On Some Monogenic Semigroups with an Associate Subgroup

Let S be any semigroup and a, s ∈ S. If a = asa, then s is an associate of a. A subgroup G of S is an associate subgroup of S if every a ∈ S has a unique associate a* in G. It turns out that G = H _z for some idempotent z, the zenith of S. The mapping a → a* is a unary operation on S. We say that S is monogenic if S is generated, as a unary semigroup, by a single element.

We embark upon the problem of the structure of monogenic semigroups in this sense by characterizing monogenic ones belonging to completely simple semigroups, normal cryptogroups, orthogroups, combinatorial semigroups, cryptic medial semigroups, cryptic orthodox semigroups, and orthodox monoids. In each of these cases, except one, we construct a free object. The general problem remains open.