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Infinite antichains in semilattices

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Abstract

In this paper we consider infinite antichains and the semilattices that they generate, mainly in the context of continuous semilattices. Conditions are first considered that lead to the antichain generating a copy of a countable product of the two-element semilattice. Then a special semilattice, called Δ, is defined, its basic properties developed, and it is shown in our main result that a continuous semilattice with an infinite antichain converging to a larger element contains a semilattice copy of Δ. The paper closes with a consideration of countable antichains that converge to a lower element or a parallel element and the kinds of semilattices generated in this context.

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Authors and Affiliations

  1. Mathematics Department, Louisiana State University, 70803, Baton Rouge, LA, USA

    J. D. Lawson

  2. Mathematics Department, Tulane University, 70118, New Orleans, LA, USA

    Michael Mislove

  3. Maths. Institute, 24/29 St. Giles, OX1 3LB, Oxford, England

    Hilary A. Priestley

Authors
  1. J. D. Lawson
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  2. Michael Mislove
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  3. Hilary A. Priestley
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Additional information

Communicated by K. Keimel

The first two authors gratefully acknowledge the partial support of the National Science Foundation and the hospitality of Oxford University during the preparation of this paper.

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Lawson, J.D., Mislove, M. & Priestley, H.A. Infinite antichains in semilattices. Order 2, 275–290 (1985). https://doi.org/10.1007/BF00333134

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  • DOI: https://doi.org/10.1007/BF00333134

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