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Transactions of the American Mathematical Society

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



Residually small varieties with modular congruence lattices

Authors: Ralph Freese and Ralph McKenzie
Journal: Trans. Amer. Math. Soc. 264 (1981), 419-430
MSC: Primary 08B10; Secondary 06B10
MathSciNet review: 603772
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Abstract: We focus on varieties $ \mathcal{V}$ of universal algebras whose congruence lattices are all modular. No further conditions are assumed. We prove that if the variety $ \mathcal{V}$ is residually small, then the following law holds identically for congruences over algebras in $ \mathcal{V}:\beta \cdot [\delta ,\delta ] \leqslant [\beta ,\delta ]$. (The symbols in this formula refer to lattice operations and the commutator operation defined over any modular variety, by Hagemann and Herrmann.) We prove that a finitely generated modular variety $ \mathcal{V}$ is residually small if and only if it satisfies this commutator identity, and in that case $ \mathcal{V}$ is actually residually $ < n$ for some finite integer $ n$. It is further proved that in a modular variety generated by a finite algebra $ A$ the chief factors of any finite algebra are bounded in cardinality by the size of $ A$, and every simple algebra in the variety has a cardinality at most that of $ A$.

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Keywords: Residually small variety, subdirect product, modular congruence lattice, commutator of congruences, chief factors of universal algebras, groups with abelian Sylow subgroups
Article copyright: © Copyright 1981 American Mathematical Society

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