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Distributive lattices have the intersection property
Mathematica Bohemica, Tome 146 (2021) no. 1, pp. 7-17
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Distributive lattices form an important, well-behaved class of lattices. They are instances of two larger classes of lattices: congruence-uniform and semidistributive lattices. Congruence-uniform lattices allow for a remarkable second order of their elements: the core label order; semidistributive lattices naturally possess an associated flag simplicial complex: the canonical join complex. In this article we present a characterization of finite distributive lattices in terms of the core label order and the canonical join complex, and we show that the core label order of a finite distributive lattice is always a meet-semilattice.
Distributive lattices form an important, well-behaved class of lattices. They are instances of two larger classes of lattices: congruence-uniform and semidistributive lattices. Congruence-uniform lattices allow for a remarkable second order of their elements: the core label order; semidistributive lattices naturally possess an associated flag simplicial complex: the canonical join complex. In this article we present a characterization of finite distributive lattices in terms of the core label order and the canonical join complex, and we show that the core label order of a finite distributive lattice is always a meet-semilattice.
DOI : 10.21136/MB.2019.0156-18
Classification : 06D05
Keywords: distributive lattice; congruence-uniform lattice; canonical join complex; core label order; intersection property 
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Mühle, Henri. Distributive lattices have the intersection property. Mathematica Bohemica, Tome 146 (2021) no. 1, pp. 7-17. doi: 10.21136/MB.2019.0156-18

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