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. 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≤ a ∨ b, there are elements x and y of S such that c≤ x ∨ y, μ≤ a, and μ≤ b.
Examples:
For an algebra B and a reduct A of B, the canonical -homomorphism from Conc A to Conc B is weakly distributive. Here, Conc A denotes the -semilattice of all compact congruences of A.
For a convex sublattice K of a lattice L, the canonical -homomorphism from Conc K to Conc L is weakly distributive.
