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The automorphism group of a function lattice: A problem of Jónsson and McKenzie

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Abstract

It is shown that Aut(LQ) is naturally isomorphic to Aut(L) × Aut(Q) whenL is a directly and exponentially indecomposable lattice,Q a non-empty connected poset, and one of the following holds:Q is arbitrary butL is ajm-lattice,Q is finitely factorable and L is complete with a join-dense subset of completely join-irreducible elements, orL is arbitrary butQ is finite. A problem of Jónsson and McKenzie is thereby solved. Sharp conditions are found guaranteeing the injectivity of the natural mapv P,Q from Aut(P) × Aut(Q) to Aut(PQ)P andQ posets), correcting misstatements made by previous authors. It is proven that, for a bounded posetP and arbitraryQ, the Dedekind-MacNeille completion ofPQ,DM(PQ), is isomorphic toDM(P)Q. This isomorphism is used to prove that the natural mapv P,Q is an isomorphism ifvDM(P),Q is, reducing a poset problem to a more tractable lattice problem.

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  1. J. D. Farley

    Present address: Mathematical Sciences Research Institute, 1000 Centennial Drive, 94720, Berkeley, CA, USA

Authors and Affiliations

  1. Mathematical Institute, University of Oxford, 24-29 St. Giles', OX1 3LB, Oxford, UK

    J. D. Farley

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  1. J. D. Farley
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The author would like to thank his supervisor, Dr. H. A. Priestley, for her direction and advice as well as his undergraduate supervisor, Prof. Garrett Birkhoff, and Dr. P. M. Neumann for comments regarding the paper. This material is based upon work supported under a (U.S.) National Science Foundation Graduate Research Fellowship and a Marshall Aid Commemoration Commission Scholarship.

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Farley, J.D. The automorphism group of a function lattice: A problem of Jónsson and McKenzie. Algebra Universalis 36, 8–45 (1996). https://doi.org/10.1007/BF01192707

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  • DOI: https://doi.org/10.1007/BF01192707

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