Complete semilattices

Abbreviation: CSlat

Definition

A complete semilattice is a directed complete partial order P = (P, ≤ ) such that every nonempty subset of P has a greatest lower bound: ∀S ⊆ P (S ≠ Ø  ⇒  ∃z ∈ P(z = ∧S)).

Morphisms

Let P and Q be complete semilattices. A morphism from P to Q is a function f : P→Q that preserves all nonempty meets and all directed joins: z = ∧S  ⇒  f(z) = ∧f[S] for all nonempty S ⊆ P and z = ∨D  ⇒  f(z) = ∨f[D] for all directed sets D ⊆ P.

Some results

Examples

Properties

Finite members

Subclasses

Superclasses