PROPN cana-1091 134 5 ) ) PUNCT cana-1091 134 6 , , PUNCT cana-1091 134 7 𝑇𝐷((𝛽5 𝑇𝐷((𝛽5 NUM cana-1091 134 8 β†’ β†’ SYM cana-1091 134 9 𝛾5 𝛾5 NOUN cana-1091 134 10 ) ) PUNCT cana-1091 134 11 β†’ β†’ SYM cana-1091 134 12 ( ( PUNCT cana-1091 134 13 ( ( PUNCT cana-1091 134 14 𝛼5 𝛼5 NOUN cana-1091 134 15 β†’ β†’ SYM cana-1091 134 16 𝛽5 𝛽5 NUM cana-1091 134 17 ) ) PUNCT cana-1091 134 18 β†’ β†’ SYM cana-1091 134 19 ( ( PUNCT cana-1091 134 20 𝛼5 𝛼5 NOUN cana-1091 134 21 β†’ β†’ SYM cana-1091 134 22 𝛾5 𝛾5 NOUN cana-1091 134 23 ) ) PUNCT cana-1091 134 24 ) ) PUNCT cana-1091 134 25 ) ) PUNCT cana-1091 134 26 } } PUNCT cana-1091 135 1 = = SYM cana-1091 135 2 𝑇𝐷(𝛽5 𝑇𝐷(𝛽5 PROPN cana-1091 135 3 β†’ β†’ SYM cana-1091 135 4 𝛾5 𝛾5 NOUN cana-1091 135 5 ) ) PUNCT cana-1091 135 6 β‰₯ β‰₯ NOUN cana-1091 135 7 min{𝑇𝐷(𝛼5 min{𝑇𝐷(𝛼5 NOUN cana-1091 135 8 β†’ β†’ SYM cana-1091 135 9 𝛽5 𝛽5 NUM cana-1091 135 10 ) ) PUNCT cana-1091 135 11 , , PUNCT cana-1091 135 12 𝑇𝐷((𝛼5 𝑇𝐷((𝛼5 NUM cana-1091 135 13 β†’ β†’ SYM cana-1091 135 14 𝛽5 𝛽5 NUM cana-1091 135 15 ) ) PUNCT cana-1091 135 16 β†’ β†’ SYM cana-1091 135 17 ( ( PUNCT cana-1091 135 18 𝛽5 𝛽5 PROPN cana-1091 135 19 β†’ β†’ SYM cana-1091 135 20 𝛾5 𝛾5 NOUN cana-1091 135 21 ) ) PUNCT cana-1091 135 22 ) ) PUNCT cana-1091 135 23 } } PUNCT cana-1091 135 24 similarly similarly ADV cana-1091 135 25 , , PUNCT cana-1091 135 26 we we PRON cana-1091 135 27 can can AUX cana-1091 135 28 prove prove VERB cana-1091 135 29 for for ADP cana-1091 135 30 𝐼𝐷 𝐼𝐷 PROPN cana-1091 135 31 , , PUNCT cana-1091 135 32 𝐹𝐷. 𝐹𝐷. ADV cana-1091 135 33 hence hence ADV cana-1091 135 34 , , PUNCT cana-1091 135 35 𝐷 𝐷 PROPN cana-1091 135 36 is be AUX cana-1091 135 37 a a DET cana-1091 135 38 neutrosophic neutrosophic ADJ cana-1091 135 39 transitive transitive ADJ cana-1091 135 40 filter filter NOUN cana-1091 135 41 . . PUNCT cana-1091 136 1 communications communication NOUN cana-1091 136 2 on on ADP cana-1091 136 3 applied apply VERB cana-1091 136 4 nonlinear nonlinear ADJ cana-1091 136 5 analysis analysis NOUN cana-1091 136 6 issn issn NOUN cana-1091 136 7 : : PUNCT cana-1091 136 8 1074 1074 NUM cana-1091 136 9 - - PUNCT cana-1091 136 10 133x 133x NUM cana-1091 136 11 vol vol NOUN cana-1091 136 12 31 31 NUM cana-1091 136 13 no no NOUN cana-1091 136 14 . . PUNCT cana-1091 137 1 5s 5s NUM cana-1091 137 2 ( ( PUNCT cana-1091 137 3 2024 2024 NUM cana-1091 137 4 ) ) PUNCT cana-1091 137 5 568 568 NUM cana-1091 137 6 https://internationalpubls.com https://internationalpubls.com X cana-1091 137 7 4 4 NUM cana-1091 137 8 . . X cana-1091 137 9 neutrosophic neutrosophic ADJ cana-1091 137 10 absorbent absorbent ADJ cana-1091 137 11 filter filter NOUN cana-1091 137 12 here here ADV cana-1091 137 13 , , PUNCT cana-1091 137 14 we we PRON cana-1091 137 15 put put VERB cana-1091 137 16 forward forward ADV cana-1091 137 17 the the DET cana-1091 137 18 conception conception NOUN cana-1091 137 19 of of ADP cana-1091 137 20 a a DET cana-1091 137 21 neutrosophic neutrosophic ADJ cana-1091 137 22 absorbent absorbent ADJ cana-1091 137 23 filter filter NOUN cana-1091 137 24 and and CCONJ cana-1091 137 25 confer confer VERB cana-1091 137 26 its its PRON cana-1091 137 27 features feature NOUN cana-1091 137 28 with with ADP cana-1091 137 29 illustrations illustration NOUN cana-1091 137 30 . . PUNCT cana-1091 138 1 definition definition NOUN cana-1091 138 2 4.1 4.1 NUM cana-1091 138 3 let let VERB cana-1091 138 4 𝐢 𝐢 PROPN cana-1091 138 5 be be AUX cana-1091 138 6 called call VERB cana-1091 138 7 a a DET cana-1091 138 8 neutrosophic neutrosophic ADJ cana-1091 138 9 absorbent absorbent ADJ cana-1091 138 10 filter filter NOUN cana-1091 138 11 of of ADP cana-1091 138 12 𝒒 𝒒 PROPN cana-1091 138 13 , , PUNCT cana-1091 138 14 if if SCONJ cana-1091 138 15 it it PRON cana-1091 138 16 persuades persuade VERB cana-1091 138 17 the the DET cana-1091 138 18 subsequent subsequent ADJ cana-1091 138 19 requirements requirement NOUN cana-1091 138 20 for for ADP cana-1091 138 21 all all DET cana-1091 138 22 𝛼5 𝛼5 NOUN cana-1091 138 23 , , PUNCT cana-1091 138 24 𝛽5 𝛽5 NOUN cana-1091 138 25 , , PUNCT cana-1091 138 26 𝛾5 𝛾5 PROPN cana-1091 138 27 ∈ ∈ PROPN cana-1091 138 28 𝒒 𝒒 PROPN cana-1091 138 29 , , PUNCT cana-1091 138 30 ( ( PUNCT cana-1091 138 31 i i NOUN cana-1091 138 32 ) ) PUNCT cana-1091 139 1 𝑇𝐢(1 𝑇𝐢(1 PROPN cana-1091 139 2 ) ) PUNCT cana-1091 139 3 β‰₯ β‰₯ NOUN cana-1091 139 4 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 NOUN cana-1091 139 5 ) ) PUNCT cana-1091 139 6 , , PUNCT cana-1091 139 7 𝐼𝐢(1 𝐼𝐢(1 PROPN cana-1091 139 8 ) ) PUNCT cana-1091 139 9 ≀ ≀ NOUN cana-1091 140 1 𝐼𝐢(𝛼5 𝐼𝐢(𝛼5 ADP cana-1091 140 2 ) ) PUNCT cana-1091 140 3 , , PUNCT cana-1091 140 4 𝐹𝐢(1 𝐹𝐢(1 PROPN cana-1091 140 5 ) ) PUNCT cana-1091 140 6 ≀ ≀ NOUN cana-1091 141 1 𝐹𝐢(𝛼5 𝐹𝐢(𝛼5 NUM cana-1091 141 2 ) ) PUNCT cana-1091 141 3 . . PUNCT cana-1091 142 1 ( ( PUNCT cana-1091 142 2 ii ii NOUN cana-1091 142 3 ) ) PUNCT cana-1091 142 4 min{𝑇𝐢(𝛼5 min{𝑇𝐢(𝛼5 ADP cana-1091 142 5 β†’ β†’ SYM cana-1091 142 6 𝛽5 𝛽5 NUM cana-1091 142 7 ) ) PUNCT cana-1091 142 8 , , PUNCT cana-1091 142 9 𝑇𝐢(𝛼5)}≀ 𝑇𝐢(𝛼5)}≀ PROPN cana-1091 142 10 𝑇𝐢(𝛽5),min{𝐼𝐢 𝑇𝐢(𝛽5),min{𝐼𝐢 X cana-1091 142 11 ( ( PUNCT cana-1091 142 12 𝛼5 𝛼5 NOUN cana-1091 142 13 β†’ β†’ SYM cana-1091 142 14 𝛽5 𝛽5 NUM cana-1091 142 15 ) ) PUNCT cana-1091 142 16 , , PUNCT cana-1091 142 17 𝐼𝐢 𝐼𝐢 PROPN cana-1091 142 18 ( ( PUNCT cana-1091 142 19 𝛼5 𝛼5 NOUN