On the lattice of weak congruence relations (Q1110553)

A weak congruence relation \(\rho\) on an algebra \({\mathcal A}\) is a symmetric, transitive relation, satisfying the substitution property, and a weak reflexivity: if c is a constant, then \(c\rho\) c. The authors prove that the lattice of all weak congruences of an algebra has the congruence lattice as a sublattice, and the subalgebra lattice is a retract. If \(\rho\) is a weak congruence on an algebra \({\mathcal A}\) then \(\rho_ A\) denotes the smallest congruence relation having the property \(\rho \leq \rho_ A\). \({\mathcal A}\) has the Congruence Intersection Property (CIP) if for all weak congruences \(\rho\), \(\Theta\), \((\rho \wedge \Theta)_ A=\rho_ A\wedge \Theta_ A\). An algebra has a modular (distributive) lattice of weak congruences iff it has modular (distributive) lattices of congruences and of subalgebras, and satisfies the CEP and CIP.