left and right euclidean;
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:
\forall (x, y, z: A),\; x \sim z \;\Rightarrow\; x \sim y \;\vee\; y \sim z
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 a comparison, even though its negation is not constructively total. (Indeed, is a linear order, even though is not constructively a total order.)