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

Commentationes Mathematicae Universitatis Carolinae (2006)

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

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topRůžička, 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

top- Bergman G.M., Von Neumann regular rings with tailor-made ideal lattices, unpublished notes, October 1986.
- 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
- Goodearl K.R., Von Neumann Regular Rings, Pitman, London, 1979, xvii + 369 pp. Zbl0841.16008MR0533669
- 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
- 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
- Goodearl K.R., Wehrung F., Representations of distributive semilattice in ideal lattices of various algebraic structures, Algebra Universalis 45 (2001), 71-102. (2001) MR1809858
- Grätzer G., General Lattice Theory, second edition, Birkhäuser, Basel, 1998, xix + 663 pp. MR1670580
- 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
- Schmidt E.T., Zur Charakterisierung der Kongruenzverbände der Verbände, Mat. Časopis Sloven. Akad. Vied 18 (1968), 3-20. (1968) MR0241335
- Wehrung F., A uniform refinement property for congruence lattices, Proc. Amer. Math. Soc. 127 (1999), 363-370. (1999) Zbl0902.06006MR1468207
- Wehrung F., Representation of algebraic distributive lattices with ${\aleph}_{1}$ compact elements as ideal lattices of regular rings, Publ. Mat. (Barcelona) 44 (2000), 419-435. (2000) Zbl0989.16010MR1800815
- 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
- Wehrung F., Forcing extensions of partial lattices, J. Algebra 262 1 (2003), 127-193. (2003) Zbl1030.03039MR1970805

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