Rel, bicategory of relations, allegory
left and right euclidean;
extensional, well-founded relations.
A comparison on a set is a (binary) relation on such that in every pair of related elements, any other element is related to one of the original elements in the same order as the original pair:
which generalises from to any (finite, positive) number of elements. To include the case where , we must explicitly state that the relation is irreflexive.
Comparisons are most often studied in constructive mathematics. In particular, the relation on the (located Dedekind) real numbers is an irreflexive comparison, even though its negation is not constructively total. (Indeed, is a linear order, even though is not constructively a total order.)
A comparison is a cartesian monoidal semicategory enriched on the co-Heyting algebra , where is the Heyting algebra of truth values.
Last revised on May 27, 2021 at 13:48:32. See the history of this page for a list of all contributions to it.