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partially ordered set

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partially orderedset (pluralpartially ordered sets)

1. (set theory, order theory,loosely) Aset that has a given, elsewhere specifiedpartial order.
2. (set theory, order theory, formally) Theordered pair comprising a set and its partial order.
   • 1959 [D. Van Nostrand], Edward James McShane, Truman Arthur Botts,Real Analysis, 2005, Dover,page 28,

     Apartially 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 thepartially ordered set${\displaystyle P}$.

     The ordered fields defined earlier are easily seen to be examples ofpartially ordered sets.

   • 1994, I. V. Evstigneev, P. E. Greenwood,Markov Fields over Countable Partially Ordered Sets: Extrema and Splitting,American Mathematical Society,page35:

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

   • 2000, David Arnold,Abelian Groups and Representations of FinitePartially Ordered Sets, Springer,page45:

     The invention of a derivative of a finitepartially ordered set by Nazarova and Roiter in the late 1960s or early 1970s was a seminal event in the subject of representations of finitepartially 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 theground set andpartial order.

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

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