The existence of free products in existence varieties of regular semigroups (Q1902529)

It has been known for many years that the class of all regular semigroups does not possess all free products. The author shows that all free products exist within a given e-variety \({\mathcal V}\) if and only if \({\mathcal V}\) is comprised either of locally inverse semigroups or of \(E\)-solid semigroups. In fact it turns out that outside these two cases no nontrivial free products can exist at all. This parallels the situation involving the existence of all `bifree' semigroups in \({\mathcal V}\), according to \textit{Y. T. Yeh} [Int. J. Algebra Comput. 2, No. 4, 471-484 (1992; Zbl 0765.20030)]\ and, not surprisingly, invokes some of the ideas of that paper.