Minimal group-universal varieties of semigroups

Abstract

An object X of a category qr is said to represent a group G if Aut X-~ G. A category ~ is group-universal ff its members represent all groups. There is quite a number of papers establishing group-universality of various categories. The group-universality of the graphs due to Frucht [6] and of the distributive lattices due to Birkhoff [3] count for classical results; among more recent ones we may point out the group-universality of the fields due to Fried and Kollfir [5]. An excellent survey of representations of groups by automorphisms and of both positive and negative results on group-universality can be found in Babai [2]. Babai points out that characteristic for the proofs of countless groupuniversality results is that the structure of the groups plays little role. In many cases group-universal categories proved to be monoid-universal, in the sense that every monoid M can be represented as End X by a suitable object X of the category in question, or even concrete-category-universal (shortly: binding), in the sense of full embeddability of any concrete category into the category in question (cf. [2]). Babai asks for "the healthy balance between the group structure of Aut X and the combinatorial nature of X" . From this point of view, it might be interesting to consider various groupuniversal varieties of algebras, containing non-trivial subvarieties which are not group-universal, and to find out where exactly, when moving down along the lattice of subvarieties, the group-universality is being lost. Put otherwise, we may want to describe the minimal group-universal subvarieties of a given variety. Of course, the question is interesting only when the variety has non-trivial non-group-universal subvarieties. By Birkhoff's result, every non-trivial variety of lattices is group-universal, hence the variety of all lattices is not interesting for us. (Let us note by passing that the boolean lattices are not group-universal, as established by McKenzie and Monk [9]; unfortunately, they do not form a variety of lattices.) Likewise uninteresting, as we point out in the closing of this paper, is