Definition:Non-Comparable

Definition

Let $\left({S, \mathcal R}\right)$ be a relational structure.

Two elements $x, y \in S, x \ne y$ are non-comparable if neither $x \mathrel {\mathcal R} y$ nor $y \mathrel {\mathcal R} x$.

If $x$ and $y$ are not non-comparable then they are comparable, but the latter term is not so frequently encountered.

Also known as

Sometimes this can be found without the hyphen: noncomparable.

Some use the term incomparable.

Also see

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.

Sources

• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 13$: Arithmetic
• 1968: A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis ... (previous) ... (next): $\S 3.3$: Ordered sets. Order types