cana-1091 142 20 ) ) PUNCT cana-1091 142 21 } } PUNCT cana-1091 142 22 β‰₯ β‰₯ AUX cana-1091 142 23 𝐼𝐢 𝐼𝐢 PROPN cana-1091 142 24 ( ( PUNCT cana-1091 142 25 𝛽5 𝛽5 NUM cana-1091 142 26 ) ) PUNCT cana-1091 142 27 and and CCONJ cana-1091 142 28 min min NOUN cana-1091 142 29 { { PUNCT cana-1091 142 30 𝐹𝐢 𝐹𝐢 PROPN cana-1091 142 31 ( ( PUNCT cana-1091 142 32 𝛼5 𝛼5 NOUN cana-1091 142 33 β†’ β†’ SYM cana-1091 142 34 𝛽5 𝛽5 NUM cana-1091 142 35 ) ) PUNCT cana-1091 142 36 , , PUNCT cana-1091 142 37 𝐹𝐢(𝛼5)}β‰₯ 𝐹𝐢(𝛼5)}β‰₯ NUM cana-1091 143 1 𝐹𝐢(𝛽5 𝐹𝐢(𝛽5 NUM cana-1091 143 2 ) ) PUNCT cana-1091 143 3 } } PUNCT cana-1091 143 4 . . PUNCT cana-1091 144 1 ( ( PUNCT cana-1091 144 2 iii iii X cana-1091 144 3 ) ) PUNCT cana-1091 144 4 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 NOUN cana-1091 144 5 ) ) PUNCT cana-1091 144 6 β‰₯ β‰₯ NOUN cana-1091 144 7 𝑇𝐢((𝛼5 𝑇𝐢((𝛼5 NUM cana-1091 144 8 β†’ β†’ SYM cana-1091 144 9 𝛽5 𝛽5 NUM cana-1091 144 10 ) ) PUNCT cana-1091 144 11 β†’ β†’ SYM cana-1091 144 12 𝛼5 𝛼5 NOUN cana-1091 144 13 ) ) PUNCT cana-1091 144 14 , , PUNCT cana-1091 144 15 𝐼𝐢(𝛼5 𝐼𝐢(𝛼5 ADP cana-1091 144 16 ) ) PUNCT cana-1091 144 17 ≀ ≀ NOUN cana-1091 144 18 𝐼𝐢((𝛼5 𝐼𝐢((𝛼5 NUM cana-1091 144 19 β†’ β†’ SYM cana-1091 144 20 𝛽5 𝛽5 ADJ cana-1091 144 21 ) ) PUNCT cana-1091 144 22 β†’ β†’ SYM cana-1091 144 23 𝛼5 𝛼5 NOUN cana-1091 144 24 ) ) PUNCT cana-1091 144 25 , , PUNCT cana-1091 144 26 𝐹𝐢(𝛼5 𝐹𝐢(𝛼5 PROPN cana-1091 144 27 ) ) PUNCT cana-1091 144 28 ≀ ≀ NOUN cana-1091 144 29 𝐹𝐢((𝛼5 𝐹𝐢((𝛼5 NUM cana-1091 144 30 β†’ β†’ SYM cana-1091 144 31 𝛽5 𝛽5 NUM cana-1091 144 32 ) ) PUNCT cana-1091 144 33 β†’ β†’ SYM cana-1091 144 34 𝛼5 𝛼5 NOUN cana-1091 144 35 ) ) PUNCT cana-1091 144 36 example example NOUN cana-1091 144 37 4.2 4.2 NUM cana-1091 144 38 let let VERB cana-1091 144 39 𝐢 𝐢 PROPN cana-1091 144 40 = = SYM cana-1091 144 41 { { PUNCT cana-1091 144 42 0 0 NUM cana-1091 144 43 , , PUNCT cana-1091 144 44 πœ–1 πœ–1 PROPN cana-1091 144 45 , , PUNCT cana-1091 144 46 πœ‡1 πœ‡1 PROPN cana-1091 144 47 , , PUNCT cana-1091 144 48 𝜌1 𝜌1 NOUN cana-1091 144 49 , , PUNCT cana-1091 144 50 1 1 NUM cana-1091 144 51 } } PUNCT cana-1091 144 52 . . PUNCT cana-1091 145 1 the the DET cana-1091 145 2 bi bi ADJ cana-1091 145 3 - - ADJ cana-1091 145 4 fold fold ADJ cana-1091 145 5 operations operation NOUN cana-1091 145 6 are be AUX cana-1091 145 7 given give VERB cana-1091 145 8 by by ADP cana-1091 145 9 the the DET cana-1091 145 10 subsequent subsequent ADJ cana-1091 145 11 tables table NOUN cana-1091 145 12 ( ( PUNCT cana-1091 145 13 4.1 4.1 NUM cana-1091 145 14 ) ) PUNCT cana-1091 145 15 and and CCONJ cana-1091 145 16 ( ( PUNCT cana-1091 145 17 4.2 4.2 NUM cana-1091 145 18 ) ) PUNCT cana-1091 145 19 . . PUNCT cana-1091 146 1 table table NOUN cana-1091 146 2 4.1 4.1 NUM cana-1091 146 3 : : PUNCT cana-1091 146 4 β€² β€² NUM cana-1091 146 5 Β° Β° NUM cana-1091 146 6 β€² β€² NOUN cana-1091 146 7 operationtable operationtable ADJ cana-1091 146 8 4.2 4.2 NUM cana-1091 146 9 : : PUNCT cana-1091 146 10 β€² β€² NUM cana-1091 146 11 β†’ β†’ SYM cana-1091 146 12 β€² β€² NUM cana-1091 146 13 operation operation NOUN cana-1091 146 14 then then ADV cana-1091 146 15 , , PUNCT cana-1091 146 16 ( ( PUNCT cana-1091 146 17 𝒒 𝒒 PROPN cana-1091 146 18 , , PUNCT cana-1091 146 19 ∨ ∨ NOUN cana-1091 146 20 , , PUNCT cana-1091 146 21 ∧ ∧ PROPN cana-1091 146 22 , , PUNCT cana-1091 146 23 ∘ ∘ ADJ cana-1091 146 24 , , PUNCT cana-1091 146 25 β†’ β†’ SYM cana-1091 146 26 , , PUNCT cana-1091 146 27 0 0 NUM cana-1091 146 28 , , PUNCT cana-1091 146 29 1 1 NUM cana-1091 146 30 ) ) PUNCT cana-1091 146 31 is be AUX cana-1091 146 32 a a DET cana-1091 146 33 blalgebra blalgebra NOUN cana-1091 146 34 . . PUNCT cana-1091 147 1 consider consider VERB cana-1091 147 2 a a DET cana-1091 147 3 neutrosophic neutrosophic ADJ cana-1091 147 4 set set NOUN cana-1091 147 5 𝐢 𝐢 PROPN cana-1091 147 6 : : PUNCT cana-1091 147 7 𝐢 𝐢 PROPN cana-1091 147 8 = = PRON cana-1091 147 9 { { PUNCT cana-1091 147 10 ( ( PUNCT cana-1091 147 11 0 0 NUM cana-1091 147 12 , , PUNCT cana-1091 147 13 [ [ X cana-1091 147 14 0.5,0.7,0.7 0.5,0.7,0.7 NOUN cana-1091 147 15 ] ] PUNCT cana-1091 147 16 ) ) PUNCT cana-1091 147 17 , , PUNCT cana-1091 147 18 ( ( PUNCT cana-1091 147 19 πœ–1 πœ–1 PROPN cana-1091 147 20 , , PUNCT cana-1091 147 21 [ [ X cana-1091 147 22 0.5,0.7,0.7 0.5,0.7,0.7 NOUN cana-1091 147 23 ] ] PUNCT cana-1091 147 24 ) ) PUNCT cana-1091 147 25 , , PUNCT cana-1091 147 26 ( ( PUNCT cana-1091 147 27 πœ‡1 πœ‡1 ADJ cana-1091 147 28 , , PUNCT cana-1091 147 29 [ [ X cana-1091 147 30 0.5,0.7,0.7 0.5,0.7,0.7 NOUN cana-1091 147 31 ] ] PUNCT cana-1091 147 32 ) ) PUNCT cana-1091 147 33 , , PUNCT cana-1091 147 34 ( ( PUNCT cana-1091 147 35 𝜌1 𝜌1 NOUN cana-1091 147 36 , , PUNCT cana-1091 147 37 [ [ X cana-1091 147 38 0.5,0.7,0.7 0.5,0.7,0.7 NOUN cana-1091 147 39 ] ] PUNCT cana-1091 147 40 ) ) PUNCT cana-1091 147 41 , , PUNCT cana-1091 147 42 ( ( PUNCT cana-1091 147 43 1 1 X cana-1091 147 44 , , PUNCT cana-1091 147 45 [ [ X cana-1091 147 46 0.6,0.7,0.7 0.6,0.7,0.7 NOUN cana-1091 147 47 ] ] PUNCT cana-1091 147 48 ) ) PUNCT cana-1091 147 49 } } PUNCT cana-1091 147 50 . . PUNCT cana-1091 148 1 it it PRON cana-1091 148 2 is be AUX cana-1091 148 3 evident evident ADJ cana-1091 148 4 that that SCONJ cana-1091 148 5 𝐢 𝐢 PROPN cana-1091 148 6 assures assure VERB cana-1091 148 7 the the DET cana-1091 148 8 definition definition NOUN cana-1091 