Richman, Fred Butler groups, valuated vector spaces, and duality. (English) Zbl 0576.20033 Rend. Sem. Mat. Univ. Padova 72, 13-19 (1984). 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. Reviewer: Ladislav Bican (Praha) Cited in 8 Documents 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 Keywords:valuated vector spaces; finite distributive lattice; finite-dimensional vector space; duality; completely decomposable T-space; Butler groups; semi-balanced extensions; pure subgroup; rank one subgroups; quasi- homomorphisms × Cite Format Result Cite Review PDF Full Text: Numdam EuDML References: [1] Arnold D.M. , Pure subgroups of completely decomposable groups, Abelian group theory , Lecture Notes in Mathematics 874 , Springer-Verlag , Heidelberg ( 1981 ), pp. 1 - 31 . MR 645913 | Zbl 0466.20030 · Zbl 0466.20030 [2] Butler M.C.R. , A class of torsion-free abelian groups of finite rank , Proc. London Math. Soc. , 15 ( 1965 ), pp. 680 - 698 . MR 218446 | Zbl 0131.02501 · Zbl 0131.02501 · doi:10.1112/plms/s3-15.1.680 [3] Butler M.C.R. , Torsion-free modules and diagrams of vector spaces , Proc. London Math. Soc. , 18 ( 1968 ), pp. 635 - 652 . MR 230767 | Zbl 0179.32603 · Zbl 0179.32603 · doi:10.1112/plms/s3-18.4.635 [4] Crawley P. - R.P. Dilworth , Algebraic theory of lattices , Prentice-Hall ( 1973 ). Zbl 0494.06001 · Zbl 0494.06001 [5] Lady E.L. , Extensions of scalars for torsion free modules over Dedekind domains , Ist. Naz. Alta Mat. Symp. Math. , 23 ( 1979 ), pp. 287 - 305 . MR 565611 | Zbl 0425.13001 · Zbl 0425.13001 [6] Richman F. - E.A. Walker , Ext in pre-abelian categories , Pacific J. Math. , 71 ( 1977 ), pp. 521 - 535 . Article | MR 444742 | Zbl 0354.18018 · Zbl 0354.18018 · doi:10.2140/pjm.1977.71.521 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.