FREE \((V;n,m)\)-GROUPOIDS - Markovski, Smile (Matematichki Bilten, 1992)
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 \(m≥ 2\) 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.