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Butler groups, valuated vector spaces, and duality. (English) Zbl 0576.20033

If \(T\) is a finite distributive lattice then a \(T\)-space is a finite-dimensional vector space (over a field \(F\)) valuated by \(T\). An anti-isomorphism between lattices induces a duality between categories of corresponding spaces (Th. 1.1). Every \(T\)-space is a quotient space and a subspace of a completely decomposable \(T\)-space (Th. 1.4). The second part deals with Butler groups. Among other results it is shown that the class of Butler groups is closed under semi-balanced extensions (Th. 2.3 – \(A\to B\to C\) is semi-balanced if every rank one pure subgroup of \(C\) is the image of a sum of finitely many rank one subgroups of \(B\)), and that the category of \(T\)-spaces over \(\mathbb Q\) is isomorphic to the category of Butler \(T\)-groups under quasi-homomorphisms.

MSC:

20K15 Torsion-free groups, finite rank
20K25 Direct sums, direct products, etc. for abelian groups
20K27 Subgroups of abelian groups
20K35 Extensions of abelian groups

References:

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