Definition:Non-Transitive Relation

Definition

Let $\RR \subseteq S \times S$ be a relation in $S$.

$\RR$ is non-transitive if and only if it is neither transitive nor antitransitive.

Also known as

Some sources use the term intransitive.

However, as intransitive is also found in other sources to mean antitransitive, it is better to use the clumsier, but less ambiguous, non-transitive.

Also see

• Definition:Transitivity (Relation Theory)

• Definition:Transitive Relation
• Definition:Antitransitive Relation

• Results about non-transitive relations can be found here.

Sources

• 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $4$: The Predicate Calculus $2$: $5$ Properties of Relations
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): non-transitive (of a relation)