nLab comparison

 Definition

A comparison or cotransitive relation or weakly linear relation on a set $A$ is a (binary) relation $\sim$ on $A$ 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:

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

which generalises from $3$ to any (finite, positive) number of elements. To include the case where $n = 1$, we must explicitly state that the relation is irreflexive.

Alternatively, this is the same condition as

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

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 $TV^\op$, where $TV$ is the Heyting algebra of truth values.

 Related concepts

• pseudo-order

• total relation

References

• Wikipedia, Apartness relation

• Wikipedia, Pseudo-order

• Univalent Foundations Project, Homotopy Type Theory – Univalent Foundations of Mathematics (2013)

• Auke Booij, Analysis in univalent type theory (2020) $[$etheses:10411, pdf$]$