Structure of the weakly left \(C\)-semigroups
Let \(S\) be a regular semigroup and \(E\) the set of all idempotents of \(S\). \(S\) is called a quasi-\(C\)-semigroup if for every \(e\in E\), \(\zeta_e:S^1\to eS^1e\) defined by \(x\mapsto exe''\) is a semigroup homomorphism. The author calls a regular semigroup \(S\) weakly left \(C\)-semigroup if \(\eta_e:S\to eS\) defined by \(x\mapsto ex'\) for all \(e\in E\) is a semigroup homomorphism. The relationship between weakly left \(C\)-semigroups and quasi-\(C\)-semigroups is then found, in particular, the author shows that a semigroup \(S\) is a weakly left \(C\)-semigroup if and only if \(S\) is a quasi-\(C\)-semigroup and the right Clifford component of \(S\) is a strong semilattice of right groups. This result is useful in studying the structure of quasi-\(C\)-semigroups.
scientific article; zbMATH DE number 908266