Definition:Non-comparable
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Definition
Let $\left({S, \mathcal R}\right)$ be a relational structure.
Two elements $x, y \in S$ are non-comparable (or noncomparable) if neither $x \mathcal R y$ nor $y \mathcal R x$.
The definition is usually used in the context of orderings and preorderings: such a relation is referred to as a partial preordering or partial ordering.
If $x$ and $y$ are not non-comparable then they are comparable, but the latter term is not so frequently encountered.
Sources
- Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 13$: Arithmetic
- A.N. Kolmogorov and S.V. Fomin‎: Introductory Real Analysis (1968): $\S 3.3$
