algebraic poset

— 1985 October, Rudolf-E. Hoffmann, The Injective Hull and the 𝒞ℒ-Compactification of a Continuous Poset, Canadian Journal of Mathematics, 37:5, Canadian Mathematical Society, page 833, A poset (P,<) is said to be algebraic if and only if i) P is up-complete, i.e., for every non-empty up-directed subset D,thesupremum operatorname supd exists, ii) for every x∈P, the set K_X:=y∈P|ycompact,y<x is non-empty and up-directed, and x= operatorname supK_x. A poset P is an algebraic poset if and only if it is a continuous poset in which, for every x,y∈P,x≪y (if and) only if x<c<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).