Partial ordering

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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)

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.

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]

is a strict partial ordering

  • 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 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

  1. Jump up Here \sqsubseteq is the name of the relation, so (x,y)\in \sqsubseteq means x\sqsubseteq y - as usual for relations

Warnings

  1. 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

  1. Jump up Analysis - Part 1: Elements - Krzysztof Maurin
  2. Jump up Set Theory - Thomas Jech - Third millennium edition, revised and expanded
  3. Jump up Real and Abstract Analysis - Edwin Hewitt & Karl Stromberg
  4. Jump up An Introduction to Category Theory - Harold Simmons - 1st September 2010 edition