Meaning of algebraic poset | Babel Free

Definitions

A partially ordered set (poset) in which every element is the supremum of the compact elements below it.

Examples

“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).”

“ALG is the category whose objects are algebraic posets and whose morphisms are continuous functions. A structure (D,#92;le#95;D,#92;cup#95;D) where (D,#92;le#95;D) is an algebraic poset and #92;cup#95;D is a binary operation on D which is continuous (in both variables), commutative, associative and absorptive (i.e., d#92;cup#95;Dd#61;d for all d#92;inD) will be called a nondeterministic algebraic poset.”

“It is important to note that we have not assumed an algebraic poset is a cpo. The canonical example of an algebraic poset is #123;A#42;#92;cupA#42;#125;#92;sqrt#123;#125; in the prefix order; this poset enjoys the added condition that every element is compact.”

“The correspondence between primes and occurrences suggests that given an abstract prime algebraic poset, we may construct a behavioural presentation from it.”

CEFR level