partially ordered set

Noun

 * 1)  A set that has a given, elsewhere specified partial order.
 * 2)  The ordered pair comprising a set and its partial order.
 * 3) * 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.

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 and.

Synonyms

 * See also Thesaurus:partially ordered set
 * See also Thesaurus:partially ordered set
 * See also Thesaurus:partially ordered set

Translations

 * Arabic: (Arabic Academy term) فِئَةٌ مُرَتَّبَةٌ جُزْئِيًّا, (common in educational communities) مَجْمُوعَةٌ مُرَتَّبَةٌ جُزْئِيًّا
 * Czech: uspořádaná množina
 * Finnish: osittain järjestetty joukko
 * German: partiell geordnete Menge
 * Japanese: 半順序集合
 * Swedish: