Residually small varieties with modular congruence lattices
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- by Ralph Freese and Ralph McKenzie
- Trans. Amer. Math. Soc. 264 (1981), 419-430
- DOI: https://doi.org/10.1090/S0002-9947-1981-0603772-9
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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 $A$.References
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Bibliographic Information
- © Copyright 1981 American Mathematical Society
- 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