Partially ordered set

English

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

Noun

1. Template:Lb A set that has a given, elsewhere specified partial order.
2. Template:Lb 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.

   • Template:Quote-book
   • Template:Quote-book

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 Template:M and Template:M.

Synonyms

• Template:Sense Template:L, Template:L
• Template:Sense Template:L
• See also Thesaurus:partially ordered set

Hypernyms

• Template:Sense Template:L

Hyponyms

• Template:Sense Template:L, Template:L

Translations

• Arabic: (Arabic Academy term) Template:T, (common in educational communities) Template:T
• Czech: Template:T
• Finnish: Template:T
• German: Template:T
• Japanese: Template:T
• Polish: Template:T
• Swedish: Template:T+

See also

• Template:L
• Template:L

Further reading

• Template:Pedia
• Template:Pedia
• Template:Pedia
• Template:Pedia