Just as preorders generalise equivalence relations and total orders, irreflexive comparisons should generalise apartness relations and linear orders
An irreflexive comparison on a set is a (binary) relation on that is both irreflexive and a comparison. That is:
An irreflexive comparison that is also a connected relation (if is false and is false, then ) is a connected irreflexive comparison.
If the set is an inequality space, then an irreflexive comparison is strongly connected if implies or .
If an irreflexive comparison satisfies symmetry (if then then it is an apartness relation.
If a connected irreflexive relation is also symmetric (if , then ), then it is a tight apartness relation, and if it is transitive (if and , then ), then it is a linear order.
Thus, irreflexive comparisons are dual to preorders while connected irreflexive comparisons are dual to partial orders.
A set equipped with an irreflexive comparison is a category (with as the set of objects) enriched over the cartesian monoidal category , that is the opposite of the poset of truth values, made into a monoidal category using disjunction. is a co-Heyting algebra.
Created on May 27, 2021 at 13:43:29. See the history of this page for a list of all contributions to it.