FREE \((V;n,m)\)-GROUPOIDS

Abstract

The class of \((V;n,m)\)-groupoids, where \(V\) is a nontrivial variety of semigroups, and \(n,m\) are positive integers, is defined in [5]. Here we give a convenient description of free \((V;n,m)\)-groupoids. Almost all usual properties of "freeness" hold, but some exceptions also arise. For example, if \(m\geq 2\) and if \(V\) is the variety of commutative semigroups, then the set of automorphisms of a free \((V;n,m)\)-groupoid with basis \(B\), which induce the identity transformation on \(B\), is infinite.