Distributive homomorphism

A congruence θ of a join-semilattice "S" is "monomial", if the θ-equivalence class of any element of "S" has a largest element. We say that θ 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 
"μ : S → T" between join-semilattices "S" and "T" is "weakly distributive", if for all "a, b" in "S" and all "c" in "T" such that "μ(c)≤ a ∨ b", there are elements "x" and "y" of "S" such that "c≤ x ∨ y", "μ(x)≤ a", and "μ(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 (∨, 0)-homomorphism from Conc A to Conc B is weakly distributive. Here, Conc A denotes the (∨, 0)-semilattice of all compact congruences of "A".
(2) For a convex sublattice "K" of a lattice "L", the canonical (∨, 0)-homomorphism from Conc "K" to Conc "L" is weakly distributive.
References.
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), 363–370.
F. Wehrung, "A solution to Dilworth's congruence lattice problem", preprint 2006.