PlanetMath: partial order

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partial order

(Definition)

A partial order (often simply referred to as an order or ordering) is a relation $\leq\:\subset A\times A$ that satisfies the following three properties:

Reflexivity: $a \leq a$ for all $a\in A$
Antisymmetry: If $a \leq b$ and $b \leq a$ for any $a, b\in A$ , then
Transitivity: If $a \leq b$ and $b \leq c$ for any $a, b, c\in A$ , then $a \leq c$

A total order is a partial order that satisfies a fourth property known as comparability:

Comparability: For any $a,b\in A$ , either $a\leq b$ or $b\leq a$ .

A set and a partial order on that set define a poset.

"partial order" is owned by mathcam. [ full author list (2) | owner history (1) ]

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Other names:

order, partial ordering, ordering

Keywords:

relation, total order, transitivity, reflexivity, antisymmetry

Attachments:: inductively ordered (Definition) by rspuzio

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Cross-references: poset, comparability, total order, transitivity, antisymmetry, reflexivity, properties, relation
There are 199 references to this entry.

This is version 14 of partial order, born on 2001-10-06, modified 2007-11-04.
Object id is 123, canonical name is PartialOrder.
Accessed 25929 times total.

Classification:

AMS MSC:

06A06 (Order, lattices, ordered algebraic structures :: Ordered sets :: Partial order, general)

Pending Errata and Addenda

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