A congruence ø of a join-semilattice S is monomial, if the ø-equivalence class of any element of S has a largest element. We say that ø is distributive, if it is a join, in the congruence lattice Con S of S, of monomial join-congruences of S.
The following definition originates in Schmidt's 1968 work and was subsequently adjusted by Wehrung.
Definition (weakly distributive homomorphisms). A homomorphism ü : S â T between join-semilattices S and T is weakly distributive, if for all a, b in S and all c in T such that ü(c) ⤠a ⨠b, there are elements x and y of S such that c ⤠x ⨠y, ü(x) ⤠a, and ü(y) ⤠b.
Examples:
(1) For an algebra B and a reduct A of B (that is, an algebra with same underlying set as B but whose set of operations is a subset of the one of B), the canonical from Con<sub>c</sub> A to Con<sub>c</sub> B is weakly distributive. Here, Con<sub>c</sub> A denotes the of all compact congruences of A.
(2) For a convex sublattice K of a lattice L, the canonical from Con<sub>c</sub> K to Con<sub>c</sub> L is weakly distributive.
E.T. Schmidt, Zur Charakterisierung der Kongruenzverbände der Verbände, Mat. Casopis Sloven. Akad. Vied. 18 (1968), 3--20.
F. Wehrung, A uniform refinement property for congruence lattices, Proc. Amer. Math. Soc. 127, no. 2 (1999), 363âÂÂ370.
F. Wehrung, A solution to Dilworth's congruence lattice problem, preprint 2006.