148 9 4.1 4.1 NUM cana-1091 148 10 . . PUNCT cana-1091 149 1 hence hence ADV cana-1091 149 2 , , PUNCT cana-1091 149 3 𝐢 𝐢 PROPN cana-1091 149 4 is be AUX cana-1091 149 5 a a DET cana-1091 149 6 neutrosophic neutrosophic ADJ cana-1091 149 7 absorbent absorbent ADJ cana-1091 149 8 filter filter NOUN cana-1091 149 9 . . PUNCT cana-1091 150 1 proposition proposition NOUN cana-1091 150 2 4.3 4.3 NUM cana-1091 150 3 every every DET cana-1091 150 4 neutrosophic neutrosophic ADJ cana-1091 150 5 associative associative ADJ cana-1091 150 6 filter filter NOUN cana-1091 150 7 of of ADP cana-1091 150 8 𝒒 𝒒 PROPN cana-1091 150 9 is be AUX cana-1091 150 10 a a DET cana-1091 150 11 neutrosophic neutrosophic ADJ cana-1091 150 12 absorbent absorbent ADJ cana-1091 150 13 filter filter NOUN cana-1091 150 14 . . PUNCT cana-1091 151 1 proof proof NOUN cana-1091 151 2 : : PUNCT cana-1091 151 3 let let VERB cana-1091 151 4 𝐢 𝐢 PROPN cana-1091 151 5 be be AUX cana-1091 151 6 a a DET cana-1091 151 7 neutrosophic neutrosophic ADJ cana-1091 151 8 associative associative ADJ cana-1091 151 9 filter filter NOUN cana-1091 151 10 of of ADP cana-1091 151 11 𝒒. 𝒒. PROPN cana-1091 151 12 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 PROPN cana-1091 151 13 ) ) PUNCT cana-1091 151 14 = = PUNCT cana-1091 151 15 𝑇𝐢(1 𝑇𝐢(1 PROPN cana-1091 151 16 β†’ β†’ SYM cana-1091 151 17 𝛼5 𝛼5 NOUN cana-1091 151 18 ) ) PUNCT cana-1091 151 19 = = SYM cana-1091 151 20 𝑇𝐢(((𝛼5 𝑇𝐢(((𝛼5 PROPN cana-1091 151 21 β†’ β†’ SYM cana-1091 151 22 𝛽5 𝛽5 PROPN cana-1091 151 23 ) ) PUNCT cana-1091 151 24 β†’ β†’ SYM cana-1091 151 25 1 1 NUM cana-1091 151 26 ) ) PUNCT cana-1091 151 27 β†’ β†’ SYM cana-1091 151 28 𝛼5 𝛼5 NOUN cana-1091 151 29 ) ) PUNCT cana-1091 151 30 β‰₯ β‰₯ NOUN cana-1091 151 31 𝑇𝐢((𝛼5 𝑇𝐢((𝛼5 NUM cana-1091 151 32 β†’ β†’ SYM cana-1091 151 33 𝛽5 𝛽5 NUM cana-1091 151 34 ) ) PUNCT cana-1091 151 35 β†’ β†’ SYM cana-1091 151 36 ( ( PUNCT cana-1091 151 37 1 1 NUM cana-1091 151 38 β†’ β†’ SYM cana-1091 151 39 𝛼5 𝛼5 NOUN cana-1091 151 40 ) ) PUNCT cana-1091 151 41 ) ) PUNCT cana-1091 152 1 = = PUNCT cana-1091 152 2 𝑇𝐢(1 𝑇𝐢(1 NOUN cana-1091 152 3 β†’ β†’ PUNCT cana-1091 152 4 ( ( PUNCT cana-1091 152 5 ( ( PUNCT cana-1091 152 6 𝛼5 𝛼5 NOUN cana-1091 152 7 β†’ β†’ SYM cana-1091 152 8 𝛽5 𝛽5 ADJ cana-1091 152 9 ) ) PUNCT cana-1091 152 10 β†’ β†’ SYM cana-1091 152 11 𝛼5 𝛼5 NOUN cana-1091 152 12 ) ) PUNCT cana-1091 152 13 ) ) PUNCT cana-1091 153 1 = = PUNCT cana-1091 153 2 𝑇𝐢((𝛼5 𝑇𝐢((𝛼5 NUM cana-1091 153 3 β†’ β†’ SYM cana-1091 153 4 𝛽5 𝛽5 ADJ cana-1091 153 5 ) ) PUNCT cana-1091 153 6 β†’ β†’ SYM cana-1091 153 7 𝛼5 𝛼5 NOUN cana-1091 153 8 ) ) PUNCT cana-1091 153 9 therefore therefore ADV cana-1091 153 10 , , PUNCT cana-1091 153 11 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 PROPN cana-1091 153 12 ) ) PUNCT cana-1091 153 13 β‰₯ β‰₯ NOUN cana-1091 154 1 𝑇𝐢((𝛼5 𝑇𝐢((𝛼5 NUM cana-1091 154 2 β†’ β†’ SYM cana-1091 154 3 𝛽5 𝛽5 NUM cana-1091 154 4 ) ) PUNCT cana-1091 154 5 β†’ β†’ SYM cana-1091 154 6 𝛼5 𝛼5 NOUN cana-1091 154 7 ) ) PUNCT cana-1091 155 1 Β° Β° PROPN cana-1091 155 2 0 0 NUM cana-1091 156 1 πœ–1 πœ–1 PROPN cana-1091 156 2 πœ‡1 πœ‡1 PROPN cana-1091 156 3 1 1 NUM cana-1091 156 4 0 0 NUM cana-1091 156 5 0 0 NUM cana-1091 156 6 πœ–1 πœ–1 PROPN cana-1091 156 7 πœ‡1 πœ‡1 PROPN cana-1091 156 8 πœ‡1 πœ‡1 PROPN cana-1091 156 9 πœ–1 πœ–1 PROPN cana-1091 156 10 0 0 NUM cana-1091 156 11 0 0 NUM cana-1091 156 12 0 0 NUM cana-1091 156 13 πœ–1 πœ–1 PROPN cana-1091 156 14 πœ‡1 πœ‡1 PROPN cana-1091 156 15 0 0 NUM cana-1091 156 16 0 0 SYM cana-1091 156 17 0 0 NUM cana-1091 156 18 πœ‡1 πœ‡1 NOUN cana-1091 156 19 1 1 NUM cana-1091 156 20 0 0 NUM cana-1091 156 21 1 1 NUM cana-1091 156 22 1 1 NUM cana-1091 156 23 0 0 NUM cana-1091 156 24 β†’ β†’ SYM cana-1091 156 25 0 0 NUM cana-1091 156 26 πœ–1 πœ–1 PROPN cana-1091 156 27 πœ‡1 πœ‡1 PROPN cana-1091 156 28 1 1 NUM cana-1091 156 29 0 0 NUM cana-1091 156 30 0 0 NUM cana-1091 156 31 0 0 NUM cana-1091 156 32 0 0 NUM cana-1091 156 33 0 0 NUM cana-1091 156 34 πœ–1 πœ–1 PROPN cana-1091 156 35 πœ–1 πœ–1 PROPN cana-1091 156 36 0 0 NUM cana-1091 156 37 0 0 NUM cana-1091 156 38 πœ–1 πœ–1 PROPN cana-1091 156 39 πœ‡1 πœ‡1 PROPN cana-1091 156 40 πœ–1 πœ–1 PROPN cana-1091 156 41 1 1 NUM cana-1091 156 42 0 0 NUM cana-1091 156 43 πœ‡1 πœ‡1 NOUN cana-1091 156 44 1 1 NUM cana-1091 156 45 1 1 NUM cana-1091 156 46 1 1 NUM cana-1091 156 47 0 0 NUM cana-1091 156 48 0 0 NUM cana-1091 156 49 communications communication NOUN cana-1091 156 50 on on ADP cana-1091 156 51 applied apply VERB cana-1091 156 52 nonlinear nonlinear ADJ cana-1091 156 53 analysis analysis NOUN cana-1091 156 54 issn issn NOUN cana-1091 156 55 : : PUNCT cana-1091 156 56 1074 1074 NUM cana-1091 156 57 - - PUNCT cana-1091 156 58 133x 133x NUM cana-1091 156 59 vol vol NOUN cana-1091 156 60 31 31 NUM cana-1091 156 61 no no NOUN cana-1091 156 62 . . PUNCT cana-1091 157 1 5s 5s NUM cana-1091 157 2 ( ( PUNCT cana-1091 157 3 2024 2024 NUM cana-1091 157 4 ) ) PUNCT cana-1091 157 5 569 569 NUM cana-1091 157 6 https://internationalpubls.com https://internationalpubls.com X cana-1091 157 7 similarly similarly ADV cana-1091 157 8 , , PUNCT cana-1091 157 9 we we PRON cana-1091 157 10 can can AUX cana-1091 157 11 prove prove VERB cana-1091 157 12 for for ADP cana-1091 157 13 𝐼𝐢 𝐼𝐢 PROPN cana-1091 157 14 , , PUNCT cana-1091 157 15 𝐹𝐢 𝐹𝐢 PROPN cana-1091 157 16 . . PUNCT cana-1091 158 1 hence hence