# Preorder

(Redirected from Preordering)

## Definition

A preorder, [ilmath]\preceq[/ilmath], on a set [ilmath]X[/ilmath] is a relation in [ilmath]X[/ilmath], so [ilmath]\preceq\subseteq X\times X[/ilmath], that is both[1]:

 Reflexive [ilmath]\forall x\in X[(x,x)\in\preceq][/ilmath] or equivalently[ilmath]\forall x\in X[x\preceq x][/ilmath] [ilmath]\forall x,y,z\in X[((x,y)\in\preceq\wedge(y,z)\in\preceq)\implies(x,z)\in\preceq][/ilmath] or equivalently[ilmath]\forall x,y,z\in X[(x\preceq y\wedge y\preceq z)\implies x\preceq z][/ilmath]
Note: all 3 of [ilmath]\preceq[/ilmath], [ilmath]\le[/ilmath] and [ilmath]\sqsubseteq[/ilmath][Note 1] are used for preorders.

## Terminology

A tuple, consisting of a set [ilmath]X[/ilmath], equipped with a preorder [ilmath]\preceq[/ilmath] is called a preset[1], then we may say:

• [ilmath](X,\preceq)[/ilmath] is a preset

## Notation

Be careful, as [ilmath]\preceq[/ilmath], [ilmath]\le[/ilmath] and [ilmath]\sqsubseteq[/ilmath] are all used to denote both partial and preorders, so always be clear which one you mean at the point of definition. That is to say write:

• Let [ilmath](X,\preceq)[/ilmath] be a preordering of [ilmath]X[/ilmath]. Or
• Given any [ilmath]\preceq[/ilmath] that is a preorder of [ilmath]X[/ilmath]

So forth