Definition:Strict Partial Ordering

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Let $\struct {S, \prec}$ be a relational structure.

Let $\prec$ be a strict ordering.

Then $\prec$ is a strict partial ordering on $S$ if and only if $\prec$ is not connected.

That is, if and only if $\struct {S, \prec}$ has at least one pair which is non-comparable:

$\exists x, y \in S: x \nprec y \land y \nprec x$

Also known as

Some sources call this an antireflexive partial ordering.