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Uniformly hyperarchimedean lattice-ordered groups. (English) Zbl 1128.06007

Authors’ summary: An abelian \(\ell\)-group with strong unit \((\mathcal{L}_1\)-object) \(G\) is hyperarchimedean \((HA)\) iff \(G \leq C (YG)\) (the \(\ell\)-group of real continuous functions on the maximal ideal space, \(YG\)) with \(\lambda (g)=\inf\{g (x)\not= 0\}>0\) for each \(0\not= g \in G\). In case \(\inf\{\lambda (g):0\not= g \in G\}>0\), we call \(G\) uniformly hyperarchimedean (UHA). This paper examines the structure of the UHA groups in detail, shows that UHA solves the problem when an \(\mathcal{L}_1\)-product is HA, and describes completely the \(\mathcal{L}_1\)-HSP -classes which are contained in HA. Final remarks detail the connection with MV-algebras.

MSC:

06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
06D35 MV-algebras
08B15 Lattices of varieties
54C40 Algebraic properties of function spaces in general topology
03B50 Many-valued logic
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