nLab comparison

A comparison on a set AA is a (binary) relation \sim on AA 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:

(x,y,z:A),xzxyyz\forall (x, y, z: A),\; x \sim z \;\Rightarrow\; x \sim y \;\vee\; y \sim z

which generalises from 33 to any (finite, positive) number of elements. To include the case where n=1n = 1, we must explicitly state that the relation is irreflexive.

Comparisons are most often studied in constructive mathematics. In particular, the relation <\lt on the (located Dedekind) real numbers is an irreflexive comparison, even though its negation \geq is not constructively total. (Indeed, <\lt is a linear order, even though \geq is not constructively a total order.)

A comparison is a cartesian monoidal semicategory enriched on the co-Heyting algebra TV opTV^\op, where TVTV 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.