# Partially ordered set

(Redirected from Poset)

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:

- Reflexivity
- for all .
- Transitivity
- If and then , for all .
- Antisymmetry
- 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]

- Wikipedia contributors. "Partially ordered set".
- Weisstein, Eric W. "Partially Ordered Set" at MathWorld.
- nLab contributors. "Partial order" on nLab.

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