Definition:Non-Comparable Elements

Definition

Let $\RR$ be a relation.

Two elements $x \in \Dom \RR$, $y \in \Img \RR$ such that $x \ne y$ are non-comparable if neither $x \mathrel \RR y$ nor $y \mathrel \RR x$.

The definition is usually used in the context of orderings and preorderings.

Such a relation with non-comparable pairs is referred to as a partial preordering or partial ordering.

Also known as

Non-comparable elements can be presented without the hyphen: noncomparable.

Some sources use the term incomparable.

Also see

• Definition:Comparable Elements elements that are not non-comparable

• Results about comparability of elements can be found here.

Sources

• 1968: A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis ... (previous) ... (next): $\S 3.3$: Ordered sets. Order types