nLab
irreflexive comparison

Irreflexive comparison

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 SS is a (binary) relation <\lt on SS that is both irreflexive and a comparison. That is:

  • x<xx \lt x is always false;
  • If x<zx \lt z, then x<yx \lt y or y<zy \lt z

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

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

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

If a connected irreflexive relation is also symmetric (if x<yx \lt y, then y<xy \lt x), then it is a tight apartness relation, and if it is transitive (if x<yx \lt y and y<zy \lt z, then x<zx \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 SS equipped with an irreflexive comparison is a category (with SS as the set of objects) enriched over the cartesian monoidal category TV opTV^\op, that is the opposite of the poset of truth values, made into a monoidal category using disjunction. TV opTV^\op is a co-Heyting algebra.

Created on May 27, 2021 at 09:43:29. See the history of this page for a list of all contributions to it.