GROUPOID POWERS

Abstract

The following statement is the main result of the paper. If \(\mathcal{V}\) is the variety of groupoids (commutative groupoids), or \(\mathcal{V}\) is the variety of \(n\)-idempotent groupoids (commutative \(n\)-idempotent groupoids), i.e. groupoids (commutative groupoids) with an axiom \(x^{n+1}=x\), \(n\geq 2\), then the monoid of powers is free with a countable infinite basis.