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Transactions of the American Mathematical Society

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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
DOI: https://doi.org/10.1090/S0002-9947-1981-0603772-9
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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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1981-0603772-9
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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