Difference between revisions of "Preorder"
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==See also== | ==See also== | ||
| + | * [[Preset]] - a tuple consisting of a set and a preorder on that set | ||
* [[Partial order]] - a preorder that is also ''[[Relation#Types of relation|anti-symmetric]]'' ({{AKA}} ''identitive'') | * [[Partial order]] - a preorder that is also ''[[Relation#Types of relation|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 | ||
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==Notes== | ==Notes== | ||
<references group="Note"/> | <references group="Note"/> | ||
==References== | ==References== | ||
<references/> | <references/> | ||
| − | {{Order theory navbox}} | + | {{Order theory navbox|plain}} |
{{Definition|Set Theory|Abstract Algebra|Order Theory}} | {{Definition|Set Theory|Abstract Algebra|Order Theory}} | ||
Latest revision as of 10:13, 20 February 2016
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
- Preset - a tuple consisting of a set and a preorder on that set
- 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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