Free groupoids with x^{2}x^{2}=x^{3}x^{3}

Abstract

A description of free objects in the variety V of groupoids defined by the identity x^{2}x^{2}=x^{3}x^{3} is obtained. The following method is used: one of the sides of the identity is considered as "suitable" and the other as "unsuitable" one. First, the left-hand side x^{2}x^{2}is chosen as "suitable" and the set of elements of F (F being an absolutely free groupoid with a basis B) containing no parts that have the form x^{3}x^{3} is taken as a "candidate" for the carrier of the desired free object in V. Continuing this procedure, a V-free object is obtained. Another construction of V-free object is obtained by choosing the right-hand side x^{3}x^{3} as "suitable" one.