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Sibirskii Matematicheskii Zhurnal, 2001, Volume 42, Number 6, Pages 1361–1374
(Mi smj1393)
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Decidability of equational theories of coverings of semigroup varieties
V. Yu. Popov Ural State University
Abstract:
For every proper semigroup variety $\mathfrak X$, there exists a semigroup variety $\mathfrak Y$ satisfying the following three conditions: (1) $\mathfrak Y$ covers $\mathfrak X$, (2) $\mathfrak X$ is finitely based then so is $\mathfrak Y$, and (3) the equational theory of $\mathfrak X$ is decidable if and only if so is the equational theory of $\mathfrak Y$. If $\mathfrak X$ is an arbitrary semigroup variety defined by identities depending on finitely many variables and such that all periodic groups of $\mathfrak X$ are locally finite, then one of the following two conditions holds: (1) all nilsemigroups of $\mathfrak X$ are locally finite and (2) $\mathfrak X$ includes a subvariety $\mathfrak Y$ whose equational theory is undecidable and which has infinitely many covering varieties with undecidable equational theories.
Received: 25.01.2001
Citation:
V. Yu. Popov, “Decidability of equational theories of coverings of semigroup varieties”, Sibirsk. Mat. Zh., 42:6 (2001), 1361–1374; Siberian Math. J., 42:6 (2001), 1132–1141
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https://www.mathnet.ru/eng/smj1393 https://www.mathnet.ru/eng/smj/v42/i6/p1361
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