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A new local criterion for the lattice property

Abstract

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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Ziegler, G.M. A new local criterion for the lattice property. Algebra Universalis 31, 608–610 (1994). https://doi.org/10.1007/BF01236510

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Keywords

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