Algebraic poset

English

Noun

1. Template:Lb A partially ordered set (poset) in which every element is the supremum of the compact elements below it.
   • 1985 October, Rudolf-E. Hoffmann, The Injective Hull and the $𝒞ℒ$-Compactification of a Continuous Poset, Template:W, 37:5, Template:W, 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 ${\displaystyle }$D, the supremum $\sup d$ exists,

     ii) for every $x\in P$, the set

     $K_X:=\{y\in P|y\text{ compact},y\le x\}$

     is non-empty and up-directed, and

     $x=\sup K_x$.

     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).

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See also

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Further reading

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• Algebraic lattice on Template:W