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Representing homomorphisms of distributive lattices as restrictions of congruences of rectangular lattices

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Abstract

Let \({\varphi}\) be a {0, 1}-homomorphism of a finite distributive lattice D into the congruence lattice Con L of a rectangular (whence finite, planar, and semimodular) lattice L. We prove that L is a filter of an appropriate rectangular lattice K such that ConK is isomorphic with D and \({\varphi}\) is represented by the restriction map from Con K to Con L. The particular case where \({\varphi}\) is an embedding was proved by E.T. Schmidt. Our result implies that each {0, 1}-lattice homomorphism between two finite distributive lattices can be represented by the restriction of congruences of an appropriate rectangular lattice to a rectangular filter.

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  1. University of Szeged, Bolyai Institute, Aradi vértanúk tere 1, Szeged, 6720, Hungary

    Gábor Czédli

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  1. Gábor Czédli
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Correspondence to Gábor Czédli.

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Presented by G. Grätzer.

This research was supported by the NFSR of Hungary (OTKA), grant no. K77432, and by TÁMOP-4.2.1/B-09/1/KONV-2010-0005.

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Czédli, G. Representing homomorphisms of distributive lattices as restrictions of congruences of rectangular lattices. Algebra Univers. 67, 313–345 (2012). https://doi.org/10.1007/s00012-012-0190-3

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