ON n-fold FILTERS IN BL-ALGEBRAS

Transcription

1 Jourl of Alger Numer Theor: Adves d Applitios Volume 2 Numer 29 Pges ON -FOLD FILTERS IN BL-ALGEBRAS M. SHIRVANI-GHADIKOLAI A. MOUSSAVI A. KORDI 2 d A. AHMADI 2 Deprtmet of Mthemtis Trit Modres Uiversit P. O. Bo: 45-7 Tehr Ir e-mil: moussvi.@modres..ir 2 Mthemtis d Iformtis Reserh Group ACECR Trit Modres Uiversit P. O. Bo: Tehr Ir Astrt The otios of -fold ftsti si logi d the relted lgers -fold ftsti BL-lgers re itrodued. We lso defie -fold ftsti filters d prove some reltios etwee these filters d ostrut quotiet lgers vi these filters.. Itrodutio The oept of BL-lgers ws itrodued Hjek [5] i order to provide lgeri proof of the ompleteess theorem of si logi (BL for short). Soo fter Cigoli et l. i [2] proved tht Hjek s logi rell is the logi of otiuous t-orms s ojetured Hjek. At the sme time strted sstemti stud of BL-lgers too. Ideed 2 Mthemtis Sujet Clssifitio: 6F35 3G25. Kewords d phrses: BL-lger -fold (positive) implitive filter -fold ftsti si logi -fold ftsti BL-lger (wek) -fold ftsti filter. Reeived Septemer 29; Revised Septemer Sietifi Adves Pulishers

2 28 M. SHIRVANI-GHADIKOLAI et l. Turue i [] pulished where BL-lgers were studied dedutive sstems. Dedutive sstems orrespod to susets losed with respet to Modus Poes d the re lled filters too. I [] Boole dedutive sstems d implitive dedutive sstems were itrodued. Moreover it ws proved tht these dedutive sstems oiide. I Hveshki et l. i [7] otiued lgeri lsis of BL-lgers d the itrodued e.g. implitive filters of BL-lgers. MV-lgers [] produt lgers d Gödel lgers re the most lsses of BL-lgers. Filters theor pl importt role i studig these lgers. From logil poit of view vrious filters orrespod to vrious sets of provle formule. Hjek i [5] itrodued the oepts of (prime) filters of BL-lgers. Usig prime filters of BL-lgers he proved the ompleteess of si logi. The lguge of propositiol Hjek si logi [5] otis the ir oetives ο d d the ostt. Aioms of BL re: (A) ( ϕ v/ ) (( v/ w) ( ϕ w) ) (A2) ( ϕ ο v / ) ϕ (A3) ( ϕ ο v / ) ( v/ ο ϕ) (A4) ( ϕ ο( ϕ v / )) ( v/ ο( v/ ϕ) ) (A5) ( ϕ ( v/ w) ) (( ϕ ο v/ ) w) (A5) (( ϕ ο v/ ) w) ( ϕ ( v/ w) ) (A6) (( ϕ v/ ) w) ((( v/ ϕ) w) w) (A7) w. The ove otio is geerlied to lgeri sstem i whih the required oditios re fulfilled (Defiitio ). Filters i BL-lgers re lso itrodued i [5]. The otios of implitive d positive implitive filters were itrodued i [7]. I Setio 2 we give some defiitios d theorems whih re eeded i the rest of the pper. We the itrodue the otios of -fold ftsti si logi d -fold ftsti BL-lger. We lso itrodue

3 ON -FOLD FILTERS IN BL-ALGEBRAS 29 the otio of -fold ftsti filters. We prove tht ever -fold ftsti BL-lger is lso ( + )- fold ftsti BL-lger ut emple we show tht the overse is ot true. Some hrteritio for BL-lger to e -fold ftsti is give. We defie -fold ftsti filters fter tht we stte the equivlet oditios for -fold ftsti BL-lgers. B [8] ever -fold positive implitive filter is -fold implitive filter ut the overse is ot true. We show tht uder some oditios -fold implitive filter is -fold positive implitive filter. 2. -Fold Ftsti BL-Algers Defiitio 2.. A BL-lger is lger ( A ) of tpe ( ) stisfig the followig ioms: (BL) ( A ) is ouded lttie (BL2) ( A ) is ommuttive mooid if (BL3) d form djoit pir; i.e. if d ol for ll (BL4) ( ) (BL5) ( ) ( ). Propositio 2.2 [3 4 5]. I eh BL-lger A the followig reltios hold for ll A: (p) ( ) (p2) ( ( ) ) (p3) if d ol if (p4) ( ) ( ) (p5) If the d (p6) ( )

