partially ordered set

partially ordered set

  1. (set theory, order theory, loosely) A set that has a given, elsewhere specified partial order.
  2. (set theory, order theory, formally) The ordered pair comprising a set and its partial order.
    • 1959 [D. Van Nostrand], Edward James McShane, Truman Arthur Botts, Real Analysis, 2005, Dover, page 28 ↗,
      A partially ordered set means a pair (P,\succ) consisting of a set P and a partial order \succ in P. As usual, when the meaning is clear, we may suppress the notation of "\succ" and speak of the partially ordered set P.
      The ordered fields defined earlier are easily seen to be examples of partially ordered sets.
    • 1994, I. V. Evstigneev, P. E. Greenwood, Markov Fields over Countable Partially Ordered Sets: Extrema and Splitting, American Mathematical Society, page 35 ↗,
      In sections 7-10 we shall consider random fields over some subsets T of the partially ordered set TM.
    • 2000, David Arnold, Abelian Groups and Representations of Finite Partially Ordered Sets, Springer, page 45 ↗,
      The invention of a derivative of a finite partially ordered set by Nazarova and Roiter in the late 1960s or early 1970s was a seminal event in the subject of representations of finite partially ordered sets (see [Simson 92]).

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