Dependence relations in a semi-modular lattice
R. P. Dilworth · Duke Mathematical Journal · 1944
Let L be an upper semi-modular lattice of finite dimensions. Then, by definition, L satisfies the Birkhoff condition: If a covers a ∩ b, then a ∪ b covers b. L is also characterized by the existence of a rank function ρ(a) which takes on integer values and has the properties: R1: ρ(z) = 0, where z is the null element of L. R2: ρ(a) = ρ(b) + 1 if a covers b. R3: ρ(a ∪ b) + ρ(a ∩ b) ≤ ρ(a) + ρ(b).