\(*\)-congruences on abundant semigroups with \(SQ\)-adequate transversals.
A congruence \(\rho\) on an abundant semigroup \(S\) is called a \(*\)-congruence if for all \(a,b\in S\), \(a\rho b\) implies \(a^*\rho b^*\) and \(a^\dag\rho b^\dag\). Let \(T\) be an adequate semigroup (that is, an abundant semigroup in which idempotents commute) with semilattice \(Y\), \(L=[Y;L_\alpha,\varphi_{\alpha,\beta}]\) be a strong semilattice of left zero bands \(L_\alpha\) and \(R=[Y;R_\alpha,\phi_{\alpha,\beta}]\) be a strong semilattice of right zero bands \(R_\alpha\). Then the set \(SQ=\{(x,s,y)\in L\times T\times R,\;x\in L_{s^\dag},\;y\in R_{s^*}\}\) with a suitable multiplication, becomes an abundant semigroup. A triple \((\rho^L,\pi^T,\rho^R)\), where \(\rho^L\) and \(\rho^R\) are congruences on \(L\) and \(R\), respectively, and \(\pi^T\) is a \(*\)-congruence on \(T\), satisfying certain conditions, is called a \(*\)-congruence triple on \(SQ\). The sets of all \(*\)-congruences and \(*\)-congruence triples on \(SQ\) are denoted by \(C^*(SQ)\) and \(CT^*(SQ)\), respectively. Defining an order relation \(\leq\) on \(CT^*(SQ)\) by \((\rho_1^L,\pi_1^T,\rho_1^R)\leq(\rho_2^L,\pi_2^T,\rho_2^R)\Leftrightarrow\rho_1^L\subseteq\rho_2^L,\;\pi_1^T\subseteq\pi_2^T,\;\rho_1^R\subseteq\rho_2^R\), it is proved that \(CT^*(SQ)\) forms a complete lattice with respect to \(\leq\) and \(C^*(SQ)\) is isomorphic to \(CT^*(SQ)\) as complete lattice.
scientific article; zbMATH DE number 6385398
scientific article