algebraic poset

Noun

 * 1)  A partially ordered set (poset) in which every element is the supremum of the compact elements below it.
 * 2) * 1985 October, Rudolf-E. Hoffmann, The Injective Hull and the $$\mathcal{CL}$$-Compactification of a Continuous Poset, , 37:5,, page 833,
 * A poset $$(P, \le)$$ is said to be algebraic if and only if
 * i) $$P$$ is up-complete, i.e., for every non-empty up-directed subset $$$$D, the supremum $$\operatorname{sup}\ d$$ exists,
 * ii) for every $$x\in P$$, the set
 * $$K_X:=\left \{y\in P\vert\ y \text{ compact},y\le x \right\}$$
 * is non-empty and up-directed, and
 * $$x = \operatorname{sup} K_x$$.
 * A poset $$P$$ is an algebraic poset if and only if it is a continuous poset in which, for every $$x, y\in P, x \ll y$$ (if and) only if $$x\le c\le y$$ for some compact element $$c$$ of $$P$$.
 * Concerning the definition of an algebraic poset, a caveat may be in order (which, mutatis mutandis, applies to continuous posets): it may happen that all of the axioms for an algebraic poset are satisfied except that the sets $$K_x$$ fail to be up-directed ([50], 4.2 or [49], 4.5). Even when "enough" compact sets are readily available, it sometimes remains a delicate problem to verify the up-directedness of the sets $$K_x$$.
 * The concept of an algebraic poset arose in theoretical computer science ([50], [54], cf. also [14]). It is a natural extension of he familiar notion of a (complete) "algebraic lattice" (cf. [9], [20], I-4).