4 3 M. SHIRVANI-GHADIKOLAI et l. (p7) ( ) ( ) (p8) ( ) ( ) (p9) [( ) ] [( ) ]. I wht follows let deotes positive iteger d A BL-lger uless otherwise speified. For elemet of A let deotes K i whih ours times d. Defiitio 2.3. A filter of BL-lger A is oempt suset F of A suh tht for ll A (f) F implies F (f2) F d impl F. F sie for ever F d F is oempt suset F. It is proved i [] tht if F is filter the it stisfies: (f3) F (f4) F d F impl F. A suset D of A is lled dedutive sstem i [] if it stisfies the ove two oditios. It is ovious tht for oempt suset D D is dedutive sstem if d ol if it is filter. Defiitio 2.4. A oempt suset F of A is lled -fold implitive filter of A if it stisfies F d: (f5) ( ) F F impl F for ll A. Theorem 2.5 [8 Theorem 4.6]. Let F e filter of A. The for ll A the followig oditios re equivlet: (i) F is -fold implitive filter of A 2 (ii) F for ll A

5 ON -FOLD FILTERS IN BL-ALGEBRAS 3 (iii) + F implies F (iv) ( ) F implies ( ) ( ) F. Defiitio 2.6. A oempt suset F of A is lled -fold positive implitive filter if it stisfies F d: (f6) (( ) ) F d F impl F for ll A. Theorem 2.7 [8 Theorem 6.2]. Ever -fold positive implitive filter of A is filter of A. Theorem 2.8 [8 Theorem 6.3]. Let F e filter of A. The for ll A the followig oditios re equivlet: (i) F is -fold positive implitive filter (ii) ( ) F implies F for ll A (iii) ( ) F implies F for ll A. Defiitio 2.9. A oempt suset F of A is lled ftsti filter if it stisfies F d: (f7) A ( ) F F impl (( ) ) F. Theorem 2. [7 Theorem 4.2]. Ever ftsti filter of A is filter of A. Defiitio 2.. Aioms of -fold ftsti si logi re those of BL plus (( ϕ v/ ) v/ ) ϕ v/ ϕ where ϕ ϕ ο ϕ ο K ο ϕ for times d where ϕ v/ deotes ϕ v/ plus /v ϕ.

6 32 M. SHIRVANI-GHADIKOLAI et l. Defiitio 2.2. A BL-lger A is lled to e -fold ftsti if it stisfies the equlit (( ) ) for ll A. Emple 2.3. Let A { }. Defie d s follows:. It is es to see tht A is m-fold ftsti BL-lger for ever m 2. Theorem 2.4. The -fold ftsti si logi is omplete for eh formul ϕ the followig oditios re equivlet: (i) The -fold ftsti si logi proves ϕ (ii) ϕ is A-tutolog for eh lierl ordered -fold ftsti BL-lger A (iii) ϕ is A-tutolog for eh -fold ftsti BL-lger A. Proof. It follows from [5 Theorem ]. This is euse -fold ftsti si logi is shemti etesio of BL. Theorem 2.5. A -fold ftsti BL-lger is m-fold ftsti BL-lger for ever m >. Proof. Let A e -fold ftsti BL-lger d m >. The we hve (( ) ) for ll A. Sie we get m m (( ) ) (( ) ). O the other hd m ( )

7 ON -FOLD FILTERS IN BL-ALGEBRAS 33 d so (( ) ). m Thus (( ) ) + whih shows tht A is m-fold ftsti BL-lger. The followig emple shows tht the overse of Theorem 2.5 is ot true. Emple 2.6. Let { }. B Defie d s follows:. The ( ) B is 2-fold ftsti BL-lger. But we hve ( ) ( ) so B is ot -fold ftsti BL-lger. Theorem 2.7. For eh BL-lger A the followig oditios re equivlet: (i) A is -fold ftsti (ii) ( ) ( ) for ll A (iii) (iv) (v) ( ).

8 M. SHIRVANI-GHADIKOLAI et l. 34 Proof. (i) (ii). Let A e -fold ftsti. The we hve (( ) ) (( ) ) ( ) ((( ) ) ) ( ) ( ). Hee for ll (( ) ) ( ). A Coversel ssume tht the iequlit (( ) ) ( ) holds for ll A. The ( ) ((( ) ) ) (( ) ) ( ) ( ) (( ) ) (( ) ). Hee ( ) ((( ) ) ) i.e. (( ) ). Now we hve ((( ) ) ) ( ) (( ) ) ( ) ( ) ( ) d so ((( ) ) ) ( ) tht is (( ) ). Hee A is -fold ftsti. (ii) (iii). Let A e suh tht d. Usig (p4) (p5) d oditio (ii) we hve ( ) ( ) (( ) ) ( ) ( ) ( ) ( )

