partially ordered set

plural partially ordered sets

1. broadlymathematicsA set that has a given, elsewhere specified partial order.
2. formalmathematicsThe ordered pair comprising a set and its partial order.

   In sections 7-10 we shall consider random fields over some subsets T of the partially ordered set T_M.

   1994, I. V. Evstigneev, P. E. Greenwood, Markov Fields over Countable Partially Ordered Sets: Extrema and Splitting, American Mathematical Society, page 35

   2 more examples

   1959 [D. Van Nostrand], Edward James McShane, Truman Arthur Botts, Real Analysis, 2005, Dover, page 28, A partially ordered set means a pair (P,≻) consisting of a set P and a partial order ≻ in P. As usual, when the meaning is clear, we may suppress the notation of "≻" and speak of the partially ordered set P. The ordered fields defined earlier are easily seen to be examples of partially ordered sets.

   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]).

   2000, David Arnold, Abelian Groups and Representations of Finite Partially Ordered Sets, Springer, page 45