Algebraic complete partial order

A complete partial order is algebraic when all of its elements can be written as least upper bounds of compact elements. In other words, every element is equal to a limit of its 'finite approximations.'

Definition

Let (D, \sqsubseteq) be a cpo, and let \mathsf{K}(D) be its set of compact elements.

D is algebraic just if for all d \in D it is true that

d = \bigsqcup \{ x \in \mathsf{K}(D) \mid x \sqsubseteq d)

Other uses

In mathematics this notion most commonly occurs not in the context of cpo's, but in complete lattices, which are stronger than cpo's. See the PlanetMath link below.

References

• PlanetMath: Algebraic lattice