9 ON -FOLD FILTERS IN BL-ALGEBRAS 35 d so i.e.. (iii) (iv). It is trivil. (iv) (v). Let A e suh tht. Note tht ( ) d (( ) ). It follows from (iv) tht ( ). (v) (ii). Sie ( ) we hve ( ) ( ) idutio. Sie ( ) it follows from (p5) d (v) tht ( ) ((( ) ) ) ( ). This ompletes the proof. 3. -Fold Ftsti Filter Defiitio 3.. A oempt suset F of A is lled -fold ftsti filter if it stisfies F d: (f8) ( ) (( ) ) F F A F. Defiitio 3.2. A oempt suset F of A is lled wek -fold ftsti filter of A if it stisfies F d: (f9) (( ) ) F F A impl ( ) F. Emple 3.3. Let { }. C Defie d s follows:.

10 36 M. SHIRVANI-GHADIKOLAI et l. The C is BL-lger. It is es to see tht F { } is 2-fold ftsti filter of C. Theorem 3.4. A -fold ftsti filter of BL-lger A is filter of A. Proof. Let F. Hee ( ) F. Sie F is -fold ftsti filter we get (( ) ) F tht is F is filter. The followig emple shows tht the overse of Theorem 3.4 is ot true. Emple 3.5. I Emple 3.3 { } is filter. Sie { } d 2 ( ) { } d (( ) ) {} {} is ot 2-fold ftsti filter. Theorem 3.6. Let F e filter of BL-lger A. The (i) F is -fold ftsti filter of A if d ol if (( ) ) F for ll A with F. (ii) F is wek -fold ftsti filter of A if d ol if (( ) ) F for ll A with ( ) F. Proof. Assume tht F is -fold ftsti filter of A d let A e suh tht F. The ( ) F d F. It follows from (f6) tht (( ) ) F. Coversel let F e filter of A suh tht (( ) ) F for ll A with F. Let A e suh tht ( ) F d F. The F (f4) d hee (( ) ) F ssumptio. Thus F is -fold ftsti filter of A. Similr rgumet idues the seod prt. Theorem 3.7. Let F d G e filters of BL-lger A suh tht F G. If F is -fold ftsti filter the so is G.

11 ON -FOLD FILTERS IN BL-ALGEBRAS 37 Proof. Let the A e suh tht G. Settig k : ( ) k (( ) ) ( ) ( ) F. Sie F is -fold ftsti filter it follows from Theorem 3.6 (i) d (p4) tht ( ) ((( k ) ) ) (( k ) ) (( ) ) (( k ) ) k F G. This implies from (f4) tht (( ) ) G. Sie k we k hve k so (( k ) ) (( ) ). Usig (f2) we kow tht (( ) ) G d hee G is - fold ftsti filter of A Theorem 3.6 (i). Corollr 3.8. Ever filter of BL-lger A is -fold ftsti filter if d ol if the filter { } is -fold ftsti. Theorem 3.9. A -fold ftsti filter is m-fold ftsti filter for ever m >. Proof. Let F e -fold ftsti filter of A d for A F d m >. Sie F is -fold ftsti filter we get (( m ) ) F d sie we hve m (( ) ) (( ) ). Hee (f2) (( ) ) F i.e. F is m-fold ftsti filter. m The followig emple shows tht the overse of Theorem 3.9 is ot true. Emple 3.. I Emple 2.6 it is ler tht F { } is 2-fold ftsti filter of B ut it is ot -fold ftsti filter. This is euse F. But (( ) ) F.

12 38 M. SHIRVANI-GHADIKOLAI et l. Let F e filter of A. We defie ir reltio o A s follows: For ever A if d ol if F d F. The is ogruee reltio o A. Deote F : { A} where [ ] : { A } A [ ] d defie ir opertios d o A F s follows: [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] d [ ] [ ] [ ] respetivel. The ( A F ) is BL-lger. Theorem 3.. A filter F of A is -fold ftsti if d ol if ever filter of the quotiet lger A F is -fold ftsti. Proof. Assume tht F is -fold ftsti filter of A d let A e suh tht [ ] [ ]. ) F Theorem 3.6 (i). Hee The F d so (( ) (([ ] [ ]) [ ]) [ ] [(( ) ) ] [ ] whih proves tht { } is -fold ftsti filter of A F. B Corollr ever filter of filter of A F is -fold ftsti. Coversel suppose tht ever A F is -fold ftsti d let A e suh tht F. The [ ] [ ] [ ]. Sie { } filter of A F it follows from Theorem 3.6 (i) tht is -fold ftsti [(( ) ) ] (([ ] [ ]) [ ]) [ ] [ ] tht is (( ) ) F. Hee Theorem 3.6 (i) F is - fold ftsti filter of A. Theorem 3.2. A BL-lger A is -fold ftsti if d ol if its trivil filter { } is -fold ftsti filter.

