Inverse semigroup homomorphisms via partial group actions (Q2748276)

One can say that the purpose of the paper is the consideration of inverse semigroup homomorphisms such that their domain or codomain is \(E\)-unitary. Let \(S\) be an inverse semigroup. The author gives an explicit construction of all homomorphisms from \(S\) which factor through an \(E\)-unitary inverse semigroup. In particular, it is proved that such homomorphisms have a unique factorization \(\beta\alpha\) with \(\alpha\) preserving the maximal group image, \(\beta\) idempotent separating and the domain of \(\beta\) \(E\)-unitary. These results are specialized to the case of inverse semigroups in which every \(\sigma\)-class has a maximum. Some results concern the existence of \(E\)-unitary covers over a group. An efficient notion is the notion of a prehomomorphism, especially of a dual prehomomorphism. A map \(\varphi\colon S\to T\) is called a dual prehomomorphism if \(\varphi(s)\varphi(t)\leq\varphi(st)\). Partial actions of a group are based on this notion.NEWLINENEWLINENEWLINEReviewer's remark: It is to mention that the notion of a dual prehomomorphism (named a restrictive representation) and partial group actions have been used in the reviewer's paper [Teor. Polugrupp Prilozh. 4, 19-40 (1978; Zbl 0402.08012)] to construct extensions of idempotent semigroups by means of groups and (as a consequence) for a description of \(E\)-unitary inverse semigroups.