The structure of \(\mathcal Q^*\)-inverse semigroups.
A semigroup \(S\) is called abundant if every \(\mathcal L^*\)-class and every \(\mathcal R ^*\)-class contains at least one idempotent of \(S\). An abundant semigroup \(S\) is called idempotent connected (IC) if for each element \(a\in S\) and for some \(a^+\in R_a^*(S)\cap E\), \(a^*\in L_a^*(S)\cap E\) there exists a bijection \(\alpha\colon\langle a^+\rangle\to\langle a^*\rangle\) such that \(xa=a(x\alpha)\) for all \(x\in\langle a^+\rangle\). An IC abundant semigroup \(S\) is called \(\mathcal Q\)-inverse semigroup if the set of all its idempotents \(E\) forms a regular band, that is, \(efege=efge\) for all \(e,f,g\in E\). Thus, the class of \(\mathcal Q\)-inverse semigroups is a generalization of the class of quasi-inverse semigroups. A construction theorem of \(\mathcal Q\)-inverse semigroups is given by using the wreath product of some semigroups. It is proved that a semigroup \(S\) is a \(\mathcal Q\)-inverse semigroup if and only if \(S\) is a spined product of an \(\mathcal L^*\)-inverse semigroup and an \(\mathcal R^*\)-inverse semigroup. Here a spined product of semigroups \(S_1\) and \(S_2\) is a semigroup \(S\) defined by means of two homomorphisms \(\theta_1\colon S_1\to H\) and \(\theta_2\colon S_2\to H\) onto their common homomorphic image \(H\): \(S=\{(s_1,s_2)\in S_1\times S_2\mid s_1\theta_1=s_2\theta_2\}\).
scientific article; zbMATH DE number 5869002
scientific article