Brusnitsyna, S. G.; Sukhanov, E. V. Countably representable varieties of semigroups. (Russian) Zbl 0675.20046 Mat. Zap. 14, No. 3, 14-22 (1988). 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. Keywords:countably representable; countable semigroup; representable by matrices; variety of semigroups PDFBibTeX XMLCite \textit{S. G. Brusnitsyna} and \textit{E. V. Sukhanov}, Mat. Zap. 14, No. 3, 14--22 (1988; Zbl 0675.20046)