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Title: Countable chains of distributive lattices as maximal semilattice quotients of positive cones of dimension groups (English)
Author: Růžička, Pavel
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 47
Issue: 1
Year: 2006
Pages: 11-20
Category: math
Summary: 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. (English)
Keyword: semilattice
Keyword: lattice
Keyword: distributive
Keyword: dimension group
Keyword: direct limit
MSC: 06A12
MSC: 06B15
MSC: 06D05
MSC: 06F20
MSC: 20K25
idZBL: Zbl 1138.06003
idMR: MR2223963
Date available: 2009-05-05T16:55:12Z
Last updated: 2012-04-30
Stable URL:
Reference: [1] Bergman G.M.: Von Neumann regular rings with tailor-made ideal lattices.unpublished notes, October 1986.
Reference: [2] Effros E.G., Handelman D.E., Shen C.-L.: Dimension groups and their affine representations.Amer. J. Math. 120 (1980), 385-407. Zbl 0457.46047, MR 0564479
Reference: [3] Goodearl K.R.: Von Neumann Regular Rings.Pitman, London, 1979, xvii + 369 pp. Zbl 0841.16008, MR 0533669
Reference: [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. Zbl 0589.06008, MR 0845783
Reference: [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. MR 0845669
Reference: [6] Goodearl K.R., Wehrung F.: Representations of distributive semilattice in ideal lattices of various algebraic structures.Algebra Universalis 45 (2001), 71-102. MR 1809858
Reference: [7] Grätzer G.: General Lattice Theory.second edition, Birkhäuser, Basel, 1998, xix + 663 pp. MR 1670580
Reference: [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. Zbl 1025.06003, MR 2005289
Reference: [9] Schmidt E.T.: Zur Charakterisierung der Kongruenzverbände der Verbände.Mat. Časopis Sloven. Akad. Vied 18 (1968), 3-20. MR 0241335
Reference: [10] Wehrung F.: A uniform refinement property for congruence lattices.Proc. Amer. Math. Soc. 127 (1999), 363-370. Zbl 0902.06006, MR 1468207
Reference: [11] 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. Zbl 0989.16010, MR 1800815
Reference: [12] Wehrung F.: Semilattices of finitely generated ideals of exchange rings with finite stable rank.Trans. Amer. Math. Soc. 356 5 (2004), 1957-1970. Zbl 1034.06007, MR 2031048
Reference: [13] Wehrung F.: Forcing extensions of partial lattices.J. Algebra 262 1 (2003), 127-193. Zbl 1030.03039, MR 1970805


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