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Congruence lattices of function lattices

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Abstract

Thefunction lattice L P is the lattice of all isotone maps from a posetP into a latticeL.

D. Duffus, B. Jónsson, and I. Rival proved in 1978 that for afinite poset P, the congruence lattice ofL P is a direct power of the congruence lattice ofL; the exponent is |P|.

This result fails for infiniteP. However, utilizing a generalization of theL P construction, theL[D] construction (the extension ofL byD, whereD is a bounded distributive lattice), the second author proved in 1979 that ConL[D] is isomorphic to (ConL) [ConD] for afinite lattice L.

In this paper we prove that the isomorphism ConL[D]≅(ConL)[ConD] holds for a latticeL and a bounded distributive latticeD iff either ConL orD is finite.

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References

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Authors and Affiliations

  1. Department of Mathematics, University of Manitoba, R3T 2N2, Winnipeg, Man., Canada

    G. Grätzer

  2. Department of Mathematics, Transport Engineering Faculty, Technical University of Budapest, Müegyetem rkp. 9, 1111, Budapest, Hungary

    E. T. Schmidt

Authors
  1. G. Grätzer
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  2. E. T. Schmidt
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Additional information

Communicated by I. Rival

The research of the first author was supported by the NSERC of Canada.

The research of the second author was supported by the Hungarian National Foundation for Scientific Research, under Grant No. 1903.

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Grätzer, G., Schmidt, E.T. Congruence lattices of function lattices. Order 11, 211–220 (1994). https://doi.org/10.1007/BF02115812

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  • DOI: https://doi.org/10.1007/BF02115812

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