Quizzes for review
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).
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