Strict partial order

Question

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?

Answer 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.

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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 $X$.