algebraic poset

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English

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Noun

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algebraic poset (plural algebraic posets)

  1. (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 -Compactification of a Continuous Poset, Canadian Journal of Mathematics, 37:5, Canadian Mathematical Society, page 833,
      A poset is said to be algebraic if and only if
      i) is up-complete, i.e., for every non-empty up-directed subset D, the supremum exists,
      ii) for every , the set
      is non-empty and up-directed, and
      .
      A poset is an algebraic poset if and only if it is a continuous poset in which, for every (if and) only if for some compact element of .
      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 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 .
      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 where is an algebraic poset and is a binary operation on which is continuous (in both variables), commutative, associative and absorptive (i.e., for all ) 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 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.

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Further reading

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