Contraction of "
partially
ordered
set", although in the
decades since their introduction,
posets have proven to be
fundamental enough to
deserve their own
word, and not have to be a
partial anything.
A poset is a set E equipped with a partial order, that is, a binary relation L (usually given the symbol "less than or equal to") which is
- reflexive
- x L x for every x ∈ E.
- transitive
- if x L y and y L z, then x L z.
- (weakly) antisymmetric
- if x L y and y L x, then x = y.
As Einar Hille put it in his nice little book
Ordinary differential equations in the complex domain, "the
fowl in a
hen-yard are
partially ordered under the
pecking order."
Posets are important in several areas of mathematics and computer science, including logic, set theory, functional analysis, combinatorics, semantics and type theory, and the study of algorithms.