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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Finitely decidable congruence modular varieties

Author: Joohee Jeong
Journal: Trans. Amer. Math. Soc. 339 (1993), 623-642
MSC: Primary 08B10; Secondary 03B25
MathSciNet review: 1150016
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Abstract: 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}$.

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Article copyright: © Copyright 1993 American Mathematical Society

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