Beyond orthodox semigroups.
Let \(S\) be a semigroup and \(\emptyset\neq B\subseteq E(S)\). Let \(\widetilde{\mathcal L}_B\) be an equivalence relation, so that for \(a,b\in S\), \(a\widetilde{\mathcal L}_Bb\) if and only if \(\{e\in B:ae=a\}=\{e\in B:be=b\}\). A semigroup \(S\) is said to be \textit{weakly B-abundant} if every \(\widetilde{\mathcal L}_B\)-class and every \(\widetilde{\mathcal R}_B\)-class contains an idempotent of \(B\). If \(\widetilde{\mathcal L}_B\) is a right congruence and \(\widetilde{\mathcal R}_B\) is a left congruence, then \(S\) is said to be satisfying the \textit{Congruence Condition} (C). A weakly \(B\)-abundant semigroup is said to be \textit{weakly B-orthodox} if it has (C) and \(B\) is a band. The authors define a \textit{generalised category} and an \textit{inductive generalised category} (over a band \(B\)) and prove that the category of weakly \(B\)-orthodox semigroups and admissible morphisms is isomorphic to the category of inductive generalised categories over bands and pseudo-functors. Special cases of the categories of orthodox and some other semigroups are discussed as well.
scientific article; zbMATH DE number 6154887