Strict partial order

I know what is a partial order: for example the power set of a set or the natural numbers.

But a strict partial order is a set with a binary relation $R$ so that $R$ is transitive, irreflexive (not $x < x$) but antisymmetric ($x < y$ and $y < x$ implies $x=y$).

I can't find any example of such. If it is strict relation the third property, $x < y$ and $y < x$ implies $x=y$ seems impossible. What is an example of a strict partial order?

Solution 1:

The third property is superfluous. If $\prec$ is a transitive and irrreflexive relation on a set $X$, then it is also asymmetric: for all $x, y \in X$ either $x \not\prec y$ or $y \not\prec x$.

If $x \prec y$ and $y \prec x$, then by transitivity $x \prec x$, contradicting irreflexivity.

This means that the hypothesis of the antisymmetry relation can never hold for transitive irreflexive relations, and so that property is vacuously satisfied by such relations.

Given any partial order $\preceq$ on a set $X$, the relation $\prec$ on $X$ defined by $$x \prec y \quad \Longleftrightarrow \quad x \preceq y \;\&\; x \neq y$$ is a strict partial order on $X$.

So, for example:

• The $<$ relation on the real/rational/integer/natural numbers.
• The $\subsetneq$ relation on $\mathcal{P} (X)$ for any set $X$.