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Finitely decidable congruence modular varieties. (English) Zbl 0788.08004

Summary: A class \(\mathcal V\) of algebras of the same type is said to be finitely decidable iff the first-order theory of the class of finite members of \(\mathcal V\) is decidable. Let \(\mathcal V\) be a congruence modular variety. In this paper we prove that if \(\mathcal V\) is finitely decidable, then the following hold. (1) Each finitely generated subvariety of \(\mathcal V\) has a finite bound on the cardinality of its subdirectly irreducible members. (2) Solvable congruences in any locally finite member of \(\mathcal V\) are Abelian. In addition we obtain various necessary conditions on the congruence lattices of finite subdirectly irreducible algebras in \(\mathcal V\).

MSC:

08B10 Congruence modularity, congruence distributivity
03B25 Decidability of theories and sets of sentences
08B26 Subdirect products and subdirect irreducibility
08A30 Subalgebras, congruence relations
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[1] M. H. Albert, A sufficient condition for finite decidability, Algebra Universalis 36 (1996), no. 4, 467 – 482. · Zbl 0901.03012 · doi:10.1007/BF01233917
[2] Stanley Burris and Ralph McKenzie, Decidability and Boolean representations, Mem. Amer. Math. Soc. 32 (1981), no. 246, viii+106. · Zbl 0483.03019 · doi:10.1090/memo/0246
[3] Stanley Burris, Ralph McKenzie, and Matthew Valeriote, Decidable discriminator varieties from unary varieties, J. Symbolic Logic 56 (1991), no. 4, 1355 – 1368. · Zbl 0747.08008 · doi:10.2307/2275480
[4] Stanley Burris and H. P. Sankappanavar, A course in universal algebra, Graduate Texts in Mathematics, vol. 78, Springer-Verlag, New York-Berlin, 1981. · Zbl 0478.08001
[5] J. Ershov, I. Lavrov, A. Tatmanov, and M. Taitslin, Elementary theories, Russian Math. Surveys 20 (1965), 35-105.
[6] Ralph Freese and Ralph McKenzie, Commutator theory for congruence modular varieties, London Mathematical Society Lecture Note Series, vol. 125, Cambridge University Press, Cambridge, 1987. · Zbl 0636.08001
[7] David Hobby and Ralph McKenzie, The structure of finite algebras, Contemporary Mathematics, vol. 76, American Mathematical Society, Providence, RI, 1988. · Zbl 0721.08001
[8] PawełM. Idziak, Reduced subpowers and the decision problem for finite algebras in arithmetical varieties, Algebra Universalis 25 (1988), no. 3, 365 – 383. · Zbl 0671.03004 · doi:10.1007/BF01229982
[9] PawełM. Idziak, Varieties with decidable finite algebras. I. Linearity, Algebra Universalis 26 (1989), no. 2, 234 – 246. , https://doi.org/10.1007/BF01236870 PawełM. Idziak, Varieties with decidable finite algebras. II. Permutability, Algebra Universalis 26 (1989), no. 2, 247 – 256. · Zbl 0679.08003 · doi:10.1007/BF01236871
[10] PawełM. Idziak, Varieties with decidable finite algebras. I. Linearity, Algebra Universalis 26 (1989), no. 2, 234 – 246. , https://doi.org/10.1007/BF01236870 PawełM. Idziak, Varieties with decidable finite algebras. II. Permutability, Algebra Universalis 26 (1989), no. 2, 247 – 256. · Zbl 0679.08003 · doi:10.1007/BF01236871
[11] -, Characterization of finitely decidable congruence modular varieties, preprint, 1992.
[12] PawełM. Idziak and Matthew Valeriote, A property of the solvable radical in finitely decidable varieties, Fund. Math. 170 (2001), no. 1-2, 69 – 86. Dedicated to the memory of Jerzy Łoś. · Zbl 0996.08007 · doi:10.4064/fm170-1-4
[13] Joohee Jeong, Finitary decidability implies congruence permutability for congruence modular varieties, Algebra Universalis 29 (1992), no. 3, 441 – 448. · Zbl 0761.08003 · doi:10.1007/BF01212443
[14] -, On finitely decidable varieties, Ph. D. thesis, Univ. of California, Berkeley, 1991.
[15] Ralph McKenzie, Narrowness implies uniformity, Algebra Universalis 15 (1982), no. 1, 67 – 85. · Zbl 0505.08004 · doi:10.1007/BF02483709
[16] R. McKenzie, G. McNulty, and W. Taylor, Algebras, lattices, varieties, vol. 1, Wadsworth & Brooks/Cole, Monterey, Calif., 1987. · Zbl 0611.08001
[17] Ralph McKenzie and Matthew Valeriote, The structure of decidable locally finite varieties, Progress in Mathematics, vol. 79, Birkhäuser Boston, Inc., Boston, MA, 1989. · Zbl 0702.08001
[18] Michael O. Rabin, A simple method for undecidability proofs and some applications, Logic, Methodology and Philos. Sci. (Proc. 1964 Internat. Congr.), North-Holland, Amsterdam, 1965, pp. 58 – 68.
[19] Matthew A. Valeriote, Decidable unary varieties, Algebra Universalis 24 (1987), no. 1-2, 1 – 20. · Zbl 0607.08004 · doi:10.1007/BF01188378
[20] Matthew A. Valeriote and Ross Willard, Some properties of finitely decidable varieties, Internat. J. Algebra Comput. 2 (1992), no. 1, 89 – 101. · Zbl 0759.08007 · doi:10.1142/S0218196792000074
[21] R. Willard, Manuscript, 1990.
[22] A. Zamyatin, A prevariety of semigroups whose elementary theory is solvable, Algebra and Logic 12 (1973), 233-241. · Zbl 0289.02034
[23] -, Varieties of associative rings whose elementary theory is decidable, Soviet Math. Dokl. 17 (1976), 996-999. · Zbl 0356.02041
[24] -, A non-abelian variety of groups has an undecidable elementary theory, Algebra and Logic 17 (1978), 13-17. · Zbl 0427.03009
[25] A. P. Zamjatin, Prevarieties of associative rings whose elementary theory is decidable, Sibirsk. Mat. Zh. 19 (1978), no. 6, 1266 – 1282, 1437 (Russian).
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