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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
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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
Email: marula@gmx.at

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
Email: shelah@math.huji.ac.il

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11566-2
PII: S 0002-9939(2013)11566-2
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