Congruences associated with inverse transversals
An inverse transversal of a regular semigroup \(S\) is a regular subsemigroup \(S^\circ\) such that \(S^\circ\) contains a unique inverse \(x^\circ\) of every \(x\in S\). The properties of the congruence lattices \(\overline{\text{Con }}S\) of the algebra \((S,\cdot,{}^\circ)\) and the properties of the lattice \(\text{Con }S\) of congruences on \(S\) are considered. An inverse transversal \(S^\circ\) is said to be multiplicative if \(\Lambda I=E(S^\circ)\), where \(E(S^\circ)\) is the set of idempotents of \(S^\circ\), \(\Lambda=\{x^\circ x, x\in S\}\) and \(I=\{x x^\circ, x\in S\}\). If the inverse transversal \(S^\circ\) is multiplicative, then \(\overline{\text{Con}} S\overline{\sim}\text{Con }S(x)\),where \(\sim\) is the dual closure equivalence defined by \(\vartheta\sim\varphi\Leftrightarrow\vartheta|_I=\varphi|_I\), \(\vartheta|_\Lambda=\varphi|_\Lambda\), \(\vartheta|_{S^\circ}=\varphi|_{S^\circ}\).
scientific article; zbMATH DE number 869942