# Preorder

## 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

• Preset - a tuple consisting of a set and a preorder on that set
• Partial order - a preorder that is also anti-symmetric (AKA identitive)
• Poset (is to partial order as preset is to preorder) - a tuple consisting of a set and a partial order on that set

## Notes

1. Don't use [ilmath]\subseteq[/ilmath] unless you really have to, prefer things like [ilmath]\preceq_A[/ilmath] and [ilmath]\preceq_B[/ilmath] if you run out, as any work involving implies (by the implies-subset relation will probably involve subsets at some point (the converse is true too, if you use subset you'll probably have implies at some point, but this isn't relevant to the warning)