The least inverse and orthodox congruences on a completely regular semigroup
The congruences mentioned in the title are described based on the fact that every completely regular semigroup \(S\) is a semilattice \(Y\) of completely simple semigroups \(S_{\alpha}\), \(\alpha\in Y\). Denote by \(\nu\), \(o\) and \(\sigma\) the least inverse, orthodox and group congruences on \(S\) and by \(\nu_{\alpha}\), \(o_{\alpha}\), \(\sigma_{\alpha}\) those on each \(S_{\alpha}\), \(\alpha\in Y\). Let \(E(S)\) be the set of idempotents of \(S\), and \(\ker\rho\) be the kernel of a congruence \(\rho\), i.e. the union of its idempotent \(\rho\)-classes. Theorem. (i) \(\ker o=\ker\nu =\cup_{\alpha \in Y}\ker \sigma_{\alpha}\), (ii) \(\nu =\cup_{\alpha \in Y}\sigma_{\alpha}\), (iii) \((a,b)\in o \Leftrightarrow aa^{-1}\pi bb^{-1}\) \& \(a\sigma_{\alpha}b\), where \(a,b\in S_{\alpha}\) and \(\pi =o|_{E(S)}.\) It is known that each \(\sigma_{\alpha}\) can be described in terms of any Rees matrix representation for \(S_{\alpha}\). Let \(F\) (resp. \(F_{\alpha}\)) be the least full self-conjugate subsemigroup of \(S\) (resp. \(S_{\alpha}\)). It is known that \(\ker o=F\). The author obtains \(F=\cup_{\alpha \in Y}F_{\alpha}\).
scientific article; zbMATH DE number 3853365
scientific article