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Liftings of diagrams of semilattices by diagrams of dimension groups. (English) Zbl 1040.06002

The authors study problems of lifting diagrams of \(\langle\vee,0\rangle\)-semilattices and homomorphisms to corresponding diagrams of dimension groups (i.e., unperforated, directed abelian groups satisfying Riesz interpolation) and positive homomorphisms, related to questions of representing \(\langle\vee,0\rangle\)-semilattices as compact ideal semilattices of dimension groups and other structures. Here a diagram indexed by a partially ordered set \(P\) means a functor from \(P\) to some category, where \(P\) is made into a category in the canonical fashion (with a unique morphism \(x\rightarrow y\) corresponding to each relation \(x\leq y\) in \(P\)). The authors extend the concept of dismantlability from finite lattices to finite posets and prove that any diagram \(\Phi\) of finite Boolean \(\langle\vee,0\rangle\)-semilattices indexed by a finite dismantlable poset \(P\) can be lifted to a diagram \(\Psi\) of dimension groups indexed by \(P\), that is, up to a natural equivalence, \(\Phi\) equals the composition of \(\Psi\) with the functor sending any dimension group to its semilattice of compact (i.e., finitely generated) ideals. In fact, \(\Psi\) takes values in the category of dimension vector spaces, i.e., directed partially ordered \(\mathbb Q\)-vector spaces with interpolation.
The second main theorem of the paper states that any \(\langle\vee,0\rangle\)-homomorphism from a countable distributive \(\langle\vee,0\rangle\)-semilattice \(S\) to the compact ideal semilattice of a countable dimension vector space \(H\) can be lifted to a positive homomorphism from some countable dimension vector space \(G\) to \(H\). Via ordered K-theory, the latter theorem is extended to an analogous result with dimension vector spaces replaced by ultramatricial algebras over a field, and from that setting to congruence lattices of locally finite, sectionally complemented modular lattices. (If \(R\) is an ultramatricial algebra, then \(K_0(R)\) is a dimension group, \(R\) is a von Neumann regular ring, and the principal right ideals of \(R\) form a locally finite, sectionally complemented modular lattice.)
The paper concludes with a discussion of various counterexamples to alternative strategies and of several open problems. In the meantime, the second author has solved the first of these open problems [Trans. Am. Math. Soc. 356, 1957–1970 (2004; Zbl 1034.06007)] – there exists a \(\langle\vee,0\rangle\)-semilattice of cardinality \(\aleph_1\) which is not isomorphic to the compact ideal semilattice of any dimension group. This improves on the counterexample of cardinality \(\aleph_2\) given by P. Ruzicka [J. Algebra 268, No. 1, 290–300 (2003; Zbl 1025.06003)], and is of minimal cardinality since there is a positive answer in the countable case, as follows from unpublished work of G. M. Bergman [cf. the discussion by the second author and the reviewer in Algebra Univers. 45, 71–102 (2001; Zbl 1039.06003)].

MSC:

06A12 Semilattices
06C20 Complemented modular lattices, continuous geometries
16E20 Grothendieck groups, \(K\)-theory, etc.
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
19A49 \(K_0\) of other rings
19K14 \(K_0\) as an ordered group, traces
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
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