Ordered set

Ordered set

In order theory in mathematics, a set with a binary relation R on its elements that is reflexive (for all a in the set, aRa), antisymmetric (if aRb and bRa, then a = b) and transitive (if aRb and bRc, then aRc) is described as a partially ordered set or poset. If the binary relation is antisymmetric, transitive and also total (for all a and b in the set, aRb or bRa), then the set is a totally ordered set. If every non-empty subset has a least element, then the set is a well-ordered set.

In information theory, an ordered set is a non-data carrying set of bits as used in 8b/10b encoding.

See also


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  • Partially ordered set — The Hasse diagram of the set of all subsets of a three element set {x, y, z}, ordered by inclusion. In mathematics, especially order theory, a partially ordered set (or poset) formalizes and generalizes the intuitive concept of an ordering,… …   Wikipedia

  • Special ordered set — In discrete optimization, a special ordered set is an ordered set of variables. There are two sorts of Special Ordered Sets.Special Ordered Sets of type 1 (SOS1 or S1) are a set of variables, at most one of which can take a strictly positive… …   Wikipedia

  • totally ordered set — Math. a set in which a relation, as less than or equal to, holds for all pairs of elements of the set. Also called chain, linearly ordered set, simply ordered set. Cf. partially ordered set, well ordered set. * * * …   Universalium

  • totally ordered set — Math. a set in which a relation, as less than or equal to, holds for all pairs of elements of the set. Also called chain, linearly ordered set, simply ordered set. Cf. partially ordered set, well ordered set …   Useful english dictionary

  • well-ordered set — Math. a totally ordered set in which every nonempty subset has a smallest element with the property that there is no element in the subset less than this smallest element. Cf. partially ordered set, totally ordered set. * * * …   Universalium

  • well-ordered set — Math. a totally ordered set in which every nonempty subset has a smallest element with the property that there is no element in the subset less than this smallest element. Cf. partially ordered set, totally ordered set …   Useful english dictionary

  • partially ordered set — Math. a set in which a relation as less than or equal to holds for some pairs of elements of the set, but not for all. Cf. totally ordered set, well ordered set. [1970 75] * * * …   Universalium

  • partially ordered set — Math. a set in which a relation as less than or equal to holds for some pairs of elements of the set, but not for all. Cf. totally ordered set, well ordered set. [1970 75] …   Useful english dictionary

  • linearly ordered set — Math. See totally ordered set. * * * …   Universalium

  • simply ordered set — Math. See totally ordered set. * * * …   Universalium

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