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Algebra universalis

, 80:10 | Cite as

The core label order of a congruence-uniform lattice

  • Henri MühleEmail author
Article
  • 4 Downloads

Abstract

We investigate the alternate order on a congruence-uniform lattice \({\mathcal {L}}\) as introduced by N. Reading, which we dub the core label order of \({\mathcal {L}}\). When \({\mathcal {L}}\) can be realized as a poset of regions of a simplicial hyperplane arrangement, the core label order is always a lattice. For general \({\mathcal {L}}\), however, this fails. We provide an equivalent characterization for the core label order to be a lattice. As a consequence we show that the property of the core label order being a lattice is inherited to lattice quotients. We use the core label order to characterize the congruence-uniform lattices that are Boolean lattices, and we investigate the connection between congruence-uniform lattices whose core label orders are lattices and congruence-uniform lattices of biclosed sets.

Keywords

Congruence-uniform lattices Interval doubling Semidistributive lattices Crosscut theorem Möbius function Biclosed sets 

Mathematics Subject Classification

06B05 06A07 

Notes

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institut für AlgebraTechnische Universität DresdenDresdenGermany

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