partially ordered set

English

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

Noun

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 $(P,\succ )$ 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 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]).

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.

Synonyms

• (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

Hypernyms

• (order theory): category

Hyponyms

• (order theory): lattice, totally ordered set

Translations

set having a specified partial order

• Arabic: (Arabic Academy term) فِئَةٌ مُرَتَّبَةٌ جُزْئِيًّا f (fiʔatun murattabatun juzʔiyyan), (common in educational communities) مَجْمُوعَةٌ مُرَتَّبَةٌ جُزْئِيًّا f (majmūʕatun murattabatun juzʔiyyan)
• Czech: uspořádaná množina f
• Finnish: osittain järjestetty joukko
• German: partiell geordnete Menge f
• Japanese: 半順序集合 (はんじゅんじょしゅうごう, hanjunjo-shūgō)
• Swedish: partiellt ordnad mängd (sv) c

See also

• complete partial order
• partial order

Further reading

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