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The core label order of a congruence-uniform lattice

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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.

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  1. Institut für Algebra, Technische Universität Dresden, Zellescher Weg 12-14, 01069, Dresden, Germany

    Henri Mühle

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  1. Henri Mühle
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Correspondence to Henri Mühle.

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Mühle, H. The core label order of a congruence-uniform lattice. Algebra Univers. 80, 10 (2019). https://doi.org/10.1007/s00012-019-0585-5

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