nLab comparison



A comparison or cotransitive relation or weakly linear relation 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.

Alternatively, this is the same condition as

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

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 pseudo-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 December 26, 2023 at 05:36:08. See the history of this page for a list of all contributions to it.