Partial ordering
From Maths
- Note to reader: this page defines ⊑ as the partial ordering under study, this is to try and make the concept distinct from \le, which the reader should have been familiar with from a young age (and thus can taint initial study)
Contents
[hide]Definition
Given a relation, \sqsubseteq in X (mathematically: \sqsubseteq\subseteq X\times X[Note 1]) we say \sqsubseteq is a partial order[1][2][3] if:
- The relation \sqsubseteq is all 3 of the following:
Name | Definition | |
---|---|---|
1 | Reflexive | \forall x\in X[(x,x)\in\sqsubseteq] or equivalently \forall x\in X[x\sqsubseteq x] |
2 | Identitive (AKA: antisymmetric) | \forall x,y\in X[((x,y)\in\sqsubseteq\wedge(y,x)\in\sqsubseteq)\implies (x=y)] or equivalently \forall x,y\in X[(x\sqsubseteq y\wedge y\sqsubseteq x)\implies(x=y)] |
3 | Transitive | \forall x,y,z\in X[((x,y)\in\sqsubseteq\wedge(y,z)\in\sqsubseteq)\implies(x,z)\in\sqsubseteq] or equivalently \forall x,y,z\in X[(x\sqsubseteq y\wedge y\sqsubseteq z)\implies(x\sqsubseteq z)] |
- Note: \le, \preceq or \subseteq[Warning 1] are all commonly used for partial relations too.
- The corresponding strict partial orderings are <, \prec and \subset
Alternative definition
Alternatively, a partial order is simply a preorder that is also anti-symmetric (AKA Identitive), that is to say[4]:
- Given a preorder in X, so a \preceq such that \preceq\subseteq X\times X, then \preceq is also a partial order if:
- \forall x,y\in X[((x,y)\in\preceq\wedge(y,x)\in\preceq)\implies (x=y)] or equivalently
- \forall x,y\in X[(x\preceq y\wedge y\preceq x)\implies(x=y)]
Induced strict partial ordering
Here, let \preceq be a partial ordering as defined above, then the relation, \prec defined by:
- (x,y)\in\prec\iff[x\ne y\wedge x\preceq y]
- Note: every strict partial ordering induces a partial ordering, given a strict partial ordering, <, we can define a relation \le as:
- x\le y\iff[x=y\vee x<y] or equivalently (in relational form): (x,y)\in\le\iff[x=y\vee (x,y)\in<]
In fact there is a 1:1 correspondence between partial and strict partial orderings, this is why the term "partial ordering" is used so casually, as given a strict you have a partial, given a partial you have a strict.
- See Overview of partial orders for more information
See also
- Preorder - like a partial order except it need not be anti-symmetric (AKA identitive)
- Strict partial order - which induces and is induced by the same partial order, thus an equivalent statement to a partial order
Notes
- Jump up ↑ Here \sqsubseteq is the name of the relation, so (x,y)\in \sqsubseteq means x\sqsubseteq y - as usual for relations
Warnings
- Jump up ↑ I avoid using \subseteq for anything other than denoting subsets, the relation and the set it relates on will go together, so you'll already be using \subseteq to mean subset
References
- Jump up ↑ Analysis - Part 1: Elements - Krzysztof Maurin
- Jump up ↑ Set Theory - Thomas Jech - Third millennium edition, revised and expanded
- Jump up ↑ Real and Abstract Analysis - Edwin Hewitt & Karl Stromberg
- Jump up ↑ An Introduction to Category Theory - Harold Simmons - 1st September 2010 edition
|