On the lattice of weak congruence relations

Abstract

We consider the set Cw(A) of weak congruences on an algebra A, i.e. of all symmetric and transitive subalgebras of A ×A. (Some other generalizations of compatible relations were given in [2] and [6]). Cw(A) coincides with the set of all congruences on all subalgebras of A. We prove here that (iCw(A),≤) is an algebraic lattice having as a sublattice the lattice of all congruences on A, and as a retract the lattice of all its subalgebras. In the second part we give necessary and sufficient conditions (one is the CEP) for modularity as well as for distributivity of the lattice of weak congruences. © 1988 Birkhäuser Verlag.