Definition:Non-Comparable
(Redirected from Definition:Non-comparable)
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Definition
Let $\struct {S, \RR}$ be a relational structure.
Two elements $x, y \in S, x \ne y$ are non-comparable if neither $x \mathrel \RR y$ nor $y \mathrel \RR x$.
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