Transitive

A relation R on a set S is transitive if it chains: whenever aRb and bRc, then aRc.

Transitivity is one of the three defining properties of a partial order and one of the two defining properties of a preorder. It captures the idea that the relation is coherent across intermediate steps — if a ≤ b and b ≤ c, then a ≤ c directly, without needing b as a witness.

In a poset viewed as a category, transitivity is composition: a morphism a → b followed by b → c composes to a morphism a → c.

In modal logic, transitivity of the accessibility relation validates the 4 axiom: □φ → □□φ. If accessibility chains (w accesses v, v accesses u, therefore w accesses u), then what is necessary is necessarily necessary. The modality j in the semiotic universe is idempotent (j(j(a)) = j(a)), which is the order-theoretic expression of transitivity: closing and then closing again yields nothing new.