ADV cana-1091 158 2 , , PUNCT cana-1091 158 3 𝐢 𝐢 PROPN cana-1091 158 4 is be AUX cana-1091 158 5 a a DET cana-1091 158 6 neutrosophic neutrosophic ADJ cana-1091 158 7 absorbent absorbent ADJ cana-1091 158 8 filter filter NOUN cana-1091 158 9 of𝒒. of𝒒. ADP cana-1091 158 10 proposition proposition NOUN cana-1091 158 11 4.4 4.4 NUM cana-1091 158 12 a a DET cana-1091 158 13 neutrosophic neutrosophic ADJ cana-1091 158 14 filter filter NOUN cana-1091 158 15 𝐢 𝐢 PROPN cana-1091 158 16 of of ADP cana-1091 158 17 a a DET cana-1091 158 18 bl bl NOUN cana-1091 158 19 - - PUNCT cana-1091 158 20 algebra algebra NOUN cana-1091 158 21 𝒒 𝒒 PROPN cana-1091 158 22 is be AUX cana-1091 158 23 a a DET cana-1091 158 24 neutrosophic neutrosophic ADJ cana-1091 158 25 positive positive ADJ cana-1091 158 26 implicative implicative ADJ cana-1091 158 27 filter filter NOUN cana-1091 158 28 if if SCONJ cana-1091 158 29 and and CCONJ cana-1091 158 30 only only ADV cana-1091 158 31 if if SCONJ cana-1091 158 32 it it PRON cana-1091 158 33 is be AUX cana-1091 158 34 a a DET cana-1091 158 35 neutrosophic neutrosophic ADJ cana-1091 158 36 absorbent absorbent ADJ cana-1091 158 37 filter filter NOUN cana-1091 158 38 . . PUNCT cana-1091 159 1 proof proof NOUN cana-1091 159 2 : : PUNCT cana-1091 159 3 assume assume VERB cana-1091 159 4 that𝐢 that𝐢 PROPN cana-1091 159 5 is be AUX cana-1091 159 6 a a DET cana-1091 159 7 neutrosophic neutrosophic ADJ cana-1091 159 8 positive positive ADJ cana-1091 159 9 implicative implicative ADJ cana-1091 159 10 filter filter NOUN cana-1091 159 11 of of ADP cana-1091 159 12 𝒒. 𝒒. PROPN cana-1091 159 13 from from ADP cana-1091 159 14 the the DET cana-1091 159 15 definition definition NOUN cana-1091 159 16 2.9 2.9 NUM cana-1091 159 17 , , PUNCT cana-1091 159 18 we we PRON cana-1091 159 19 have have VERB cana-1091 159 20 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 NUM cana-1091 159 21 ) ) PUNCT cana-1091 159 22 β‰₯ β‰₯ NOUN cana-1091 159 23 min{𝑇𝐢 min{𝑇𝐢 PROPN cana-1091 159 24 ( ( PUNCT cana-1091 159 25 1 1 NUM cana-1091 159 26 β†’ β†’ SYM cana-1091 159 27 ( ( PUNCT cana-1091 159 28 ( ( PUNCT cana-1091 159 29 𝛼5 𝛼5 NOUN cana-1091 159 30 β†’ β†’ SYM cana-1091 159 31 𝛽5 𝛽5 ADJ cana-1091 159 32 ) ) PUNCT cana-1091 159 33 β†’ β†’ SYM cana-1091 159 34 𝛼5 𝛼5 NOUN cana-1091 159 35 ) ) PUNCT cana-1091 159 36 ) ) PUNCT cana-1091 159 37 , , PUNCT cana-1091 159 38 𝑇𝐢(1 𝑇𝐢(1 PROPN cana-1091 159 39 ) ) PUNCT cana-1091 159 40 } } PUNCT cana-1091 160 1 = = PUNCT cana-1091 160 2 min{𝑇𝐢((𝛼5 min{𝑇𝐢((𝛼5 NOUN cana-1091 160 3 β†’ β†’ SYM cana-1091 160 4 𝛽5 𝛽5 ADJ cana-1091 160 5 ) ) PUNCT cana-1091 160 6 β†’ β†’ SYM cana-1091 160 7 𝛼5 𝛼5 NOUN cana-1091 160 8 ) ) PUNCT cana-1091 160 9 , , PUNCT cana-1091 160 10 𝑇𝐢(1 𝑇𝐢(1 PROPN cana-1091 160 11 ) ) PUNCT cana-1091 160 12 } } PUNCT cana-1091 161 1 = = PUNCT cana-1091 161 2 𝑇𝐢((𝛼5 𝑇𝐢((𝛼5 NUM cana-1091 161 3 β†’ β†’ SYM cana-1091 161 4 𝛽5 𝛽5 ADJ cana-1091 161 5 ) ) PUNCT cana-1091 161 6 β†’ β†’ SYM cana-1091 161 7 𝛼5 𝛼5 NOUN cana-1091 161 8 ) ) PUNCT cana-1091 161 9 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 NOUN cana-1091 161 10 ) ) PUNCT cana-1091 161 11 β‰₯ β‰₯ NOUN cana-1091 161 12 𝑇𝐢((𝛼5 𝑇𝐢((𝛼5 NUM cana-1091 161 13 β†’ β†’ SYM cana-1091 161 14 𝛽5 𝛽5 NUM cana-1091 161 15 ) ) PUNCT cana-1091 161 16 β†’ β†’ SYM cana-1091 161 17 𝛼5)for 𝛼5)for ADP cana-1091 161 18 all all DET cana-1091 161 19 𝛼5 𝛼5 NOUN cana-1091 161 20 , , PUNCT cana-1091 161 21 𝛽5 𝛽5 VERB cana-1091 161 22 ∈ ∈ NOUN cana-1091 161 23 𝒒. 𝒒. NOUN cana-1091 161 24 similarly similarly ADV cana-1091 161 25 , , PUNCT cana-1091 161 26 we we PRON cana-1091 161 27 can can AUX cana-1091 161 28 prove prove VERB cana-1091 161 29 for for ADP cana-1091 161 30 𝐼𝐢 𝐼𝐢 PROPN cana-1091 161 31 , , PUNCT cana-1091 161 32 𝐹𝐢 𝐹𝐢 PROPN cana-1091 161 33 . . PUNCT cana-1091 162 1 hence hence ADV cana-1091 162 2 , , PUNCT cana-1091 162 3 𝐢 𝐢 PROPN cana-1091 162 4 is be AUX cana-1091 162 5 a a DET cana-1091 162 6 neutrosophic neutrosophic ADJ cana-1091 162 7 absorbent absorbent ADJ cana-1091 162 8 filter filter NOUN cana-1091 162 9 of𝒒. of𝒒. ADP cana-1091 162 10 conversely conversely ADV cana-1091 162 11 , , PUNCT cana-1091 162 12 𝐢 𝐢 PROPN cana-1091 162 13 is be AUX cana-1091 162 14 a a DET cana-1091 162 15 neutrosophic neutrosophic ADJ cana-1091 162 16 absorbent absorbent ADJ cana-1091 162 17 filter filter NOUN cana-1091 162 18 of of ADP cana-1091 162 19 𝒒. 𝒒. PROPN cana-1091 162 20 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 PROPN cana-1091 162 21 ) ) PUNCT cana-1091 162 22 β‰₯ β‰₯ NOUN cana-1091 162 23 𝑇𝐢((𝛼5 𝑇𝐢((𝛼5 NUM cana-1091 162 24 β†’ β†’ SYM cana-1091 162 25 𝛽5 𝛽5 NUM cana-1091 162 26 ) ) PUNCT cana-1091 162 27 β†’ β†’ SYM cana-1091 162 28 𝛼5 𝛼5 NOUN cana-1091 162 29 ) ) PUNCT cana-1091 162 30 β‰₯ β‰₯ NOUN cana-1091 162 31 min{𝑇𝐢 min{𝑇𝐢 PROPN cana-1091 162 32 ( ( PUNCT cana-1091 162 33 𝛾5 𝛾5 PROPN cana-1091 162 34 β†’ β†’ SYM cana-1091 162 35 ( ( PUNCT cana-1091 162 36 ( ( PUNCT cana-1091 162 37 𝛼5 𝛼5 NOUN cana-1091 162 38 β†’ β†’ SYM cana-1091 162 39 𝛽5 𝛽5 ADJ cana-1091 162 40 ) ) PUNCT cana-1091 162 41 β†’ β†’ SYM cana-1091 162 42 𝛼5 𝛼5 NOUN cana-1091 162 43 ) ) PUNCT cana-1091 162 44 ) ) PUNCT cana-1091 162 45 , , PUNCT cana-1091 162 46 𝑇𝐢(𝛾5 𝑇𝐢(𝛾5 PROPN cana-1091 162 47 ) ) PUNCT cana-1091 162 48 } } PUNCT cana-1091 162 49 therefore therefore ADV cana-1091 162 50 , , PUNCT cana-1091 162 51 𝑇𝐢(𝛼5 𝑇𝐢(𝛼5 PROPN