Square extension of a groupoid. (Q1771922)

The square extension of a given idempotent groupoid \(J\) is defined here as a supergroupoid of \(J\) with a carrier being a union of some sets indexed by the elements of \(J\) such that the operation of the square extension of \(J\) is an extension of the fundamental operation of \(J\). Let \(\mathcal V\) be a variety of idempotent groupoids with equational base \(B\). The authors prove that the class of all square extensions of groupoids from \(\mathcal V\) forms a variety \({\mathcal V}^*\) with the equational base \(B^*\) which is explicitly defined in the paper. The square extension of bands is considered and it is applied for a structural characterization of semigroups with \(n+1\) different essentially \(n\)-ary term operations for every positive integer \(n\).