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A new local criterion for the lattice property


We prove that a bounded poset of finite length is a lattice if and only if the following condition holds: whenever two elementsx 1,x 2 that cover a common elementx are both smaller that two elementsy 1,y 2 that are covered by a common elementy, then there exists an elementz that is an upper bound forx 1,x 2 and a lower bound fory 1,y 2.

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  1. [1]

    Birkhoff, G.,Lattice theory, 3. ed., Amer. Math. Soc. Colloquium publ.25, 1967.

  2. [2]

    Björner, A., Edelman, P. H. andZiegler, G. M.,Hyperplane arrangements with a lattice of regions, Discrete Comput. Geometry5 (1990), 263–288.

    Google Scholar 

  3. [3]

    Stanley, R. P.,Enumerative Combinatorics, Volume I, Wadsworth, 1986.

  4. [4]

    Ziegler, G. M.,Higher Bruhat orders and cyclic hyperplane arrangements, Topology32 (1993), 259–279.

    Google Scholar 

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Ziegler, G.M. A new local criterion for the lattice property. Algebra Universalis 31, 608–610 (1994).

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  • Lattice Property
  • Finite Length
  • Local Criterion
  • Bounded Poset