# Definition:Non-Comparable

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## Contents

## 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