A Characterization of Finitely Decidable Congruence Modular Varieties

Idziak, Paweł

doi:10.1090/S0002-9947-97-01904-1

A Characterization of Finitely Decidable Congruence Modular Varieties
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by Paweł M. Idziak

Trans. Amer. Math. Soc. 349 (1997), 903-934

DOI: https://doi.org/10.1090/S0002-9947-97-01904-1

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Abstract:

For every finitely generated, congruence modular variety $\mathcal {V}$ of finite type we find a finite family $\mathcal {R}$ of finite rings such that the variety $\mathcal {V}$ is finitely decidable if and only if $\mathcal {V}$ is congruence permutable and residually small, all solvable congruences in finite algebras from $\mathcal {V}$ are Abelian, each congruence above the centralizer of the monolith of a subdirectly irreducible algebra $\mathbf {A}$ from $\mathcal {V}$ is comparable with all congruences of $\mathbf {A}$, each homomorphic image of a subdirectly irreducible algebra with a non-Abelian monolith has a non-Abelian monolith, and, for each ring $R$ from $\mathcal {R}$, the variety of $R$–modules is finitely decidable.

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