Algebraic posets

Abbreviation: APos

Definition

An algebraic poset is a directed complete partial order P = (P, ≤ ) such that
the set of compact elements below any element is directed and
every element is the join of all compact elements below it.

An element c ∈ P is compact if for every subset S ⊆ P such that c ≤ ∨S, there exists a finite subset S₀ of S such that c ≤ ∨S₀.

The set of compact elements of P is denoted by K(P).

Morphisms

Let P and Q be algebraic posets. A morphism from P to Q is a function f : P→Q that is Scott-continuous, which means that f preserves all directed joins: z = ∨D  ⇒  f(z) = ∨f[D] for all directed sets D ⊆ P.

Some results

Examples

Properties

Finite members

Subclasses

Superclasses