Free objects in the variety of groupoids defined by the identity xx^(m)=x(m+1)

Abstract

A construction of free objects in the variety V_(m) of groupoids defined by the identity xx^(m)=x^(m+1), where m is a fixed positive integer, and (k) is a transformation of a groupoid G=(G, .), defined by x^(0)=x, x^(k+1)=(x^(k))^2, is given. A class of injective groupoids in V_(m) is defined and a corresponding Bruck theorem for this variety is proved. It is shown that the class of free groupoids in V_(m) is a proper subclass of the class of injective groupoids in V_(m) .