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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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Distributive homomorphism」の詳細全文を読む スポンサード リンク
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