nLab irreflexive comparison

Idea

Just as preorders generalise equivalence relations and total orders, irreflexive comparisons should generalise apartness relations and linear orders

Definitions

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.

Properties

A set $S$ equipped with an irreflexive comparison is a category (with $S$ as the set of objects) enriched over the cartesian monoidal category $TV^\op$, that is the opposite of the poset of truth values, made into a monoidal category using disjunction. $TV^\op$ is a co-Heyting algebra.

Related concepts

• apartness relation

• tight apartness relation

• linear order

• preorder

• partial order

• quasiorder