# partially ordered set

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## English

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A partially ordered set with the relation "is divisible by"

### Noun

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 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]).

#### Usage notes

• 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.