On the lattice of weak congruence relations

We consider the setC _w (A) of weak congruences on an algebraA, i.e. of all symmetric and transitive subalgebras ofA ×A. (Some other generalizations of compatible relations were given in 2] and 6]).C _w (A) coincides with the set of all congruences on all subalgebras ofA.We prove here that (^iC _w(A),) is an algebraic lattice having as a sublattice the lattice of all congruences onA, 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.Presented by H. P. Gumm.