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Countably representable varieties of semigroups. (Russian) Zbl 0675.20046

A variety \({\mathcal V}\) of semigroups is called countably representable if each countable semigroup is isomorphically representable by matrices over some field. For a family S of identities var S denotes the variety of semigroups defined by S. Let \({\mathcal A}_ n=var\{x^ ny=y\), \(xy=yx\}\), \({\mathcal L}=var\{xy=x\}\), \({\mathcal N}=var\{xy=zu\}\), \({\mathcal R}=var\{xy=y\}.\)
The main result of this paper is the following Theorem: A variety \({\mathcal X}\) of semigroups is countably representable over a field of zero [prime] characteristic if and only if \({\mathcal X}\subseteq {\mathcal R}\vee {\mathcal L}\vee {\mathcal N}\) [\({\mathcal X}\subseteq {\mathcal R}\vee {\mathcal L}\vee {\mathcal N}\vee {\mathcal A}_{p^ m}\), p is a prime].
Reviewer: L.Martynov

MSC:

20M07 Varieties and pseudovarieties of semigroups
08B15 Lattices of varieties
20M20 Semigroups of transformations, relations, partitions, etc.
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