A glimpse of deductive systems in algebra

• Autorzy: Dumitru Buşneag , Sergiu Rudeanu
• Języki publikacji: EN

Abstrakty

EN

The concept of a deductive system has been intensively studied in algebraic logic, per se and in connection with various types of filters. In this paper we introduce an axiomatization which shows how several resembling theorems that had been separately proved for various algebras of logic can be given unique proofs within this axiomatic framework. We thus recapture theorems already known in the literature, as well as new ones. As a by-product we introduce the class of pre-BCK algebras.

Słowa kluczowe

EN

Deductive system; Filter; ⊙-filter; Strong filter; Algebra of logic; Hilbert algebra; pre-BCK algebra

Kategorie tematyczne

• 03G20: \L ukasiewicz and Post algebras
• 06F35: BCK-algebras, BCI-algebras
• 06B10: Ideals, congruence relations
• 03G25: Other algebras related to logic
• 06D20: Heyting algebras
• 06A12: Semilattices
• 03G05: Boolean algebras
• 06D35: MV-algebras

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2010
• Tom: 8
• Numer: 4
• Strony: 688-705

Daty

wydano

2010-08-01

online

2010-07-24

Twórcy

autor

Dumitru Buşneag

• Faculty of Mathematics and Informatics, University of Craiova, Craiova, Romania

autor

Sergiu Rudeanu

• Faculty of Mathematics and Informatics, University of Bucharest, Bucharest, Romania

