Generalising congruence regularity for varieties. (Q1771948)

The concept of \([s,t]\)-regularity of a variety \(\mathcal V\) is introduced for a couple \([s,t]\) of \(n_1\)-ary and \(n_2\)-ary term \(s\) and \(t\) in the signature of \(\mathcal V\). In fact this concept is not new, and it was already treated by \textit{G.\ D.\ Barbour} and \textit{J.\ G.\ Raftery} [Czech.\ Math.\ J.\ 47, 317--325 (1997; Zbl 0927.08001)] in a more general setting. The Mal'tsev condition presented by the author is also a particular case of Theorem 3 in the quoted paper. The author applies this theorem to show that the variety of \(I\)-semigroups is \(x^{-1}x\)-regular and to characterize inverse semigroups having this property.