Transitive Relation

A Relation R : A → A → Ω : 𝑅 → 𝐴 𝐴 → Ω R:A\to A\to\Omega is transitive if for all x , y , z : A : 𝑥 𝑦 𝑧 𝐴 x,y,z:A , R ⁢ ( x , y ) ∧ R ⁢ ( y , z ) ⇒ R ⁢ ( x , z ) ⇒ 𝑅 𝑥 𝑦 𝑅 𝑦 𝑧 𝑅 𝑥 𝑧 R(x,y)\land R(y,z)\Rightarrow R(x,z) . Equivalently, R 𝑅 R is transitive if R ∘ R ⊆ R 𝑅 𝑅 𝑅 R\circ R\subseteq R , where R ∘ R 𝑅 𝑅 R\circ R is the Composition of Relations; this definition works nicely in an Allegory.