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Varieties of Lattices with Geometric Descriptions

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Abstract

A lattice L is spatial if every element of L is a join of completely join-irreducible elements of L (points), and strongly spatial if it is spatial and the minimal coverings of completely join-irreducible elements are well-behaved. Herrmann et al. proved in 1994 that every modular lattice can be embedded, within its variety, into an algebraic and spatial lattice. We extend this result to n-distributive lattices, for fixed n. We deduce that the variety of all n-distributive lattices is generated by its finite members, thus it has a decidable word problem for free lattices. This solves two problems stated by Huhn in 1985. We prove that every modular (resp., n-distributive) lattice embeds within its variety into some strongly spatial lattice. Every lattice which is either algebraic modular spatial or bi-algebraic is strongly spatial. We also construct a lattice that cannot be embedded, within its variety, into any algebraic and spatial lattice. This lattice has a least and a largest element, and it generates a locally finite variety of join-semidistributive lattices.

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Authors and Affiliations

  1. Laboratoire d’Informatique Fondamentale de Marseille, Université de Provence, 39 rue F. Joliot Curie, 13453, Marseille Cedex 13, France

    Luigi Santocanale

  2. LMNO, CNRS UMR 6139 Département de Mathématiques, Université de Caen, 14032, Caen Cedex, France

    Friedrich Wehrung

Authors
  1. Luigi Santocanale
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  2. Friedrich Wehrung
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Correspondence to Friedrich Wehrung.

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Both authors were partially supported by the PEPS project TRECOLOCOCO.

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Santocanale, L., Wehrung, F. Varieties of Lattices with Geometric Descriptions. Order 30, 13–38 (2013). https://doi.org/10.1007/s11083-011-9225-1

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