Countable chains of distributive lattices as maximal semilattice quotients of positive cones of dimension groups

Pavel Růžička

Commentationes Mathematicae Universitatis Carolinae (2006)

  • Volume: 47, Issue: 1, page 11-20
  • ISSN: 0010-2628

Abstract

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We construct a countable chain of Boolean semilattices, with all inclusion maps preserving the join and the bounds, whose union cannot be represented as the maximal semilattice quotient of the positive cone of any dimension group. We also construct a similar example with a countable chain of strongly distributive bounded semilattices. This solves a problem of F. Wehrung.

How to cite

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Růžička, Pavel. "Countable chains of distributive lattices as maximal semilattice quotients of positive cones of dimension groups." Commentationes Mathematicae Universitatis Carolinae 47.1 (2006): 11-20. <http://eudml.org/doc/249867>.

@article{Růžička2006,
abstract = {We construct a countable chain of Boolean semilattices, with all inclusion maps preserving the join and the bounds, whose union cannot be represented as the maximal semilattice quotient of the positive cone of any dimension group. We also construct a similar example with a countable chain of strongly distributive bounded semilattices. This solves a problem of F. Wehrung.},
author = {Růžička, Pavel},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {semilattice; lattice; distributive; dimension group; direct limit; semilattice; direct limit},
language = {eng},
number = {1},
pages = {11-20},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Countable chains of distributive lattices as maximal semilattice quotients of positive cones of dimension groups},
url = {http://eudml.org/doc/249867},
volume = {47},
year = {2006},
}

TY - JOUR
AU - Růžička, Pavel
TI - Countable chains of distributive lattices as maximal semilattice quotients of positive cones of dimension groups
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2006
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 47
IS - 1
SP - 11
EP - 20
AB - We construct a countable chain of Boolean semilattices, with all inclusion maps preserving the join and the bounds, whose union cannot be represented as the maximal semilattice quotient of the positive cone of any dimension group. We also construct a similar example with a countable chain of strongly distributive bounded semilattices. This solves a problem of F. Wehrung.
LA - eng
KW - semilattice; lattice; distributive; dimension group; direct limit; semilattice; direct limit
UR - http://eudml.org/doc/249867
ER -

References

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  1. Bergman G.M., Von Neumann regular rings with tailor-made ideal lattices, unpublished notes, October 1986. 
  2. Effros E.G., Handelman D.E., Shen C.-L., Dimension groups and their affine representations, Amer. J. Math. 120 (1980), 385-407. (1980) Zbl0457.46047MR0564479
  3. Goodearl K.R., Von Neumann Regular Rings, Pitman, London, 1979, xvii + 369 pp. Zbl0841.16008MR0533669
  4. Goodearl K.R., Partially Ordered Abelian Groups with Interpolation, Math. Surveys and Monographs, Vol. 20, Amer. Math. Soc., Providence, R.I., 1986, xxii + 336 pp. Zbl0589.06008MR0845783
  5. Goodearl K.R., Handelman D.E., Tensor product of dimension groups and K 0 of unit-regular rings, Canad. J. Math. 38 3 (1986), 633-658. (1986) MR0845669
  6. Goodearl K.R., Wehrung F., Representations of distributive semilattice in ideal lattices of various algebraic structures, Algebra Universalis 45 (2001), 71-102. (2001) MR1809858
  7. Grätzer G., General Lattice Theory, second edition, Birkhäuser, Basel, 1998, xix + 663 pp. MR1670580
  8. Růžička P., A distributive semilattice not isomorphic to the maximal semilattice quotient of the positive cone of any dimension group, J. Algebra 268 (2003), 290-300. (2003) Zbl1025.06003MR2005289
  9. Schmidt E.T., Zur Charakterisierung der Kongruenzverbände der Verbände, Mat. Časopis Sloven. Akad. Vied 18 (1968), 3-20. (1968) MR0241335
  10. Wehrung F., A uniform refinement property for congruence lattices, Proc. Amer. Math. Soc. 127 (1999), 363-370. (1999) Zbl0902.06006MR1468207
  11. Wehrung F., Representation of algebraic distributive lattices with ℵ 1 compact elements as ideal lattices of regular rings, Publ. Mat. (Barcelona) 44 (2000), 419-435. (2000) Zbl0989.16010MR1800815
  12. Wehrung F., Semilattices of finitely generated ideals of exchange rings with finite stable rank, Trans. Amer. Math. Soc. 356 5 (2004), 1957-1970. (2004) Zbl1034.06007MR2031048
  13. Wehrung F., Forcing extensions of partial lattices, J. Algebra 262 1 (2003), 127-193. (2003) Zbl1030.03039MR1970805

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