Definition·D051
Partial order
A relation that is reflexive, antisymmetric, and transitive.
A relation
on a set
is a partial order when it is
The pair
is then a partially ordered set, or poset; one often writes
for
.
In words
A partial order is a relation that is reflexive: every element relates to itself, antisymmetric: two elements related in both directions are equal, and transitive: the relation chains through intermediates.
Rests onA02
Never needed: F02 · F03 · F04 · F05 · F06 · F08 · F09 · F10 · F11 · F12 · F13 · F14 · A01 · A03 · A04 · A05 · A06 · A07 · A08 (computed from the citation graph, not asserted).
Remarks
Partial orders abstract 'less than or equal to', 'divides', and 'is a subset of'. The word 'partial' permits incomparable elements: where an equivalence relation is symmetric, a partial order is antisymmetric instead. Requiring every pair to be comparable gives a total order; requiring a least element in every nonempty subset gives a well order. Examples with genuinely incomparable elements: subset inclusion on a power set, and divisibility on
(D026); while the order on the naturals is total.