Bibliografia

• [1] Abbott J.C., Implicational algebras, Bull. Math. Soc. Sci. Math. R. S. Roumanie, 1967, 11(59), 3–23
• [2] Abbott J.C., Semi-boolean algebra, Mat. Vesnik., 1967, 4(19), 177–198
• [3] Birkhoff G., Lattice Theory, 3rd ed., American Mathematical Society, Providence, 1967
• [4] Boicescu V., Filipoiu A., Georgescu G., Rudeanu S., Łukasiewicz-Moisil Algebras, North-Holland, Amsterdam, 1991
• [5] Buşneag D., Contribuţi la studiul algebrelor Hilbert, Ph.D. thesis, Univ. Bucharest, 1985
• [6] Buşneag D., On the maximal deductive systems of a bounded Hilbert algebra, Bull. Math. Soc. Sci. Math. R. S. Roumanie, 1987, 31(79)(1), 9–21
• [7] Buşneag D., Hertz algebras of fractions and maximal Hertz algebra of quotients, Math. Japon., 1994, 39(3), 461–469
• [8] Buşneag D., Categories of Algebraic Logic, Editura Academiei Române, Bucharest, 2006
• [9] Buşneag D., Piciu D., On the lattice of deductive systems of a BL-algebra, Cent. Eur. J. Math., 2003, 1(2), 221–237 http://dx.doi.org/10.2478/BF02476010[Crossref]
• [10] Celani S.A., Cabrer L.M., Montangie D., Representation and duality for Hilbert algebras, Cent. Eur. J. Math., 2009, 7(3), 463–478 http://dx.doi.org/10.2478/s11533-009-0032-5[Crossref][WoS]
• [11] Chajda I., The lattice of deductive systems on Hilbert algebras, Southeast Asian Bull. Math., 2002, 26(1), 21–26 http://dx.doi.org/10.1007/s100120200022[Crossref]
• [12] Chajda I., Halaš R., Algebraic properties of pre-logics, Math. Slovaca, 2002, 52(2), 157–175
• [13] Chajda I., Halaš R., Abbott groupoids, J. Mult.-Valued Logic Soft Comput., 2004, 10(4), 385–394
• [14] Chajda I., Halaš R., Deductive systems and Galois connections, In: Galois Connections and Applications, Kluwer, Dordrecht, 2004, 399–411
• [15] Chajda I., Halaš R., Distributive implication groupoids, Cent. Eur. J. Math., 2007, 5(3), 484–492 http://dx.doi.org/10.2478/s11533-007-0021-5[Crossref][WoS]
• [16] Chajda I., Halaš R., Kuhr J., Semilattice Structures, Heldermann, Lemgo, 2007
• [17] Cignoli R., Algebras de Moisil de order n, Ph.D. thesis, Universidad Nacional del Sur, Bahía Blanca, 1969
• [18] Diego A., Sobre álgebras de Hilbert, Notas de Lógica Matematica, 12, Instituto de Matemática, Univ. Nacional del Sur, Bahía Blanca, 1965
• [19] Diego A., Sur les algèbres de Hilbert, Collection de Logique Math., Sér. A, 21, Gauthier-Villars, Paris, 1966
• [20] Figallo A., Ziliani A., Remarks on Hertz algebras and implicative semilattices, Bull. Sect. Logic Univ. Łódź, 2005, 34(1), 37–42
• [21] Georgescu G., Algebra logicii - logica algebrică (I), Revista de Logică, 2009, 2, available at: http://egovbus.net/rdl/articole/No1Art34.pdf
• [22] Grätzer G., Universal Algebra, 2nd ed., Springer, New York-Heidelberg, 1979
• [23] Halaš R., Ideals and D-systems in orthoimplication algebras, J. Mult.-Valued Logic Soft Comput., 2005, 11(3–4), 309–316
• [24] Iorgulescu A., Algebras of Logic as BCK Algebras, Editura ASE, Bucharest, 2008
• [25] Iséki K., Tanaka S., Ideal theory of BCK-algebras, Math. Japon., 1976, 21(4), 351–366
• [26] Jun Y.B., Deductive systems of Hilbert algebras, Math. Japon., 1996, 43(1), 51–54
• [27] Katriňák T., Bemerkung über pseudokomplementaren halbgeordneten Mengen, Acta Fac. Rerum Natur. Univ. Comenian. Math., 1968, 19, 181–185
• [28] Liu L.Z., Li K.T., Boolean filters and positive implicative filters of residuated lattices, Inform. Sci., 2007, 177(24), 5725–5738 http://dx.doi.org/10.1016/j.ins.2007.07.014[WoS][Crossref]
• [29] Monteiro A., L’arithmétique des filtres et les espaces topologiques, In: De Segundo Symposium de Matematicas-Villavicencio (Mendoza, Buenos Aires), 21–25 July 1954, Centro di Cooperacion UNESCO para America Latina, Montevideo, 129–172; Notas de Lógica Matemática, 1974, 30, 157
• [30] Monteiro A., Sur la définition des algebres de Łukasiewicz trivalentes, Bull. Math. Soc. Sci. Math. Phys. R. P. Roumaine, 1963, 7(55), 3–12
• [31] Nemitz W.C., On the lattice of filters of an implicative semi-lattice, J. Math. Mech., 1968/69, 18, 683–688
• [32] Pałasiński M., On ideal and congruence lattices of BCK algebras, Math. Japon., 1981, 26(5), 543–544
• [33] Piciu D., Algebras of Fuzzy Logic, Ed. Universitaria Craiova, 2007
• [34] Rasiowa H., An Algebraic Approach to Non-Classical Logics, North-Holland, Amsterdam, 1974
• [35] Roman S., Lattices and Ordered Sets, Springer, New York, 2008
• [36] Rudeanu S., On relatively pseudocomplemented posets and Hilbert algebras, An. Śtiinţ. Univ. Al. I. Cuza Iaşi Secţ. I a Mat., 1985, 31(suppl.), 74–77
• [37] Rudeanu S., Localizations and fractions in algebra of logic, J. Mult.-Valued Logic Soft Comput., 2010, 16(3–5), 467–504
• [38] Turunen E., Mathematics Behind Fuzzy Logic, Physica-Verlag, Heidelberg, 1999
• [39] Turunen E., BL-algebras of basic fuzzy logic, Mathware Soft Comput., 1999, 6(1), 49–61

Identyfikatory

DOI

10.2478/s11533-010-0041-4