Universality of the lattice of transformation monoids
HTML articles powered by AMS MathViewer
- by Michael Pinsker and Saharon Shelah
- Proc. Amer. Math. Soc. 141 (2013), 3005-3011
- DOI: https://doi.org/10.1090/S0002-9939-2013-11566-2
- Published electronically: May 21, 2013
- PDF | Request permission
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.References
- Garrett Birkhoff and Orrin Frink Jr., Representations of lattices by sets, Trans. Amer. Math. Soc. 64 (1948), 299–316. MR 27263, DOI 10.1090/S0002-9947-1948-0027263-2
- Martin Goldstern and Michael Pinsker, A survey of clones on infinite sets, Algebra Universalis 59 (2008), no. 3-4, 365–403. MR 2470587, DOI 10.1007/s00012-008-2100-2
- George Grätzer, General lattice theory, 2nd ed., Birkhäuser Verlag, Basel, 1998. New appendices by the author with B. A. Davey, R. Freese, B. Ganter, M. Greferath, P. Jipsen, H. A. Priestley, H. Rose, E. T. Schmidt, S. E. Schmidt, F. Wehrung and R. Wille. MR 1670580
- Thomas Jech, Set theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded. MR 1940513
- Michael Pinsker, Algebraic lattices are complete sublattices of the clone lattice over an infinite set, Fund. Math. 195 (2007), no. 1, 1–10. MR 2314073, DOI 10.4064/fm195-1-1
- V. B. Repnitskiĭ, On the representation of lattices by lattices of subsemigroups, Izv. Vyssh. Uchebn. Zaved. Mat. 1 (1996), 60–70 (Russian); English transl., Russian Math. (Iz. VUZ) 40 (1996), no. 1, 55–64. MR 1424151
- Jiří Tůma, Intervals in subgroup lattices of infinite groups, J. Algebra 125 (1989), no. 2, 367–399. MR 1018952, DOI 10.1016/0021-8693(89)90171-3
- Philip M. Whitman, Lattices, equivalence relations, and subgroups, Bull. Amer. Math. Soc. 52 (1946), 507–522. MR 16750, DOI 10.1090/S0002-9904-1946-08602-4
Bibliographic 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: 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
- MR Author ID: 160185
- ORCID: 0000-0003-0462-3152
- Email: shelah@math.huji.ac.il
- 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: https://doi.org/10.1090/S0002-9939-2013-11566-2
- MathSciNet review: 3068953