cana-1091 162 52 ) ) PUNCT cana-1091 162 53 β‰₯ β‰₯ PROPN cana-1091 162 54 min min PROPN cana-1091 162 55 { { PUNCT cana-1091 162 56 𝑇𝐢(𝛾5 𝑇𝐢(𝛾5 PROPN cana-1091 162 57 β†’ β†’ PUNCT cana-1091 162 58 ( ( PUNCT cana-1091 162 59 ( ( PUNCT cana-1091 162 60 𝛼5 𝛼5 NOUN cana-1091 162 61 β†’ β†’ SYM cana-1091 162 62 𝛽5 𝛽5 ADJ cana-1091 162 63 ) ) PUNCT cana-1091 162 64 β†’ β†’ SYM cana-1091 162 65 𝛼5 𝛼5 NOUN cana-1091 162 66 ) ) PUNCT cana-1091 162 67 ) ) PUNCT cana-1091 162 68 , , PUNCT cana-1091 162 69 𝑇𝐢(𝛾5 𝑇𝐢(𝛾5 PROPN cana-1091 162 70 ) ) PUNCT cana-1091 162 71 } } PUNCT cana-1091 162 72 . . PUNCT cana-1091 163 1 similarly similarly ADV cana-1091 163 2 , , PUNCT cana-1091 163 3 we we PRON cana-1091 163 4 can can AUX cana-1091 163 5 prove prove VERB cana-1091 163 6 for for ADP cana-1091 163 7 𝐼𝐢 𝐼𝐢 PROPN cana-1091 163 8 , , PUNCT cana-1091 163 9 𝐹𝐢 𝐹𝐢 PROPN cana-1091 163 10 . . PUNCT cana-1091 164 1 hence hence ADV cana-1091 164 2 , , PUNCT cana-1091 164 3 𝐢 𝐢 PROPN cana-1091 164 4 is be AUX cana-1091 164 5 a a DET cana-1091 164 6 neutrosophic neutrosophic ADJ cana-1091 164 7 positive positive ADJ cana-1091 164 8 implicative implicative ADJ cana-1091 164 9 filter filter NOUN cana-1091 164 10 of of ADP cana-1091 164 11 𝒒. 𝒒. PROPN cana-1091 164 12 corollary corollary NOUN cana-1091 164 13 4.5 4.5 NUM cana-1091 164 14 let let VERB cana-1091 164 15 𝐢 𝐢 PROPN cana-1091 164 16 be be AUX cana-1091 164 17 a a DET cana-1091 164 18 neutrosophic neutrosophic ADJ cana-1091 164 19 filter filter NOUN cana-1091 164 20 of of ADP cana-1091 164 21 a a DET cana-1091 164 22 bl bl NOUN cana-1091 164 23 - - PUNCT cana-1091 164 24 algebra algebra NOUN cana-1091 164 25 𝒒. 𝒒. PROPN cana-1091 164 26 if if SCONJ cana-1091 164 27 𝐢 𝐢 PROPN cana-1091 164 28 is be AUX cana-1091 164 29 a a DET cana-1091 164 30 neutrosophic neutrosophic ADJ cana-1091 164 31 absorbent absorbent ADJ cana-1091 164 32 filter filter NOUN cana-1091 164 33 , , PUNCT cana-1091 164 34 then then ADV cana-1091 164 35 it it PRON cana-1091 164 36 is be AUX cana-1091 164 37 a a DET cana-1091 164 38 neutrosophic neutrosophic ADJ cana-1091 164 39 fantastic fantastic ADJ cana-1091 164 40 filter filter NOUN cana-1091 164 41 of of ADP cana-1091 164 42 𝒒. 𝒒. PROPN cana-1091 164 43 proof proof NOUN cana-1091 164 44 : : PUNCT cana-1091 164 45 let let VERB cana-1091 164 46 𝐢 𝐢 PROPN cana-1091 164 47 be be AUX cana-1091 164 48 a a DET cana-1091 164 49 neutrosophic neutrosophic ADJ cana-1091 164 50 absorbent absorbent ADJ cana-1091 164 51 filter filter NOUN cana-1091 164 52 of of ADP cana-1091 164 53 a a DET cana-1091 164 54 bl bl NOUN cana-1091 164 55 - - PUNCT cana-1091 164 56 algebra algebra NOUN cana-1091 164 57 𝒒. 𝒒. PROPN cana-1091 164 58 then then ADV cana-1091 164 59 , , PUNCT cana-1091 164 60 by by ADP cana-1091 164 61 the the DET cana-1091 164 62 proposition proposition NOUN cana-1091 164 63 4.4 4.4 NUM cana-1091 164 64 , , PUNCT cana-1091 164 65 𝐢 𝐢 PROPN cana-1091 164 66 is be AUX cana-1091 164 67 a a DET cana-1091 164 68 neutrosophic neutrosophic ADJ cana-1091 164 69 positive positive ADJ cana-1091 164 70 implicative implicative ADJ cana-1091 164 71 filter filter NOUN cana-1091 164 72 of of ADP cana-1091 164 73 𝒒. 𝒒. PROPN cana-1091 164 74 then then ADV cana-1091 164 75 from from ADP cana-1091 164 76 the the DET cana-1091 164 77 proposition proposition NOUN cana-1091 164 78 2.12 2.12 NUM cana-1091 164 79 , , PUNCT cana-1091 164 80 𝐢 𝐢 PROPN cana-1091 164 81 is be AUX cana-1091 164 82 a a DET cana-1091 164 83 neutrosophic neutrosophic ADJ cana-1091 164 84 fantastic fantastic ADJ cana-1091 164 85 filter filter NOUN cana-1091 164 86 of of ADP cana-1091 164 87 𝒒. 𝒒. PROPN cana-1091 164 88 5 5 NUM cana-1091 164 89 . . PUNCT cana-1091 165 1 conclusion conclusion NOUN cana-1091 165 2 in in ADP cana-1091 165 3 the the DET cana-1091 165 4 current current ADJ cana-1091 165 5 study study NOUN cana-1091 165 6 , , PUNCT cana-1091 165 7 we we PRON cana-1091 165 8 have have AUX cana-1091 165 9 put put VERB cana-1091 165 10 forward forward ADV cana-1091 165 11 the the DET cana-1091 165 12 notions notion NOUN cana-1091 165 13 of of ADP cana-1091 165 14 neutrosophic neutrosophic ADJ cana-1091 165 15 transitive transitive NOUN cana-1091 165 16 and and CCONJ cana-1091 165 17 absorbent absorbent ADJ cana-1091 165 18 filters filter NOUN cana-1091 165 19 in in ADP cana-1091 165 20 basic basic ADJ cana-1091 165 21 logic logic NOUN cana-1091 165 22 algebras algebra NOUN cana-1091 165 23 and and CCONJ cana-1091 165 24 looked look VERB cana-1091 165 25 into into ADP cana-1091 165 26 a a DET cana-1091 165 27 few few ADJ cana-1091 165 28 associated associated ADJ cana-1091 165 29 features feature NOUN cana-1091 165 30 . . PUNCT cana-1091 166 1 additionally additionally ADV cana-1091 166 2 , , PUNCT cana-1091 166 3 we we PRON cana-1091 166 4 have have AUX cana-1091 166 5 proved prove VERB cana-1091 166 6 that that SCONJ cana-1091 166 7 every every DET cana-1091 166 8 neutrosophic neutrosophic ADJ cana-1091 166 9 transitive transitive ADJ cana-1091 166 10 filter filter NOUN cana-1091 166 11 in in ADP cana-1091 166 12 bl bl NOUN cana-1091 166 13 - - PUNCT cana-1091 166 14 algebras algebras PROPN cana-1091 166 15 is be AUX cana-1091 166 16 a a DET cana-1091 166 17 neutrosophic neutrosophic ADJ cana-1091 166 18 filter filter NOUN cana-1091 166 19 and and CCONJ cana-1091 166 20 a a DET cana-1091 166 21 neutrosophic