Strict partial ordering

Note to reader: this page defines $\sqsubset$ as the strict partial ordering under study, this is to try and make the concept distinct from $<$, which the reader should have been familiar with from a young age (and thus can taint initial study), this notation corresponds with the notation in partial ordering.

Definition

Given a relation, $\sqsubset$ on $X$ (mathematically: $\sqsubset\subseteq X\times X$) we say that $\sqsubset$ is a strict partial ordering if:

• The relation $\prec$ is both of the following:

	Name	Meaning
1	Irreflexive (not reflexive)	$\forall x\in X[(x,x)\notin\sqsubset]$ or equivalently: $\forall x\in X[x\not\sqsubset x]$
2	Transitive	$\forall x,y,z\in X[((x,y)\in\sqsubset\wedge(y,z)\in\sqsubset)\implies((x,z)\in\sqsubset)]$ or equivalently $\forall x,y,z\in X[(x\sqsubset y\wedge y\sqsubset z)\implies x\sqsubset z]$

• Note: $<$, $\prec$ and $\subset$ are all commonly used for strict relations too
  • Their corresponding partial orders are $\le$, $\preceq$ and $\subseteq$

Induced partial ordering

Here, let $\prec$ be a strict partial ordering as defined above, then the relation, $\preceq$ defined by:

• $(x,y)\in\preceq\iff[x = y\vee x\prec y]$

is a partial ordering

• Note: every partial ordering induces a strict partial ordering, given a partial ordering, $\le$, we can define a relation $<$ as:
  • $x< y\iff[x\ne y\wedge x\le y]$ or equivalently (in relational form): $(x,y)\in\le\iff[x\ne y\wedge (x,y)\in\le]$

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