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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Universality of the lattice of transformation monoids
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by Michael Pinsker and Saharon Shelah PDF
Proc. Amer. Math. Soc. 141 (2013), 3005-3011 Request permission


The set of all transformation monoids on a fixed set of infinite cardinality $\lambda$, equipped with the order of inclusion, forms a complete algebraic lattice $\operatorname {Mon}(\lambda )$ with $2^\lambda$ compact elements. We show that this lattice is universal with respect to closed sublattices; i.e., the closed sublattices of $\operatorname {Mon}(\lambda )$ are, up to isomorphism, precisely the complete algebraic lattices with at most $2^\lambda$ compact elements.
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Additional Information
  • Michael Pinsker
  • Affiliation: Équipe de Logique Mathématique, Université Diderot – Paris 7, UFR de Mathématiques, 75205 Paris Cedex 13, France
  • MR Author ID: 742015
  • ORCID: 0000-0002-4727-918X
  • Email:
  • Saharon Shelah
  • Affiliation: Institute of Mathematics, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel – and – Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08854
  • MR Author ID: 160185
  • ORCID: 0000-0003-0462-3152
  • Email:
  • Received by editor(s): June 24, 2011
  • Received by editor(s) in revised form: September 3, 2011, September 12, 2011, and November 24, 2011
  • Published electronically: May 21, 2013
  • Additional Notes: The research of the first author was supported by an APART fellowship of the Austrian Academy of Sciences
    The research of the second author was supported by German-Israeli Foundation for Scientific Research & Development Grant No. 963-98.6/2007.
    The authors would like to thank an anonymous referee for valuable comments which led to significant improvements in the presentation of the paper.
  • Communicated by: Julia Knight
  • © Copyright 2013 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 141 (2013), 3005-3011
  • MSC (2010): Primary 06B15; Secondary 06B23, 20M20
  • DOI:
  • MathSciNet review: 3068953