The structure of medial weakly \(U\)-abundant semigroups.
\textit{M. Loganathan} [in Semigroup Forum 36, 69-74 (1987; Zbl 0627.20035)], described the structure of a regular semigroup \(S\) with a medial idempotent \(u\), that is an idempotent \(u\) such that \(x=xux\), for all \(x\in\langle E(S)\rangle\), in terms of the idempotent generated regular semigroup \(\langle E(S)\rangle\) and the band \(uSu\). This result has been extended, in particular to abundant semigroups with a medial idempotent [\textit{F. Jing}, Semigroup Forum 51, No. 2, 247-261 (1995; Zbl 0834.20059)]. In this paper the authors consider the wider class of medial weakly \(U\)-abundant semigroups and describe their structure by means of bands and weakly-abundant semigroups over bands (called quasi-Ehresmann [\textit{M. V. Lawson}, J. Algebra 141, No. 2, 422-462 (1991; Zbl 0747.18007)]). In a semigroup \(S\) given a set \(V\) of idempotents, elements \(a,b\in S\) are said to be \(\widetilde{\mathcal L}_V\)-related if they admit exactly the same set of right identities from \(V\). When \(S\) is regular \(\mathcal L=\widetilde{\mathcal L}_{E(S)}\). Dually, one defines \(\widetilde{\mathcal R}_V\). One says that \(S\) is \textit{weakly \(V\)-abundant} if each \(\widetilde{\mathcal L}_V\)-class and each \(\widetilde{\mathcal R}_V\)-class contains idempotents from \(V\). In general, \(\widetilde{\mathcal L}_V\) [\(\widetilde{\mathcal R}_V\)] is not right [left] compatible. If we are in the presence of the one sided compatibility conditions for \(\widetilde{\mathcal L}_V\) and \(\widetilde{\mathcal R}_V\), and there exists \(u\in V\) such that for all \(x\in V\), \(x=xux\) and \(ux,xu,uxu\in V\) for all \(x\in\langle V\rangle\), we say that \(S\) is \textit{medial} and \(u\) is called a \textit{medial projection} of \(S\) with respect to \(V\). The main aim of this paper is Theorem 4.3, where any weakly \(V\)-abundant semigroup satisfying the compatible conditions with a medial projection \(u\) is proved to be isomorphic to a semigroup \(W(V,uSu,uVu,u)\) built over the bands \(Vu\), \(uV\), and \(uVu\), together with the weakly \(uVu\) [\(uV\), \(Vu\)]-abundant semigroups \(uSu\) [\(uS\), \(Su\)].
scientific article; zbMATH DE number 6136737
scientific article