Dense Relation

A Relation R : A → A → Ω : 𝑅 → 𝐴 𝐴 → Ω R:A\to A\to\Omega is dense if for all x , y : A : 𝑥 𝑦 𝐴 x,y:A with R ⁢ ( x , y ) 𝑅 𝑥 𝑦 R(x,y) , there merely exists a z : A : 𝑧 𝐴 z:A with R ⁢ ( x , z ) ∧ R ⁢ ( y , z ) 𝑅 𝑥 𝑧 𝑅 𝑦 𝑧 R(x,z)\land R(y,z) . More concisely, R 𝑅 R is dense if R ⊆ R ∘ R 𝑅 𝑅 𝑅 R\subseteq R\circ R , where R ∘ R 𝑅 𝑅 R\circ R denotes Composition of Relations.