Abstract
Since the middle 1930’s when the work of Birkhoff and Ore stimulated the modern development of lattice theory, it has been conjectured that in a finite modular lattice the number of meet irreducibles is equal to the number of join irreducibles.1 I shall prove here the following general combinatorial theorem on finite modular lattices which includes a proof of this conjecture as a special case.
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References
G. Birkhoff. Lattice Theory, Revised edition, Amer. Math. Soc. Coll. Publications, Vol. 25, 1948.
L. Weisner, Abstract theory of inversion of finite series, Trans. Amer. Math. Soc., 38 (1935) 474–484.
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Dilworth, R.P. (1990). Proof of a Conjecture on Finite Modular Lattices. 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_21
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DOI: https://doi.org/10.1007/978-1-4899-3558-8_21
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