partially ordered set

partially ordered set (plural partially ordered sets)

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 ${\displaystyle (P,\succ )}$ consisting of a set ${\displaystyle P}$ and a partial order ${\displaystyle \succ }$ in ${\displaystyle P}$ . As usual, when the meaning is clear, we may suppress the notation of "${\displaystyle \succ }$ " and speak of the partially ordered set ${\displaystyle P}$ .

     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 T_M.

   • 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]).

• The two senses are commonly used interchangeably, there rarely being a need to distinguish between them.
• The components of the ordered pair may be referred to separately as the ground set and partial order.

• (set on which a partial order is defined): ground set, poset
• (ordered pair of set and partial order): poset
• See also Thesaurus:partially ordered set

• (order theory): category

• (order theory): lattice, totally ordered set

• complete partial order
• partial order

• Complete partial order on Wikipedia.Wikipedia
• Hasse diagram on Wikipedia.Wikipedia
• Lattice (order) on Wikipedia.Wikipedia
• Order theory on Wikipedia.Wikipedia