Semilattices of finitely generated ideals of exchange rings with finite stable rank
Author:
F. Wehrung
Journal:
Trans. Amer. Math. Soc. 356 (2004), 19571970
MSC (2000):
Primary 06A12, 20M14, 06B10; Secondary 19K14
Published electronically:
October 28, 2003
MathSciNet review:
2031048
Fulltext PDF Free Access
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Abstract: We find a distributive semilattice of size that is not isomorphic to the maximal semilattice quotient of any Riesz monoid endowed with an orderunit of finite stable rank. We thus obtain solutions to various open problems in ring theory and in lattice theory. In particular:  
 There is no exchange ring (thus, no von Neumann regular ring and no C*algebra of real rank zero) with finite stable rank whose semilattice of finitely generated, idempotentgenerated twosided ideals is isomorphic to .
 
 There is no locally finite, modular lattice whose semilattice of finitely generated congruences is isomorphic to .
These results are established by constructing an infinitary statement, denoted here by , that holds in the maximal semilattice quotient of every Riesz monoid endowed with an orderunit of finite stable rank, but not in the semilattice .
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 2.
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 3.
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 4.
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 5.
 A. Day, Characterization of finite lattices that are bounded homomorphic images of sublattices of free lattices, Canad. J. Math. 31 (1979), 6978. MR 81h:06004
 6.
 E.G. Effros, D.E. Handelman, and C.L. Shen, Dimension groups and their affine representations, Amer. J. Math. 102 (1980), no. 2, 385407. MR 83g:46061
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 , ``Partially Ordered Abelian Groups with Interpolation'', Math. Surveys and Monographs 20, Amer. Math. Soc., Providence, 1986. MR 88f:06013
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 K.R. Goodearl and D.E. Handelman, Tensor products of dimension groups and of unitregular rings, Canad. J. Math. 38, no. 3 (1986), 633658. MR 87i:16043
 11.
 K.R. Goodearl and F. Wehrung, Representations of distributive semilattices in ideal lattices of various algebraic structures, Algebra Universalis 45 (2001), 71102.MR 2002g:06008
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 G. Grätzer, ``General Lattice Theory. Second edition'', 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, Birkhäuser Verlag, Basel, 1998. xx+663 p.MR 2000b:06001
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 G. Grätzer, H. Lakser, and F. Wehrung, Congruence amalgamation of lattices, Acta Sci. Math. (Szeged) 66 (2000), 339358. MR 2001f:06010
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 G. Grätzer and F. Wehrung, On the number of joinirreducibles in a congruence representation of a finite distributive lattice, Algebra Universalis 49 (2003), 165178.
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 B. Jónsson and J.E. Kiefer, Finite sublattices of a free lattice, Canad. J. Math. 14 (1962), 487497. MR 25:1117
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 C. Moreira dos Santos, A refinement monoid whose maximal antisymmetric quotient is not a refinement monoid, Semigroup Forum 65, no. 2 (2002), 249263. MR 2003c:20074
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 E. Pardo, Monoides de refinament i anells d'intercanvi, Ph.D. Thesis, Universitat Autònoma de Barcelona, 1995.
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 P. Ruzicka, A distributive semilattice not isomorphic to the maximal semilattice quotient of the positive cone of any dimension group, J. Algebra 268, no. 1 (2003), 290300.
 19.
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 20.
 , A survey of recent results on congruence lattices of lattices, Algebra Universalis 48, no. 4 (2002), 439471.
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 , From joinirreducibles to dimension theory for lattices with chain conditions, J. Algebra Appl. 1, no. 2 (2002), 215242. MR 2003f:06007
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 , Forcing extensions of partial lattices, J. Algebra 262, no. 1 (2003), 127193.
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Additional Information
F. Wehrung
Affiliation:
Département de Mathématiques, CNRS, UMR 6139, Université de Caen, Campus II, B.P. 5186, 14032 Caen Cedex, France
Email:
wehrung@math.unicaen.fr
DOI:
http://dx.doi.org/10.1090/S0002994703033695
PII:
S 00029947(03)033695
Keywords:
Semilattice,
distributive,
monoid,
refinement,
ideal,
stable rank,
strongly separative,
exchange ring,
lattice,
congruence
Received by editor(s):
January 3, 2003
Received by editor(s) in revised form:
April 2, 2003
Published electronically:
October 28, 2003
Article copyright:
© Copyright 2003 American Mathematical Society
