Subdirect representation and semimodularity of weak congruence lattices
A weak congruence on an algebra \(\mathcal A\) is a symmetric and transitive subalgebra of \({\mathcal A}^2\). As shown by B. Šešelja and G. Vojvodič, the set \(C_w ({\mathcal A})\) of all weak congruences on \(\mathcal A\) forms an algebraic lattice with respect to inclusion. Let \(\Delta = \{ (x,x); x\in A \}\). \(\mathcal A\) satisfies the congruence intersection property (CIP for short) if \(\Delta \vee (\alpha \wedge \beta) = (\Delta \vee \alpha)\wedge (\Delta \vee \beta)\) holds for all \(\alpha , \beta \in C_w ({\mathcal A}).\) \({\mathcal A}\) satisfies the congruence extension property (CEP for short) if each congruence on every subalgebra of \(\mathcal A\) is a restriction of a congruence on \(\mathcal A\). Main result: Let \(\mathcal A\) satisfy CIP and CEP. Then \(C_w ({\mathcal A})\) is isomorphic to a subdirect product of \(\text{Sub}( {\mathcal A})\) and \(\text{Con}({\mathcal A})\). It follows that if \(\mathcal A\) satisfies CIP and CEP, then a lattice quasi-identity holds in \(C_w ({\mathcal A})\) if and only if it holds in \(\text{Con}({\mathcal A})\) and \(\text{Sub} ({\mathcal A})\); especially, \(C_w({\mathcal A})\) is lower resp. upper semimodular iff both \(\text{Con}({\mathcal A})\) and \(\text{Sub}({\mathcal A})\) are lower resp. upper semimodular.
scientific article; zbMATH DE number 1899973