\(\theta\)-modularity and the relation \(K\) on Bruck-Reilly regular extensions of monoids. (Q1430476)

In this paper, \(T\) is a regular semigroup and \(S=BR(T,\alpha)\) a Bruck-Reilly extension of \(T\). \({\mathcal C}(S)\) is the lattice of congruences on \(S\), \(\sigma\in{\mathcal C}(S)\) is the least group congruence and \(\beta\in{\mathcal C}(S)\) is induced by the canonical homomorphism of \(S\) onto the bicyclic semigroup. \(C_\alpha(T)\) is the lattice of all congruences \(\Gamma\) on \(T\) such that \(a\Gamma b\) implies as \(a\alpha\Gamma b\beta\) for all \(a,b\in T\). For congruences \(\rho,\lambda\), put \(\rho{\mathbf K}\lambda\) if \(\rho\) and \(\lambda\) have the same kernel. It is shown that \(\mathbf K\) is a congruence relation on \({\mathcal C}(S)\) if and only if \(\mathbf K\) is a congruence on the principal ideal \((\sigma\vee\beta]\) of \({\mathcal C}(S)\), and this is the case if and only if \(\mathbf K\) is a congruence on \(C_\alpha(T)\) and, \((\sigma\vee\beta]\to(\beta]\), \(\lambda\to\lambda\land\beta\) is a surjective homomorphism. The authors look at other properties (\(\theta\)-modularity, \(K\)-modularity, \(K'\)-modularity) of the lattice \({\mathcal C}(S)\) and obtain similar results for each of these situations.