13 ON -FOLD FILTERS IN BL-ALGEBRAS 39 Proof. Let A e -fold ftsti d ( ) tht is. Sie A is -fold ftsti we hve (( ) ). Hee {} is -fold ftsti filter. Coversel ssume tht {} is -fold ftsti filter of A d let k ( ) for ever A. The k (( ) ) ( ) ( ) { } d hee (( k ) ) k tht is ( ) k. Now k implies k d so ( ) ( k ). It follows tht so tht (( k ) ) k (( ) ) k (( ) ) (( ) ) (( ) ) k it mes tht ( ) ( ). Hee Theorem 2.7 A is -fold ftsti BL-lger. Theorem 3.3. Ever -fold ftsti filter is wek -fold ftsti filter. Proof. Let F e -fold ftsti filter of A. The A F is -fold ftsti. Let A e suh tht ( ) F. The [( ) ] ([ ] [ ]) [ ] k ([ ] [ ]) [ ] [( ) ] [ ] d so [( ) ] [ ] i.e. ( ) F. It follows from Theorem 3.6 (i) tht F is wek -fold ftsti filter of A. B [8] ever -fold positive implitive filter is -fold implitive filter ut the overse is ot true. Now we show uder some oditios -fold implitive filter is -fold positive implitive filter.

14 4 M. SHIRVANI-GHADIKOLAI et l. Theorem 3.4. Let F e filter of A. The F is -fold positive implitive filter if d ol if F is -fold implitive d -fold ftsti filter. Proof. Let F e -fold implitive d -fold ftsti filter d ssume tht ( 2 2 ) F. Sie ( ) ( ) Theorem 2.5 d (f2) we hve 2 ( ) ( ) F. Sie F is -fold ftsti filter d ( ) F Theorem 3.6 (i) we hve 2 (( ( )) ( )) (( ) ( )) F. 2 O the other hds sie ( ) ( ) F (f4) we hve F. Therefore Theorem 2.8 F is -fold positive implitive filter of A. Coversel let F e -fold positive implitive filter of A. Cosider A d F. Puttig k (( ) ) the Propositio (2.2) we hve ( k ) k ( k ) ((( ) ) ) (( ) ) (( k ) ) (( ) ) (( ) ) F. Hee (f2) we hve ( ) k F d sie F is -fold k positive implitive filter Theorem (2.8) k (( ) ) F i.e. F is -fold ftsti filter of A. The followig emple shows tht ever -fold ftsti filter of A is ot -fold positive implitive filter.

15 ON -FOLD FILTERS IN BL-ALGEBRAS 4 Emple 3.5. I Emple 2.6 { } is 2-fold ftsti filter ut 2 ( ) d hee { } is ot 2-fold positive implitive filter. The followig emple shows tht ever -fold ftsti filter of A is ot -fold implitive filter. Emple 3.6. I Emple 3.3 F { } is -fold ftsti filter ut F is ot -fold implitive filter sie ( ) F d F ut F. The followig emple shows tht ever -fold implitive filter of A is ot -fold ftsti filter. Emple 3.7. Let D { }. Defie d s follows:. The ( D ) is BL-lger. It is ler tht F { } is 2-fold implitive filter ut it is ot 2-fold ftsti filter. We hve 2 F ut (( ) ) F. Referees [] C. C. Chg Algeri lsis of m vlued logi Trs. Amer. Mth. So. 88 (958) [2] R. Cigol F. Estev d L. Godo Bsi fu logi is the logi of otiuous t-orm d their residul Soft Comput. 4 (2) 6-2. [3] A. Di Nol G. Georgesu d A. Iorgulesu Pseudo BL-Alger Prt I Mult. Vl. Logi. 8(5-6) (22) [4] A. Di Nol d L. Leuste Compt Represettios of BL-Alger Deprtmet of Computer Siee Uiversit Arhus BRICS Report Siee (22). [5] P. Hjek Metmthemtis of Fu Logi Kluwer Ademi Pulishers (988).

16 42 M. SHIRVANI-GHADIKOLAI et l. [6] P. Hjek Wht is mthemtil fu logi? Fu Sets d Sstems 57 (24) [7] M. Hveshki A. Borumd Seid d E. Eslmi Some tpes of filters i BLlgers Soft Comput. (26) [8] M. Hveshki d E. Eslmi -fold filters i BL-lgers Mth. Log. Qurt. 54 (28) [9] M. Kodo d W. A. Dudek Filter theor of BL-lgers Soft Comput. 2 (28) [] E. Turue BL-lgers of si fu logi Mth. Soft Comput. 6 (999) [] E. Turue Boole dedutive sstem of BL-lgers Arh. Mth. Logi. 4 (2) g