Guarded and banded semigroups. (Q2491181)

A guarded semigroup is a semigroup \((S,\cdot)\) with a unary operation \(x\to\overline x\) satisfying (i) \(\overline x\cdot x=x\), (ii) \(\overline x\cdot\overline y=\overline y\cdot\overline x\), (iii) \(\overline{\overline x\cdot y}=\overline x\cdot\overline y\), (iv) \(x\cdot\overline y=\overline{xy}\cdot x\). Such semigroups have been considered already by \textit{A. Batbedat} under the name of \(\gamma\)-semigroups [Semigroups, Proc. Conf., Oberwolfach 1978, Lect. Notes Math. 855, 1-18 (1981; Zbl 0452.20059)]. There is also an overlapping with the paper of \textit{M. Jackson} and \textit{T. Stokes} [Semigroup Forum 62, No. 2, 279-310 (2001; Zbl 0982.20051)], where they are called \(C\)-semigroups. As first results the weakly ample resp. left ample resp. inverse guarded semigroups are characterized (the generalization is motivated by these classes of semigroups). A covering theorem on inverse semigroups (due to D. McAlister) is generalized to guarded semigroups using semidirect products of monoids by semilattices. Furthermore the variety of guarded semigroups \(S\) with left-unit is described introducing a new binary operation with respect to which \(S\) forms a right normal band. This observation leads to the concept of a banded semigroup defined as a semigroup \((S,*)\) with a unary operation \(x\to\overline x\) satisfying (i), (ii), (iii) above, and (v) \(x*y=\overline x*y\). It is shown that (1) every banded semigroup is a right normal band satisfying three further identities; (2) any guarded semigroup induces a canonical banded semigroup structure on the same set; (3) on a semigroup \((S,*)\) there exists a unary operation making \((S,*)\) a banded semigroup if and only if \((S,*)\) is a split strong semilattice of right zero semigroups.