Abstract
Let \(\mathcal{K}\) be a finite collection of finite algebras of finite signature such that SP(\(\mathcal{K}\)) has meet semi-distributive congruence lattices. We prove that there exists a finite collection \(\mathcal{K}\) 1 of finite algebras of the same signature, \(\mathcal{K}_1 \supseteq \mathcal{K}\), such that SP(\(\mathcal{K}\) 1) is finitely axiomatizable.We show also that if \(HS(\mathcal{K}) \subseteq SP(\mathcal{K})\), then SP(\(\mathcal{K}\) 1) is finitely axiomatizable. We offer new proofs of two important finite basis theorems of D. Pigozzi and R. Willard. Our actual results are somewhat more general than this abstract indicates.
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References
Baker, K., ‘Primitive satisfaction and equational problems for lattices and other algebras’, Transactions Amer. Math. Soc. 190:125–150, 1974.
Baker, K., ‘Finite equational bases for finite algebras in a congruence distributive equational class’, Advances in Math. 24:207–243, 1977.
Baker, K. and Ju Wang, ‘Definable principal subcongruences’, (preprint).
Baker, K., G. McNulty, and Ju Wang, ‘An extension of Willard’s finite basis theorem: congruence meet semidistributive varieties of finite critical depth’, (preprint).
Belkin, V. P., ‘Quasi-identities of finite rings and lattices’, Algebra i Logika 17:247–259, 1978.
Bestsennyi, I. P., ‘On quasi-identities of finite nilpotent algebras’, Algebra Universalis 36:60–278, 1996 (no. 2).
Bryant, R., ‘The laws of finite pointed groups’, Bull. London Math. Soc. 14: 119–123, 1982.
Czelakowski, J. and W. Dziobiak, ‘Congruence distributive quasivarieties whose finitely subdirectly irreducible members form a universal class’, Algebra Universalis 27:128–149, 1990 (no. 1).
Dziobiak, W., ‘Finitely generated congruence distributive quasivarieties of algebras’, Fundamenta Mathematicae 133:47–57, 1989.
Dziobiak, W., ‘Finite bases for finitely generated relatively congruence distributive quasivarieties’, Algebra Universalis, 28:303–323, 1991.
Gerhard, J. A., ‘The lattice of equational classes of idempotent semigroup’, Journal of Algebra 15:195–224, 1970.
Hobby, D. and R. McKenzie, The structure of finite algebras, American Math. Soc., Contemporary Mathematics, vol. 76, 1988.
Ježek, J., M. Maróti, and R. McKenzie, ‘Quasi-equational theories of flat algebras’, to appear in Czechoslovak Math. J.
Kearnes, K. and R. McKenzie, ‘Commutator theory for relatively modular quasi-varieties’ Trans. Amer. Math. Soc. 331:465–502, 1992.
Kruse, R. L., ‘Identities satisfied by a finite ring’, Journal of Algebra 26: 298–318, 1973.
L’vov, I. V. ‘Varieties of associative rings, I, II,’ Algebra i Logika 12: 267–298, 667–668, 735, 1973.
Lawrence, J. and R. Willard, ‘On finitely based groups and non finitely based quasivarieties’, Journal of Algebra 203:9–11, 1998 (no. 1).
Lyndon, R., ‘Identities in two-valued calculi’, Transactions Amer. Math. Soc. 71:457–465, 1951.
Margolis S. and M. V. Sapir, ‘Quasi-identities of finite semigroups and symbolic dynamics’, Israel J. Math. 92:317–331, 1995 (no. 1–3).
McKenzie, R., ‘A new product of algebras and a type reduction theorem’, Algebra Universalis 18:29–69, 1984.
McKenzie, R., ‘Finite equational bases for congruence modular varieties’, Algebra Universalis 24:224–250, 1987.
McKenzie, R., ‘The residual bound of a finite algebra is not computable’, International Journal of Algebra and Computation 6:29–48, 1996.
McKenzie, R. ‘Tarski’s finite basis problem is undecidable’, International Journal of Algebra and Computation 6:49–104, 1996.
McNulty, G. and Ju Wang, ‘The class of subdirectly irreducible groups generated by a finite group is finitely axiomatizable’, (preprint).
Murskii, V. L., ‘The existence in three-valued logic of a closed class with finite basis not having a finite complete set of identities’, Dokl. Akad. Nauk. SSR 163: 815–818, 1965; English translation in Soviet Math. Dokl. 6:1020–1024,1965.
Oates, S. and M. B. Powell, ‘Identical relations in finite groups’, Journal of Algebra 1:11–39, 1964.
Ol’shanskii, A. Yu, ‘Conditional identities in finite groups’, Siber. Math. J. 15:1000–1003, 1974.
Perkins, P., ‘Bases for equational theories of semigroups’, Journal of Algebra 11:293–314, 1969.
Pigozzi, D., ‘A finite basis theorem for quasivarieties’, Algebra Universalis 22:1–13, 1986.
Pigozzi, D., ‘Finite basis theorems for relatively congruence distributive quasivarieties’, Trans. Amer. Math. Soc. 310:499–533, 1988.
Polin, S. V., ‘On the identities of finite algebras’, Siberian Math. J. 17: 1356–1366, 1976.
Sapir, M, V., ‘Inherently non-finitely based finite semigroups’, Mat. Sb. (N.S.) 133(175):154–166, 270; English translation in Math. USSR-Sb. 61: 55–166, 1988 (no. 1).
Tumanov, V. I., ‘On finite lattices not having an independent basis of quasi-identities’, Mat. Zametki 36:625–634, 1984.
Vishin, V. V., ‘Identity transformations in a four-valued logic’, Dokl. Akad. Nauk. SSSR 150:719–721, 1963; English translation in Soviet Math. Dokl. 4:724,1963.
Willard, R. ‘Tarski’s finite basis problem via A (\(\mathcal{T}\))’, Transactions Amer. Math. Soc. 349:2755–2774, 1997.
Willard, R., ‘A finite basis theorem for residually finite, congruence meet-semidistributive varieties’, Journal of Symbolic Logic 65:187–200, 2000.
Willard, R., ‘Extending Baker’s theorem’, Algebra Universalis 45:335–344, 2001.
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While working on this paper, the first author was partially supported by the Hungarian National Foundation for Scientific Research (OTKA) grant no. T37877 and the second author was supported by the US National Science Foundation grant no. DMS-0245622.
Special issue of Studia Logica: “Algebraic Theory of Quasivarieties” Presented by M. E. Adams, K. V. Adaricheva, W. Dziobiak, and A. V. Kravchenko
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Maróti, M., McKenzie, R. Finite basis problems and results for quasivarieties. Stud Logica 78, 293–320 (2004). https://doi.org/10.1007/s11225-005-3320-5
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DOI: https://doi.org/10.1007/s11225-005-3320-5