congruence Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In mod ...

θ of a

join-semilattice In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset. Dually, a meet-semilattice (or lower semilattice) is a partially ordered set which has a ...

''S'' is ''monomial'', if the θ-

equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...

of any element of ''S'' has a largest element. We say that θ is ''distributive'', if it is a

join Join may refer to: * Join (law), to include additional counts or additional defendants on an indictment *In mathematics: ** Join (mathematics), a least upper bound of sets orders in lattice theory ** Join (topology), an operation combining two topo ...

, 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 Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...

''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_c ''A'' to Con_c ''B'' is weakly distributive. Here, Con_c ''A'' denotes the of all compact congruences of ''A''. (2) For a

convex sublattice Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polyto ...

''K'' of a lattice ''L'', the canonical {{nowrap, (∨, 0)-homomorphism from Con_c ''K'' to Con_c ''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. Abstract algebra