*-bisimple type A \(\omega\)-semigroups. I
Two elements a, b of a semigroup S are said to be \({\mathcal L}^*\)-related if there is an over-semigroup T of S in which \(a{\mathcal L}b\); \({\mathcal R}^*\)-related elements are defined dually. The semigroup S is said to be a type A semigroup if each \({\mathcal L}^*\), \({\mathcal R}^*\) class contains an idempotent and if, for each \(a\in S\), \(e\in E(S)\), there exist idempotents f, g such that f\({\mathcal L}^*ea\), \(g{\mathcal R}^*ae\) and \(ea=af\), \(ae=ga\). These semigroups generalize both inverse semigroups and cancellative monoids. Following the pattern of Green's relations in semigroups, we denote by \({\mathcal H}^*\) the intersection of \({\mathcal L}^*\) and \({\mathcal R}^*\) and by \({\mathcal D}^*\) the join of \({\mathcal L}^*\) and \({\mathcal R}^*\). In general it is not true that \({\mathcal L}^*\) and \({\mathcal R}^*\) commute. Also associated with \({\mathcal L}^*\) and \({\mathcal R}^*\) is the relation \(\tilde {\mathcal D}\) defined by \(a\tilde {\mathcal D}b\) if and only if there are idempotents e,f,g,h such that a\({\mathcal L}^*e\), b\({\mathcal L}^*f\), e\({\mathcal D}f\), a\({\mathcal R}^*g\), b\({\mathcal R}^*h\), g\({\mathcal D}h\). On a type A semigroup \({\mathcal D}\subseteq\tilde {\mathcal D}\subseteq {\mathcal D}^*\) but in general \(\tilde {\mathcal D}\neq {\mathcal D}^*.\) The author gives structure and isomorphism theorems for type A semigroups consisting of a single \(\tilde {\mathcal D}\)-class in which the idempotents form an \(\omega\)-chain under the natural partial order. The results obtained are entirely analogous to \textit{N. R. Reilly's} structure theorems for \(\omega\)-bisimple semigroups [Proc. Glasg. Math. Assoc. 7, 160-167 (1966; Zbl 0138.019)]. The role played by a group in Reilly's theory is played by a cancellative monoid here.
scientific article; zbMATH DE number 3875550
scientific article