Varieties of lattices with geometric descriptions

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, Pickering, and Roddy 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. 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, join-semidistributive variety.
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Contributor : Friedrich Wehrung <>
Submitted on : Friday, July 1, 2011 - 9:04:48 PM
Last modification on : Friday, October 18, 2019 - 12:10:09 PM
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Luigi Santocanale, Friedrich Wehrung. Varieties of lattices with geometric descriptions. Order, Springer Verlag, 2013, 30 (1), pp.13--38. ⟨10.1007/s11083-011-9225-1⟩. ⟨hal-00564024v2⟩

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