# algebraic poset
## English
### Noun

algebraic poset (plural algebraic posets)
* (algebra, order theory) 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 ${\mathcal {CL}}$-Compactification of a Continuous Poset, Canadian Journal of Mathematics, 37:5, Canadian Mathematical Society, page 833,
    A poset $(P,\leq )$ 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
       $K_{X}:=\left\{y\in P\vert \ y{\text{ compact}},y\leq x\right\}$
    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 $x\leq c\leq y$ 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).
** 1988, Karel Hrbacek, “A Powerdomain Construction”, in Michael Main, Austin Melton, Michael Mislove, David Schmidt, editors, Mathematical Foundations of Programming Language Semantics: 3rd Workshop, Proceedings, Springer, page 202:ALG is the category whose objects are algebraic posets and whose morphisms are continuous functions. A structure $(D,\leq _{D},\cup _{D})$ where $(D,\leq _{D})$ is an algebraic poset and $\cup _{D}$ is a binary operation on $D$ which is continuous (in both variables), commutative, associative and absorptive (i.e., $d\cup _{D}d=d$ for all $d\in D$) will be called a nondeterministic algebraic poset.
** 1992, Stephen D. Brookes et al., editors, Mathematical Foundations of Programming Semantics: 7th International Conference, Proceedings, Springer, page 88:It is important to note that we have not assumed an algebraic poset is a cpo. The canonical example of an algebraic poset is ${A^{*}\cup A^{*}}^{\sqrt}$ in the prefix order; this poset enjoys the added condition that every element is compact.
** 1997, Michael W. Shields, Semantics of Parallelism: Non-Interleaving Representation of Behaviour, Springer, page 41:The correspondence between primes and occurrences suggests that given an abstract prime algebraic poset, we may construct a behavioural presentation from it.

### See also

* complete partial order
* up-complete poset
* up-directed

### Further reading

* Compact element on Wikipedia.Wikipedia
** Compact element on Wikipedia.Wikipedia
* Complete partial order on Wikipedia.Wikipedia
* Complete lattice on Wikipedia.Wikipedia
* Directed set on Wikipedia.Wikipedia
* Domain theory on Wikipedia.Wikipedia
* Algebraic lattice on Encyclopedia of Mathematics