Abstract
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:
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R1: ρ(z) = 0, where z is the null element of L.
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R2: ρ(a) = ρ(b) + 1 if a covers b.
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R3: ρ(a ∪ b) + ρ(a ∩ b) ≤ ρ(a) + ρ(b).
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References
R.P. Dilworth, The arithmetical theory of Birkhoff lattices, this Journal, vol. 8(1941), pp. 286–299.
M. Hall and R.P. Dilworth, The embedding problem for modular lattices, Annals of Mathematics, vol. 45(1944).
S. MacLane, A lattice formulation for transcendence degrees and p-bases, this Journal, vol. 4 (1938), pp. 455–468.
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Dilworth, R.P. (1990). Dependence Relations in a Semi-Modular Lattice. In: Bogart, K.P., Freese, R., Kung, J.P.S. (eds) The Dilworth Theorems. Contemporary Mathematicians. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-3558-8_26
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DOI: https://doi.org/10.1007/978-1-4899-3558-8_26
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