Definition:Algebraic Ordered Set

This page is about Algebraic Ordered Set. For other uses, see Algebraic.

Definition

Let $\struct {S, \preceq}$ be an ordered set.

Then $\struct {S, \preceq}$ is algebraic if and only if

(for all elements $x$ of $S$: $x^{\mathrm{compact} }$ is directed)

and:

$\struct {S, \preceq}$ is up-complete and satisfies the axiom of $K$-approximation:

where $x^{\mathrm{compact} }$ denotes the compact closure of $x$.

Sources

• 1980: G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott: A Compendium of Continuous Lattices

• Mizar article WAYBEL_8:def 4