Finitely decidable congruence modular varieties
Author:
Joohee Jeong
Journal:
Trans. Amer. Math. Soc. 339 (1993), 623642
MSC:
Primary 08B10; Secondary 03B25
MathSciNet review:
1150016
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Abstract: A class of algebras of the same type is said to be finitely decidable iff the first order theory of the class of finite members of is decidable. Let be a congruence modular variety. In this paper we prove that if is finitely decidable, then the following hold. (1) Each finitely generated subvariety of has a finite bound on the cardinality of its subdirectly irreducible members. (2) Solvable congruences in any locally finite member of are abelian. In addition we obtain various necessary conditions on the congruence lattices of finite subdirectly irreducible algebras in .
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 [1]
 M. Albert, A sufficient condition for finite decidability, Trans. Amer. Math. Soc. (to appear). MR 1419361 (97m:03024)
 [2]
 S. Burris and R. McKenzie, Decidability and Boolean representations, Mem. Amer. Math. Soc. 32 (1981). MR 617896 (83j:03024)
 [3]
 S. Burris, R. McKenzie, and M. Valeriote, Decidable discriminator varieties form unary varieties, J. Symbolic Logic 56 (1991), 13551368. MR 1136462 (93f:08006)
 [4]
 S. Burris and H. Sankappanavar, A course in universal algebra, Graduate Texts in Math., SpringerVerlag, New York, 1981. MR 648287 (83k:08001)
 [5]
 J. Ershov, I. Lavrov, A. Tatmanov, and M. Taitslin, Elementary theories, Russian Math. Surveys 20 (1965), 35105.
 [6]
 R. Freese and R. McKenzie, Commutator theory for congruence modular varieties, London Math. Soc. Lecture Note Ser., no. 125, 1987. MR 909290 (89c:08006)
 [7]
 D. Hobby and R. McKenzie, The structure of finite algebras, Contemp. Math., vol. 76, Amer. Math. Soc., Providence, R.I., 1988. MR 958685 (89m:08001)
 [8]
 P. Idziak, Reduced subpowers and the decision problem for finite algebras in arithmetical varieties, Algebra Universalis 25 (1988), 365383. MR 969157 (89k:08005)
 [9]
 , Varieties with decidable finite algebras. I: Linearity, Algebra Universalis 26 (1989), 234246. MR 999233 (90d:08006a)
 [10]
 , Varieties with decidable finite algebras. II: Permutability, Algebra Universalis 26 (1989), 247256. MR 999234 (90d:08006b)
 [11]
 , Characterization of finitely decidable congruence modular varieties, preprint, 1992.
 [12]
 P. Idziak and M. Valeriote, A property of the solvable radical in finitely decidable varieties, preprint. MR 1881369 (2002m:03009)
 [13]
 J. Jeong, Finitary decidability implies congruence permutability for congruence modular varieties, Algebra Universalis 29 (1992), 441448. MR 1170199 (93h:08010)
 [14]
 , On finitely decidable varieties, Ph. D. thesis, Univ. of California, Berkeley, 1991.
 [15]
 R. McKenzie, Narrowness implies uniformity, Algebra Universalis 15 (1982), 6785. MR 663953 (83i:08003)
 [16]
 R. McKenzie, G. McNulty, and W. Taylor, Algebras, lattices, varieties, vol. 1, Wadsworth & Brooks/Cole, Monterey, Calif., 1987.
 [17]
 R. McKenzie and M. Valeriote, The structure of decidable locally finite varieties, Progr. Math., no. 79, Birkhäuser, Boston, Mass., 1989. MR 1033992 (92j:08001)
 [18]
 M. Rabin, A simple method for undecidability proofs and some applications, Logic, Methodology Philos. Sci., NorthHolland, Amsterdam, 1965, pp. 5868. MR 0221924 (36:4976)
 [19]
 M. Valeriote, Decidable unary varieties, Algebra Universalis 27 (1987), 120. MR 921525 (89b:08008)
 [20]
 M. Valeriote and R. Willard, Some properties of finitely decidable varieties, Internat. J. Algebra Comput. 2 (1992), 89101. MR 1167529 (93g:08002)
 [21]
 R. Willard, Manuscript, 1990.
 [22]
 A. Zamyatin, A prevariety of semigroups whose elementary theory is solvable, Algebra and Logic 12 (1973), 233241.
 [23]
 , Varieties of associative rings whose elementary theory is decidable, Soviet Math. Dokl. 17 (1976), 996999.
 [24]
 , A nonabelian variety of groups has an undecidable elementary theory, Algebra and Logic 17 (1978), 1317.
 [25]
 , Prevarieties of associative rings whose elementary theory is decidable, Sibirsk. Math. Zh. 19 (1978), 890901. MR 515180 (80i:16049)
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DOI:
http://dx.doi.org/10.1090/S00029947199311500166
PII:
S 00029947(1993)11500166
Article copyright:
© Copyright 1993
American Mathematical Society