neutrosophic ADJ cana-1091 166 22 associative associative ADJ cana-1091 166 23 filter filter NOUN cana-1091 166 24 . . PUNCT cana-1091 167 1 in in ADP cana-1091 167 2 addition addition NOUN cana-1091 167 3 , , PUNCT cana-1091 167 4 we we PRON cana-1091 167 5 confer confer VERB cana-1091 167 6 some some DET cana-1091 167 7 necessary necessary ADJ cana-1091 167 8 and and CCONJ cana-1091 167 9 sufficient sufficient ADJ cana-1091 167 10 condition condition NOUN cana-1091 167 11 , , PUNCT cana-1091 167 12 extension extension NOUN cana-1091 167 13 property property NOUN cana-1091 167 14 for for ADP cana-1091 167 15 a a DET cana-1091 167 16 neutrosophic neutrosophic ADJ cana-1091 167 17 filter filter NOUN cana-1091 167 18 to to PART cana-1091 167 19 be be AUX cana-1091 167 20 a a DET cana-1091 167 21 transitive transitive ADJ cana-1091 167 22 filter filter NOUN cana-1091 167 23 . . PUNCT cana-1091 168 1 the the DET cana-1091 168 2 purpose purpose NOUN cana-1091 168 3 of of ADP cana-1091 168 4 this this DET cana-1091 168 5 article article NOUN cana-1091 168 6 is be AUX cana-1091 168 7 two two NUM cana-1091 168 8 fold fold NOUN cana-1091 168 9 , , PUNCT cana-1091 168 10 first first ADV cana-1091 168 11 is be AUX cana-1091 168 12 to to PART cana-1091 168 13 introduce introduce VERB cana-1091 168 14 the the DET cana-1091 168 15 notions notion NOUN cana-1091 168 16 in in ADP cana-1091 168 17 bl bl NOUN cana-1091 168 18 - - PUNCT cana-1091 168 19 algebra algebra PROPN cana-1091 168 20 and and CCONJ cana-1091 168 21 then then ADV cana-1091 168 22 to to PART cana-1091 168 23 explore explore VERB cana-1091 168 24 their their PRON cana-1091 168 25 relationship relationship NOUN cana-1091 168 26 among among ADP cana-1091 168 27 various various ADJ cana-1091 168 28 filters filter NOUN cana-1091 168 29 . . PUNCT cana-1091 169 1 further far ADV cana-1091 169 2 , , PUNCT cana-1091 169 3 we we PRON cana-1091 169 4 have have AUX cana-1091 169 5 obtained obtain VERB cana-1091 169 6 ( ( PUNCT cana-1091 169 7 i i NOUN cana-1091 169 8 ) ) PUNCT cana-1091 169 9 everyneutrosophic everyneutrosophic ADJ cana-1091 169 10 associative associative ADJ cana-1091 169 11 filter filter NOUN cana-1091 169 12 is be AUX cana-1091 169 13 an an DET cana-1091 169 14 absorbent absorbent ADJ cana-1091 169 15 filter filter NOUN cana-1091 169 16 . . PUNCT cana-1091 170 1 ( ( PUNCT cana-1091 170 2 ii ii X cana-1091 170 3 ) ) PUNCT cana-1091 170 4 𝐢 𝐢 PROPN cana-1091 170 5 is be AUX cana-1091 170 6 a a DET cana-1091 170 7 neutrosophic neutrosophic ADJ cana-1091 170 8 positive positive ADJ cana-1091 170 9 implicative implicative ADJ cana-1091 170 10 filter filter NOUN cana-1091 170 11 if if SCONJ cana-1091 171 1 and and CCONJ cana-1091 171 2 only only ADV cana-1091 171 3 if if SCONJ cana-1091 171 4 it it PRON cana-1091 171 5 is be AUX cana-1091 171 6 a a DET cana-1091 171 7 neutrosophic neutrosophic ADJ cana-1091 171 8 absorbent absorbent ADJ cana-1091 171 9 filter filter NOUN cana-1091 171 10 . . PUNCT cana-1091 172 1 ( ( PUNCT cana-1091 172 2 iii iii X cana-1091 172 3 ) ) PUNCT cana-1091 172 4 if if SCONJ cana-1091 172 5 𝐢 𝐢 PROPN cana-1091 172 6 is be AUX cana-1091 172 7 a a DET cana-1091 172 8 neutrosophic neutrosophic ADJ cana-1091 172 9 absorbent absorbent ADJ cana-1091 172 10 filter filter NOUN cana-1091 172 11 , , PUNCT cana-1091 172 12 then then ADV cana-1091 172 13 it it PRON cana-1091 172 14 is be AUX cana-1091 172 15 a a DET cana-1091 172 16 neutrosophic neutrosophic ADJ cana-1091 172 17 fantastic fantastic ADJ cana-1091 172 18 filter filter NOUN cana-1091 172 19 . . PUNCT cana-1091 173 1 in in ADP cana-1091 173 2 the the DET cana-1091 173 3 future future NOUN cana-1091 173 4 , , PUNCT cana-1091 173 5 the the DET cana-1091 173 6 above above ADJ cana-1091 173 7 research research NOUN cana-1091 173 8 can can AUX cana-1091 173 9 be be AUX cana-1091 173 10 extended extend VERB cana-1091 173 11 to to ADP cana-1091 173 12 ultra ultra ADJ cana-1091 173 13 and and CCONJ cana-1091 173 14 deductive deductive ADJ cana-1091 173 15 filters filter NOUN cana-1091 173 16 . . PUNCT cana-1091 174 1 references reference NOUN cana-1091 174 2 [ [ X cana-1091 174 3 1 1 NUM cana-1091 174 4 ] ] PUNCT cana-1091 174 5 xu xu PROPN cana-1091 174 6 , , PUNCT cana-1091 174 7 y. y. PROPN cana-1091 174 8 ( ( PUNCT cana-1091 174 9 1993 1993 NUM cana-1091 174 10 ) ) PUNCT cana-1091 174 11 . . PUNCT cana-1091 175 1 lattice lattice PROPN cana-1091 175 2 implication implication NOUN cana-1091 175 3 algebras algebras PROPN cana-1091 175 4 . . PUNCT cana-1091 176 1 journal journal PROPN cana-1091 176 2 of of ADP cana-1091 176 3 southwest southwest PROPN cana-1091 176 4 jiatong jiatong PROPN cana-1091 176 5 university university PROPN cana-1091 176 6 , , PUNCT cana-1091 176 7 1 1 NUM cana-1091 176 8 : : SYM cana-1091 176 9 20 20 NUM cana-1091 176 10 - - SYM cana-1091 176 11 27 27 NUM cana-1091 176 12 . . PUNCT cana-1091 177 1 [ [ X cana-1091 177 2 2 2 NUM cana-1091 177 3 ] ] PUNCT cana-1091 177 4 xu xu PROPN cana-1091 177 5 , , PUNCT cana-1091 177 6 y. y. PROPN cana-1091 177 7 and and CCONJ cana-1091 177 8 qin qin PROPN cana-1091 177 9 , , PUNCT cana-1091 177 10 k.y k.y PROPN cana-1091 177 11 . . PROPN cana-1091 177 12 ( ( PUNCT cana-1091 177 13 1993 1993 NUM cana-1091 177 14 ) ) PUNCT cana-1091 177 15 . . PUNCT cana-1091 178 1 on on ADP cana-1091 178 2 filters filter NOUN cana-1091 178 