Just as preorders generalise equivalence relations and total orders, irreflexive comparisons should generalise apartness relations and strict total orders
An irreflexive comparison on a set is a (binary) relation on that is both irreflexive and a comparison. That is:
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.
An important part of an irreflexive comparison is that it is a preorder.
The negation of an irreflexive comparison is transitive.
The contrapositive of comparison says that
for all , , and , if or is false, then is false.
By one of de Morgan's laws, that or is false is logically to equivalent to that and , and substituting this into the original statement results in
if and , then
which is precisely transitivity for the negation of the irreflexive comparison.
The negation of an irreflexive comparison is reflexive.
Irreflexivity states that is true, which is precisely reflexivity for the negation of the strict weak order.
The negation of an irreflexive comparison is a preorder.
The incomparability relation of a strict weak order, , is an equivalence relation
For every preorder, is an equivalence relation. Since is a preorder, is an equivalence relation.
If the irreflexive comparison is a connected relation, then its negation is a partial order.
The connectedness condition states that implies equality, which is precisely the antisymmetry condition for the negation of the strict weak order, implying that its negation is a partial order.
If the irreflexive comparison is an apartness relation, then its negation is an equivalence relation.
The symmetry condition of the apartness relation states that implies for all and . It’s contrapositive states that implies for all and , which is the symmetry condition for the negation of . A symmetric preorder is the same thing as an equivalence relation, which means that the negation of is an equivalence relation.
If the irreflexive comparison is a tight apartness relation, then its negation is the equality relation.
This follows from the previous two theorems.
If an irreflexive comparison satisfies symmetry (if then then it is an apartness relation.
If an irreflexive comparison is asymmetric (if , then ) or transitive (if and , then ), then it is a strict weak order.
Connected versions of the above result in tight apartness relations and strict total orders.
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 .
Last revised on December 26, 2023 at 00:42:44. See the history of this page for a list of all contributions to it.