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A distributive semilattice not isomorphic to the maximal semilattice quotient of the positive cone of any dimension group. (English) Zbl 1025.06003
Summary: We construct a distributive 0-semilattice which is not isomorphic to the maximal semilattice quotient of the positive cone of any dimension group. The size of the semilattice is $${\aleph}_2$$.

MSC:
 06A12 Semilattices
Full Text:
References:
 [1] G.M. Bergman, Von Neumann regular rings with tailor-made ideal lattices, Unpublished notes, 1986 [2] Effros, E.G.; Handelman, D.E.; Shen, C.-L., Dimension groups and their affine representations, Amer. J. math., 120, 385-407, (1980) · Zbl 0457.46047 [3] Goodearl, K.R., Partially ordered abelian groups with interpolation, Math. surveys monogr., 20, (1986), Amer. Math. Soc Providence, RI · Zbl 0589.06008 [4] Goodearl, K.R.; Handelman, D.E., Tensor product of dimension groups and K0 of unitregular rings, Canad. J. math., 38, 633-658, (1986) · Zbl 0608.16027 [5] Goodearl, K.R.; Wehrung, F., Representations of distributive semilattice in ideal lattices of various algebraic structures, Algebra universalis, 45, 71-102, (2001) · Zbl 1039.06003 [6] Grätzer, K.R., General lattice theory, (1998), Birkhäuser Basel [7] P. Růžička, Representation of distributive lattices by semilattices of finitely generated ideals of locally matricial algebras, Preprint, 1999 [8] J. Tůma, F. Wehrung, Liftings of diagrams of semilattices by diagrams of dimension groups, Proc. London Math. Soc., in press · Zbl 1040.06002 [9] Wehrung, F., Non-measurability properties of interpolation vector spaces, Israel J. math., 103, 177-206, (1998) · Zbl 0916.06018 [10] Wehrung, F., A uniform refinement property for congruence lattices, Proc. amer. math. soc., 127, 363-370, (1999) · Zbl 0902.06006 [11] Wehrung, F., Representation of algebraic distributive lattices with $$ℵ1$$ compact elements as ideal lattices of regular rings, Publ. mat., 44, 419-435, (2000) · Zbl 0989.16010
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