Minimal group-universal varieties of semigroups
A category \({\mathcal C}\) is ''group-universal'' if any group G is isomorphic to Aut X for some object X in \({\mathcal C}\), (in which case X is said to represent G). For instance group-universality of graphs was proved by \textit{R. Frucht} [Compos. Math. 6, 239-250 (1938; Zbl 0020.07804)]. Group- universality of semigroups (in fact commutative semigroups) was proven by \textit{Z. Hedrlín} and \textit{J. Lambek} [J. Algebra 11, 195-212 (1969; Zbl 0206.025)]. The authors determine here the minimal group-universal varieties of semigroups. They are (1) the variety of semilattices; (2) the variety \({\mathcal M}\) generated by a certain four-element semigroup; (3) the variety of all inflations of rectangular bands; (4) the varieties \(\sqrt{{\mathcal L}}\) and \(\sqrt{{\mathcal R}}\), consisting of all semigroups S for which \(S^ 2\) is left, respectively right, zero; and for each prime p, (5) the variety of all inflations of elementary abelian groups of exponent p, and (6) the variety generated by the \(2\times 2\) nonorthodox Rees matrix semigroup over the cyclic group of order p. In each case a representing semigroup for a given group is constructed via a representing graph. This yields bounds on the cardinalities required.
minimal group-universal varieties of semigroups
scientific article; zbMATH DE number 3948609
scientific article