3 of of ADP cana-1091 178 4 lattice lattice PROPN cana-1091 178 5 implication implication NOUN cana-1091 178 6 algebra algebra NOUN cana-1091 178 7 . . PUNCT cana-1091 179 1 journal journal NOUN cana-1091 179 2 of of ADP cana-1091 179 3 fuzzy fuzzy ADJ cana-1091 179 4 mathematics mathematic NOUN cana-1091 179 5 , , PUNCT cana-1091 179 6 2 2 NUM cana-1091 179 7 : : SYM cana-1091 179 8 251260 251260 NUM cana-1091 179 9 . . PUNCT cana-1091 180 1 [ [ X cana-1091 180 2 3 3 X cana-1091 180 3 ] ] X cana-1091 180 4 sambasiva sambasiva NOUN cana-1091 180 5 rao rao PROPN cana-1091 180 6 , , PUNCT cana-1091 180 7 m. m. NOUN cana-1091 180 8 ( ( PUNCT cana-1091 180 9 2014 2014 NUM cana-1091 180 10 ) ) PUNCT cana-1091 180 11 . . PUNCT cana-1091 181 1 transitive transitive VERB cana-1091 181 2 and and CCONJ cana-1091 181 3 absorbent absorbent ADJ cana-1091 181 4 filters filter NOUN cana-1091 181 5 of of ADP cana-1091 181 6 lattice lattice PROPN cana-1091 181 7 implication implication NOUN cana-1091 181 8 algebras algebra NOUN cana-1091 181 9 . . PUNCT cana-1091 181 10 journal journal PROPN cana-1091 181 11 of of ADP cana-1091 181 12 applied apply VERB cana-1091 181 13 maths math NOUN cana-1091 181 14 and and CCONJ cana-1091 181 15 informatics informatic NOUN cana-1091 181 16 , , PUNCT cana-1091 181 17 32(3 32(3 NUM cana-1091 181 18 - - PUNCT cana-1091 181 19 4):323 4):323 NUM cana-1091 181 20 - - PUNCT cana-1091 181 21 330 330 NUM cana-1091 181 22 . . PUNCT cana-1091 182 1 communications communication NOUN cana-1091 182 2 on on ADP cana-1091 182 3 applied apply VERB cana-1091 182 4 nonlinear nonlinear ADJ cana-1091 182 5 analysis analysis NOUN cana-1091 182 6 issn issn NOUN cana-1091 182 7 : : PUNCT cana-1091 182 8 1074 1074 NUM cana-1091 182 9 - - PUNCT cana-1091 182 10 133x 133x NUM cana-1091 182 11 vol vol NOUN cana-1091 182 12 31 31 NUM cana-1091 182 13 no no NOUN cana-1091 182 14 . . PUNCT cana-1091 183 1 5s 5s NUM cana-1091 183 2 ( ( PUNCT cana-1091 183 3 2024 2024 NUM cana-1091 183 4 ) ) PUNCT cana-1091 183 5 570 570 NUM cana-1091 183 6 https://internationalpubls.com https://internationalpubls.com X cana-1091 184 1 [ [ X cana-1091 184 2 4 4 NUM cana-1091 184 3 ] ] X cana-1091 184 4 ibrahim ibrahim PROPN cana-1091 184 5 , , PUNCT cana-1091 184 6 a. a. PROPN cana-1091 184 7 and and CCONJ cana-1091 184 8 karunya karunya PROPN cana-1091 184 9 helen helen PROPN cana-1091 184 10 gunaseeli gunaseeli PROPN cana-1091 184 11 , , PUNCT cana-1091 184 12 s. s. PROPN cana-1091 184 13 ( ( PUNCT cana-1091 184 14 2023 2023 NUM cana-1091 184 15 ) ) PUNCT cana-1091 184 16 . . PUNCT cana-1091 185 1 on on ADP cana-1091 185 2 neutrosophic neutrosophic ADJ cana-1091 185 3 filter filter NOUN cana-1091 185 4 of of ADP cana-1091 185 5 bl bl NOUN cana-1091 185 6 - - PUNCT cana-1091 185 7 algebras algebras PROPN cana-1091 185 8 . . PUNCT cana-1091 186 1 ratio ratio PROPN cana-1091 186 2 mathematica mathematica PROPN cana-1091 186 3 , , PUNCT cana-1091 186 4 47:141 47:141 NUM cana-1091 186 5 - - SYM cana-1091 186 6 150 150 NUM cana-1091 186 7 . . PUNCT cana-1091 187 1 [ [ X cana-1091 187 2 5 5 NUM cana-1091 187 3 ] ] X cana-1091 187 4 ibrahim ibrahim PROPN cana-1091 187 5 , , PUNCT cana-1091 187 6 a. a. PROPN cana-1091 187 7 and and CCONJ cana-1091 187 8 karunya karunya PROPN cana-1091 187 9 helen helen PROPN cana-1091 187 10 gunaseeli gunaseeli PROPN cana-1091 187 11 , , PUNCT cana-1091 187 12 s. s. PROPN cana-1091 187 13 ( ( PUNCT cana-1091 187 14 2023 2023 NUM cana-1091 187 15 ) ) PUNCT cana-1091 187 16 . . PUNCT cana-1091 188 1 on on ADP cana-1091 188 2 neutrosophic neutrosophic ADJ cana-1091 188 3 filter filter NOUN cana-1091 188 4 and and CCONJ cana-1091 188 5 fantastic fantastic ADJ cana-1091 188 6 filter filter NOUN cana-1091 188 7 of of ADP cana-1091 188 8 blalgebras.international blalgebras.international PROPN cana-1091 188 9 journal journal NOUN cana-1091 188 10 of of ADP cana-1091 188 11 neutrosophic neutrosophic ADJ cana-1091 188 12 science science NOUN cana-1091 188 13 , , PUNCT cana-1091 188 14 21(2 21(2 NUM cana-1091 188 15 ): ): PUNCT cana-1091 188 16 59 59 NUM cana-1091 188 17 - - SYM cana-1091 188 18 67 67 NUM cana-1091 188 19 . . PUNCT cana-1091 189 1 [ [ X cana-1091 189 2 6 6 NUM cana-1091 189 3 ] ] X cana-1091 189 4 ibrahim ibrahim PROPN cana-1091 189 5 , , PUNCT cana-1091 189 6 a. a. PROPN cana-1091 189 7 and and CCONJ cana-1091 189 8 karunya karunya PROPN cana-1091 189 9 helen helen PROPN cana-1091 189 10 gunaseeli gunaseeli PROPN cana-1091 189 11 , , PUNCT cana-1091 189 12 s. s. PROPN cana-1091 189 13 and and CCONJ cana-1091 189 14 florentinsmarandache(2024 florentinsmarandache(2024 NOUN cana-1091 189 15 ) ) PUNCT cana-1091 189 16 . . PUNCT cana-1091 190 1 on on ADP cana-1091 190 2 neutrosophic neutrosophic ADJ cana-1091 190 3 implicative implicative ADJ cana-1091 190 4 filters filter NOUN cana-1091 190 5 of of ADP cana-1091 190 6 bl bl NOUN cana-1091 190 7 - - PUNCT cana-1091 190 8 algebra algebra NOUN cana-1091 190 9 . . PUNCT cana-1091 191 1 communications communication NOUN cana-1091 191 2 in in ADP cana-1091 191 3 mathematics mathematic NOUN cana-1091 191 4 and and CCONJ cana-1091 191 5 applications application NOUN cana-1091 191 6 . . PUNCT cana-1091 192 1 [ [ X cana-1091 192 2 7 7 NUM cana-1091 192 3 ] ] PUNCT cana-1091 192 4 hajek hajek NOUN cana-1091 192 5 , , PUNCT cana-1091 192 6 p. p. NOUN cana-1091 192 7 ( ( PUNCT cana-1091 192 8 1998 1998 NUM cana-1091 192 9 ) ) PUNCT cana-1091 192 10 . . PUNCT cana-1091 193 1 metamathematics metamathematic NOUN cana-1091 193 2 of of ADP cana-1091 193 3 fuzzy fuzzy ADJ cana-1091 193 4 logic logic NOUN cana-1091 193 5 , , PUNCT cana-1091 193 6 springer springer PROPN cana-1091 193 7 netherlands netherlands PROPN cana-1091 193 8 , , PUNCT cana-1091 193 9 dordrecht dordrecht PROPN cana-1091 193 10 . . PUNCT cana-1091 194 1 https://doi.org/10.1007/978-94011-5300-3 https://doi.org/10.1007/978-94011-5300-3 PROPN cana-1091 194 2 . . PUNCT cana-1091 195 1 [ [ X cana-1091 195 2 8 8 NUM cana-1091 195 3 ] ] X cana-1091 195 4 haveshki haveshki PROPN cana-1091 195 5 , , PUNCT cana-1091 195 6 m. m. NOUN cana-1091 195 7 , , PUNCT cana-1091 195 8 borumand borumand PROPN cana-1091 195 9 saied saie VERB cana-1091 195 10 , , PUNCT cana-1091 195 11 a. a. NOUN cana-1091 195 12 and and CCONJ cana-1091 195 13 eslami eslami PROPN cana-1091 195 14 , , PUNCT cana-1091 195 15 e. e. PROPN cana-1091 195 16 ( ( PUNCT cana-1091 195 17 2006).some 2006).some NUM cana-1091 195 18 types type NOUN cana-1091 195 19 of of ADP cana-1091 195 20 filters filter NOUN cana-1091 195 21 in in ADP cana-1091 195 22 bl bl NOUN cana-1091 195 23 - - PUNCT cana-1091 195 24 algebras algebras PROPN cana-1091 195 25 . . PUNCT cana-1091 196 1 soft soft ADJ cana-1091 196 2 computing,10 computing,10 NOUN cana-1091 196 3 : : PUNCT cana-1091 196 4 657 657 NUM cana-1091 196 5 - - SYM cana-1091 196 6 664 664 NUM cana-1091 196 7 . . PUNCT cana-1091 196 8 https://doi.org/10.1007/s00500-005-0534-4 https://doi.org/10.1007/s00500-005-0534-4 NUM cana-1091 196 9 . . PUNCT cana-1091 197 1 [ [ X cana-1091 197 2 9 9 NUM cana-1091 197 3 ] ] X cana-1091 197 4 turunen turunen NOUN cana-1091 197 5 , , PUNCT cana-1091 197 6 e. e. PROPN cana-1091 197 7 ( ( PUNCT cana-1091 197 8 2001 2001 NUM cana-1091 197 9 ) ) PUNCT cana-1091 197 10 . . PUNCT cana-1091 198 1 boolean boolean ADJ cana-1091 198 2 deductive deductive ADJ cana-1091 198 3 systems system NOUN cana-1091 198 4 of of ADP cana-1091 198 5 bl bl NOUN cana-1091 198 6 - - PUNCT cana-1091 198 7 algebras algebras PROPN cana-1091 198 8 . . PUNCT cana-1091 199 1 arch.mathematical arch.mathematical PROPN cana-1091 199 2 logic,40:467473 logic,40:467473 PROPN cana-1091 199 3 . . PUNCT cana-1091 200 1 https://doi.org/10.1007/s001530100088 https://doi.org/10.1007/s001530100088 PRON cana-1091 200 2 . . PUNCT cana-1091 201 1 [ [ X cana-1091 201 2 10 10 NUM cana-1091 201 3 ] ] X cana-1091 201 4 liu liu PROPN cana-1091 201 5 , , PUNCT cana-1091 201 6 l.z l.z PROPN cana-1091 201 7 . . PROPN cana-1091 202 1 and and CCONJ cana-1091 202 2 li li PROPN cana-1091 202 3 , , PUNCT cana-1091 202 4 k.t k.t PROPN cana-1091 202 5 . . PROPN cana-1091 202 6 ( ( PUNCT cana-1091 202 7 2005 2005 NUM cana-1091 202 8 ) ) PUNCT cana-1091 202 9 . . PUNCT cana-1091 203 1 fuzzy fuzzy ADJ cana-1091 203 2 filters filter NOUN cana-1091 203 3 of of ADP cana-1091 203 4 bl-algebras.information bl-algebras.information NOUN cana-1091 203 5 sciences,173:141 sciences,173:141 X cana-1091 203 6 - - SYM cana-1091 203 7 154 154 NUM cana-1091 203 8 . . PUNCT cana-1091 204 1 https://doi.org/10.1016/j.ins.2004.07.009 https://doi.org/10.1016/j.ins.2004.07.009 ADJ cana-1091 204 2 . . PUNCT cana-1091 205 1 [ [ X cana-1091 205 2 11 11 NUM cana-1091 205 3 ] ] PUNCT cana-1091 205 4 salama salama NOUN cana-1091 205 5 , , PUNCT cana-1091 205 6 a. a. NOUN cana-1091 205 7 a. a. NOUN cana-1091 205 8 and and CCONJ cana-1091 205 9 alagamy alagamy PROPN cana-1091 205 10 , , PUNCT cana-1091 205 11 h.(2013 h.(2013 PROPN cana-1091 205 12 ) ) PUNCT cana-1091 205 13 . . PUNCT cana-1091 206 1 neutrosophic neutrosophic ADJ cana-1091 206 2 filters filter NOUN cana-1091 206 3 . . PUNCT cana-1091 207 1 journal journal NOUN cana-1091 207 2 of of ADP cana-1091 207 3 computer computer NOUN cana-1091 207 4 science science NOUN cana-1091 207 5 engineering,307 engineering,307 PROPN cana-1091 207 6 - - PUNCT cana-1091 207 7 312 312 NUM cana-1091 207 8 . . PUNCT cana-1091 207 9 https://doi.org/10.5281/zenodo.23184 https://doi.org/10.5281/zenodo.23184 NOUN cana-1091 207 10 . . PUNCT cana-1091 208 1 [ [ X cana-1091 208 2 12 12 NUM cana-1091 208 3 ] ] X cana-1091 208 4 smarandache smarandache NOUN cana-1091 208 5 , , PUNCT cana-1091 208 6 f. f. PROPN cana-1091 208 7 ( ( PUNCT cana-1091 208 8 1999 1999 NUM cana-1091 208 9 ) ) PUNCT cana-1091 208 10 . . PUNCT cana-1091 209 1 a a DET cana-1091 209 2 unifying unifying ADJ cana-1091 209 3 field field NOUN cana-1091 209 4 in in ADP cana-1091 209 5 logics logic NOUN cana-1091 209 6 : : PUNCT cana-1091 209 7 neutrosophic neutrosophic ADJ cana-1091 209 8 logic logic NOUN cana-1091 209 9 . . PUNCT cana-1091 210 1 neutrosophy neutrosophy NOUN cana-1091 210 2 , , PUNCT cana-1091 210 3 neutrosophic neutrosophic ADJ cana-1091 210 4 set set NOUN cana-1091 210 5 , , PUNCT cana-1091 210 6 neutrosophic neutrosophic ADJ cana-1091 210 7 probability probability NOUN cana-1091 210 8 , , PUNCT cana-1091 210 9 american american ADJ cana-1091 210 10 research research PROPN cana-1091 210 11 press press PROPN cana-1091 210 12 , , PUNCT cana-1091 210 13 rehoboth rehoboth PROPN cana-1091 210 14 , , PUNCT cana-1091 210 15 u.s.a u.s.a NOUN cana-1091 210 16 , , PUNCT cana-1091 210 17 1 1 NUM cana-1091 210 18 - - SYM cana-1091 210 19 144 144 NUM cana-1091 210 20 . . PUNCT