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 ${\\displaystyle {\\mathcal {CL}}}$ -Compactification of a Continuous Poset, Canadian Journal of Mathematics, 37:5, Canadian Mathematical Society, page 833,
  A poset ${\\displaystyle (P,\\leq )}$ is said to be algebraic if and only if
  i) ${\\displaystyle P}$ is up-complete, i.e., for every non-empty up-directed subset ${\\displaystyle }$ D, the supremum ${\\displaystyle \\operatorname {sup} \\ d}$ exists,
  ii) for every ${\\displaystyle x\\in P}$ , the set
  ${\\displaystyle K_{X}:=\\left\\{y\\in P\\vert \\ y{\\text{ compact}},y\\leq x\\right\\}}$
  is non-empty and up-directed, and
  ${\\displaystyle x=\\operatorname {sup} K_{x}}$ .
  A poset ${\\displaystyle P}$ is an algebraic poset if and only if it is a continuous poset in which, for every ${\\displaystyle x,y\\in P,x\\ll y}$ (if and) only if ${\\displaystyle x\\leq c\\leq y}$ for some compact element ${\\displaystyle c}$ of ${\\displaystyle P}$ .
  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 ${\\displaystyle K_{x}}$ 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 ${\\displaystyle K_{x}}$ .
  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, 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 ${\\displaystyle (D,\\leq _{D},\\cup _{D})}$ where ${\\displaystyle (D,\\leq _{D})}$ is an algebraic poset and ${\\displaystyle \\cup _{D}}$ is a binary operation on ${\\displaystyle D}$ which is continuous (in both variables), commutative, associative and absorptive (i.e., ${\\displaystyle d\\cup _{D}d=d}$ for all ${\\displaystyle d\\in D}$ ) 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 ${\\displaystyle {A^{*}\\cup A^{*}}^{\\sqrt {}}}$ 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.

单词	algebraic poset
释义	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 ${\\displaystyle {\\mathcal {CL}}}$ -Compactification of a Continuous Poset, Canadian Journal of Mathematics, 37:5, Canadian Mathematical Society, page 833, A poset ${\\displaystyle (P,\\leq )}$ is said to be algebraic if and only if i) ${\\displaystyle P}$ is up-complete, i.e., for every non-empty up-directed subset ${\\displaystyle }$ D, the supremum ${\\displaystyle \\operatorname {sup} \\ d}$ exists, ii) for every ${\\displaystyle x\\in P}$ , the set ${\\displaystyle K_{X}:=\\left\\{y\\in P\\vert \\ y{\\text{ compact}},y\\leq x\\right\\}}$ is non-empty and up-directed, and ${\\displaystyle x=\\operatorname {sup} K_{x}}$ . A poset ${\\displaystyle P}$ is an algebraic poset if and only if it is a continuous poset in which, for every ${\\displaystyle x,y\\in P,x\\ll y}$ (if and) only if ${\\displaystyle x\\leq c\\leq y}$ for some compact element ${\\displaystyle c}$ of ${\\displaystyle P}$ . 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 ${\\displaystyle K_{x}}$ 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 ${\\displaystyle K_{x}}$ . 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, 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 ${\\displaystyle (D,\\leq _{D},\\cup _{D})}$ where ${\\displaystyle (D,\\leq _{D})}$ is an algebraic poset and ${\\displaystyle \\cup _{D}}$ is a binary operation on ${\\displaystyle D}$ which is continuous (in both variables), commutative, associative and absorptive (i.e., ${\\displaystyle d\\cup _{D}d=d}$ for all ${\\displaystyle d\\in D}$ ) 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 ${\\displaystyle {A^{}\\cup A^{}}^{\\sqrt {}}}$ 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#Algebraic posets on Wikipedia.Wikipedia Complete partial order on Wikipedia.Wikipedia Complete lattice on Wikipedia.Wikipedia Directed set on Wikipedia.Wikipedia Domain theory#Way-below relation on Wikipedia.Wikipedia Algebraic lattice on Encyclopedia of Mathematics
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algebraic poset

English

Noun

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