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Universality of the lattice of transformation monoids

Authors: Michael Pinsker and Saharon Shelah
Journal: Proc. Amer. Math. Soc. 141 (2013), 3005-3011
MSC (2010): Primary 06B15; Secondary 06B23, 20M20
Published electronically: May 21, 2013
MathSciNet review: 3068953
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Abstract: 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

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

Keywords: Algebraic lattice, transformation monoid, submonoid, closed sublattice
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
Article copyright: © Copyright 2013 American Mathematical Society