Preorder
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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] |
|---|---|
| Transitive | [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
See also
- Partial order - a preorder that is also anti-symmetric (AKA identitive)
Notes
- ↑ 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)
References
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