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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The set of all transformation monoids on a fixed set of infinite cardinality , equipped with the order of inclusion, forms a complete algebraic lattice with compact elements. We show that this lattice is universal with respect to closed sublattices; i.e., the closed sublattices of are, up to isomorphism, precisely the complete algebraic lattices with at most compact elements.

**[BF48]**Garrett Birkhoff and Orrin Frink Jr.,*Representations of lattices by sets*, Trans. Amer. Math. Soc.**64**(1948), 299–316. MR**0027263**, 10.1090/S0002-9947-1948-0027263-2**[GP08]**Martin Goldstern and Michael Pinsker,*A survey of clones on infinite sets*, Algebra Universalis**59**(2008), no. 3-4, 365–403. MR**2470587**, 10.1007/s00012-008-2100-2**[Grä03]**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****[Jec03]**Thomas Jech,*Set theory*, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded. MR**1940513****[Pin07]**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**, 10.4064/fm195-1-1**[Rep96]**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****[Tům89]**Jiří T\ocirc{u}ma,*Intervals in subgroup lattices of infinite groups*, J. Algebra**125**(1989), no. 2, 367–399. MR**1018952**, 10.1016/0021-8693(89)90171-3**[Whi46]**Philip M. Whitman,*Lattices, equivalence relations, and subgroups*, Bull. Amer. Math. Soc.**52**(1946), 507–522. MR**0016750**, 10.1090/S0002-9904-1946-08602-4

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
06B15,
06B23,
20M20

Retrieve articles in all journals with MSC (2010): 06B15, 06B23, 20M20

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:
https://doi.org/10.1090/S0002-9939-2013-11566-2

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