Distributive homomorphism
Encyclopedia
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.
Congruence relation
In abstract algebra, a congruence relation is an equivalence relation on an algebraic structure that is compatible with the structure...
θ of a join-semilattice
Semilattice
In mathematics, a join-semilattice is a partially ordered set which has a join for any nonempty finite subset. Dually, a meet-semilattice is a partially ordered set which has a meet for any nonempty finite subset...
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
Compact element
In the mathematical area of order theory, the compact or finite elements of a partially ordered set are those elements that cannot be subsumed by a supremum of any non-empty directed set that does not already contain members above the compact element....
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
Universal algebra
Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....
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
Compact element
In the mathematical area of order theory, the compact or finite elements of a partially ordered set are those elements that cannot be subsumed by a supremum of any non-empty directed set that does not already contain members above the compact element....
of A.
(2) For a convex sublattice
Lattice (order)
In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...
K of a lattice L, the canonical (∨, 0)-homomorphism from Conc K to Conc L is weakly distributive.