The variety of all commutative BCK-algebras is generated by its finite members as a quasivariety

• Autorzy: Pałasiński, M.
• Języki publikacji: EN

Abstrakty

EN

We prove the result announced by the title as well as some of its consequences.

Słowa kluczowe

EN

BCK-algebra

• Czasopismo: Demonstratio Mathematica
• Rocznik: 2012
• Tom: Vol. 45, nr 3
• Strony: 495-518
• Opis fizyczny: Bibliogr. 39 poz.

Twórcy

autor

Pałasiński, M.

• University Of Information Technology And Management Rzeszów, Poland,

Bibliografia

• [1] R. Balbes, Ph. Dwinger, Distributive lattices, University of Missouri Press, 1974.
• [2] W. J. Blok, I. Ferreirim, Hoops and their implicational reducts (abstract), algebraic methods in logic and in computer science, Banach Center Publ. 28 (1993), 219–230.
• [3] W. Blok, I. Ferreirim, On the structure of hoops, Algebra Universalis 43 (2000), 233–257.
• [4] W. Blok, D. Pigozzi, On the structure of varieties with equationally definable principal congruences III, Algebra Universalis 32 (1994), 545–608.
• [5] W. J. Blok, J. G. Raftery, On the quasivariety of BCK-algebras and its subvarieties, Algebra Universalis 33 (1995), 68–90.
• [6] W. J. Blok, J. G. Raftery, Varieties of commutative residuated integral pomonoids and their residuation subreducts, J. Algebra 190 (1997), 280–328.
• [7] W. J. Blok, C. Van Alten, The finite embeddability property for residuated lattices, pocrims and BCK-algebras, Algebra Universalis 48 (2002), 253–271.
• [8] W. J. Blok, C. Van Alten, On the finite embeddability property for residuated ordered grupoids, Trans. Amer. Math. Soc. 357 (2005), 4141–4157.
• [9] C. C. Chang, Algebraic analysis of many valued logics, Trans. Amer. Math. Soc. 88 (1958), 467–490.
• [10] C. C. Chang, A new proof of the completeness of the Łukasiewicz axioms, Trans. Amer. Math. Soc. 93 (1959), 74–80.
• [11] R. Cignoli, M. L. D’Ottaviano, D. Mundici, Algebraic Foundations of Many-Valued Reasoning, Kluwer Academic Pub., Dordrecht, 2000.
• [12] W. H. Cornish, T. Sturm, T. Traczyk, Embedding of commutative BCK-algebra into distributive lattice BCK-algebras, Math. Japon. 29 (1984), 309–320.
• [13] A. Di Nola, A. Lettieri, Equational characterization of all varieties of MV-algebras, J. Algebra 221 (1999), 463–474.
• [14] I. Ferreirim, On a conjecture by Andrzej Wroński for BCK-algebras and subreducts of hoops, Sci. Math. Jpn. 53 (2001), 119–132.
• [15] J. M. Font, A. J. Rodríguez, A. Torrens, Wajsberg algebras, Stochastica 8 (1984), 5–31.
• [16] J. Gispert, D. Mundici, MV-algebras: a variety for magnitudes with Archimedean units, Algebra Universalis 53 (2005), 7–43.
• [17] G. Grätzer, Universal Algebra, Van Nostrand, 1968.
• [18] K. Iséki, S. Tanaka, An introduction to the theory of BCK-algebras, Math. Japon. 23 (1978), 1–26.
• [19] K. Iséki, BCK-algebras with condition (S), Math. Japon. 24 (1979), 107–120.
• [20] Y. Komori, Super-Łukasiewicz implicational logics, Nagoya Math. J. 72 (1978), 127–133.
• [21] J. Łukasiewicz, A. Tarski, Untersuchungen uber den Aussagenkalkul, Comptes-rendus des seances de la Societe de Sciences et de Lettres de Varsovie, Cl. III, 23 (1930), 30–50.
• [22] D. Mundici, Bounded commutative BCK-algebras have the amalgamation property, Math. Japon. 32 (1987), 279–282.
• [23] D. Mundici, Interpretation of AF C*-algebras in Łukasiewicz sentential calculus, J.Funct. Anal. 65 (1986), 15–63.
• [24] D. Mundici, MV-algebras are categorically equivalent to bounded commutative BCK-algebras, Math. Japon. 31 (1986), 889–894.
• [25] K. Pałasińska, Amalgamation property in some classes of BCK-algebras, Rep. Math. Logic 21 (1987), 73–84.
• [26] M. Pałasiński, Some remarks on BCK-algebras, Mathematics Seminar Notes, Kobe Univ. 8 (1980), 137–144.
• [27] M. Pałasiński, Representation theorem for commutative BCK-algebras, Mathematics Seminar Notes, Kobe Univ. 10 (1982), 473–478.
• [28] M. Pałasiński, On BCK-algebras with the operation (S), Polish Acad. Sci., Bull. of the Section of Logic 13 (1984), 13–20.
• [29] M. Pałasiński, A. Romanowska, Varieties of commutative BCK-algebras not generated by their finite members, Demonstratio Math. 18 (1985), 499–508.
• [30] A. Romanowska, T. Traczyk, On commutative BCK-algebras, Math. Japon. 25 (1980), 567–583.
• [31] A. Romanowska, T. Traczyk, On the structure of commutative BCK-chains, Math. Japon. 26 (1981), 433–442.
• [32] A. Romanowska, T. Traczyk, Commutative BCK-algebras. Subdirectly irreducible algebras and varieties, Math. Japon. 27 (1982), 35–48.
• [33] A. Romanowska, T. Traczyk, Simple commutative BCK-chains, Math. Japon. 29 (1984), 171–181.
• [34] W. Suchon, Definition des foncteurs modaux de Moisil dans le calcul n-valent des propositions de Łukasiewicz avec implication et négation, Rep. Math. Logic 2 (1974), 43–47.
• [35] S. Tanaka, On commutative algebras, Mathematics Seminar Notes, Kobe Univ. 3 (1975), 59–64.
• [36] H. Yutani, An axiom system for a BCK-algebra with the condition (S), Mathematics Seminar Notes, Kobe Univ. 7 (1979), 427–432.
• [37] H. Yutani, Quasi-commutative BCK-algebras and congruence relations, Math. Seminar Notes, Kobe Univ. 5 (1977), 469–480.
• [38] A. Wroński, On varieties of commutative BCK-algebras not generated by their finite members, Math. Japon. 30 (1985), 227–233.
• [39] A. Wroński, An algebraic motivation for BCK-algebras, Math. Japon. 30 (1985), 187–193.