Partial order relation

Partial order relation --- отношение частичного упорядочения (порядка).

The binary relation [math]\displaystyle{ R }[/math] is a partial order relation (or simply a partial order) on [math]\displaystyle{ V }[/math], if [math]\displaystyle{ R }[/math] is a reflexive, transitive and antisymmetric relation. Partial orders are often denoted by [math]\displaystyle{ \leq }[/math] instead of [math]\displaystyle{ R }[/math]:

[math]\displaystyle{ x \leq y\mbox{ if }(x,y) \in R,\mbox{ and }x \lt y\mbox{ and } x \neq y. }[/math]

[math]\displaystyle{ P = (V,\leq) }[/math] is then called a poset (partially ordered set). A poset [math]\displaystyle{ (V,\leq) }[/math] is finite if [math]\displaystyle{ V }[/math] is finite.

A poset [math]\displaystyle{ P = (V,\leq) }[/math] is a linear order if for all [math]\displaystyle{ u,v \in V }[/math] [math]\displaystyle{ u \leq v }[/math] or [math]\displaystyle{ v \leq u }[/math] holds.

Two posets [math]\displaystyle{ (V_{1}, \leq_{1}) }[/math], [math]\displaystyle{ (V_{2}, \leq_{2}) }[/math] are isomorphic (denoted [math]\displaystyle{ (V_{1}, \leq_{1}) \cong (V_{2}, \leq_{2}) }[/math]) if there is a bijective function [math]\displaystyle{ f }[/math] from [math]\displaystyle{ V_{1} }[/math] onto [math]\displaystyle{ V_{2} }[/math] such that [math]\displaystyle{ u \leq_{1} v }[/math] iff [math]\displaystyle{ f(u) \leq_{2} f(v) }[/math].