Distributive homomorphism (original) (raw)
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. Examples: (2) For a convex sublattice K of a lattice L, the canonical (∨, 0)-homomorphism from Conc K to Conc L is weakly distributive.
| Property | Value |
|---|---|
| dbo:abstract | 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. (en) |
| dbo:wikiPageID | 9754647 (xsd:integer) |
| dbo:wikiPageLength | 1947 (xsd:nonNegativeInteger) |
| dbo:wikiPageRevisionID | 1066526359 (xsd:integer) |
| dbo:wikiPageWikiLink | dbr:Join_(mathematics) dbr:Universal_algebra dbr:Congruence_relation dbr:Equivalence_class dbr:Compact_element dbr:Lattice_(order) dbc:Algebra dbr:Semilattice |
| dbp:wikiPageUsesTemplate | dbt:Technical |
| dcterms:subject | dbc:Algebra |
| gold:hypernym | dbr:Monomial |
| rdfs:comment | 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. Examples: (2) For a convex sublattice K of a lattice L, the canonical (∨, 0)-homomorphism from Conc K to Conc L is weakly distributive. (en) |
| rdfs:label | Distributive homomorphism (en) |
| owl:sameAs | freebase:Distributive homomorphism wikidata:Distributive homomorphism https://global.dbpedia.org/id/4j7ju |
| prov:wasDerivedFrom | wikipedia-en:Distributive_homomorphism?oldid=1066526359&ns=0 |
| foaf:isPrimaryTopicOf | wikipedia-en:Distributive_homomorphism |
| is dbo:wikiPageWikiLink of | dbr:Congruence_lattice_problem |
| is foaf:primaryTopic of | wikipedia-en:Distributive_homomorphism |