An irreflexive comparison on a set $S$ is a (binary) relation$\lt$ on $S$ that is both irreflexive and a comparison. That is:

$x \lt x$ is always false;

If $x \lt z$, then $x \lt y$ or $y \lt z$

An irreflexive comparison that is also a connected relation (if $x \lt y$ is false and $y \lt x$ is false, then $x = y$) is a connected irreflexive comparison.

If the set is an inequality space, then an irreflexive comparison is strongly connected if $x \neq y$ implies $x \lt y$ or $y \lt x$.

If an irreflexive comparison satisfies symmetry (if $x \lt y$ then $y \lt x$ then it is an apartness relation.

If a connected irreflexive relation is also symmetric (if $x \lt y$, then $y \lt x$), then it is a tight apartness relation, and if it is transitive (if $x \lt y$ and $y \lt z$, then $x \lt z$), then it is a linear order.

Thus, irreflexive comparisons are dual to preorders while connected irreflexive comparisons are dual to partial orders.