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Countable chains of distributive lattices as maximal semilattice quotients of positive cones of dimension groups. (English) Zbl 1138.06003
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 [Trans. Am. Math. Soc. 356, No. 5, 1957–1970 (2004; Zbl 1034.06007)].
MSC:
06A12 Semilattices
06B15 Representation theory of lattices
06D05 Structure and representation theory of distributive lattices
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