Partially ordered set

From Polytope Wiki
Jump to navigation Jump to search
A partial order on the elements of the cube

A partially ordered set, or poset, is a structure that formalizes and generalizes orderings on a set.

Definition[edit | edit source]

A partially ordered set is a set, S , with a binary relation, , that satisfies the following properties:

for all .
If and then , for all .
If and then , for all .

Two elements of a poset, x  and y , are said to be comparable if then either or .

Totally ordered set[edit | edit source]

A totally ordered set is a partially ordered set such that all elements are comparable with all other elements. That is if then either or .

See also[edit | edit source]

External links[edit | edit source]