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Finitely valued \(f\)-modules, an addendum. (English) Zbl 0979.06010
Summary: In an \(\ell\)-group \(M\) with an appropriate operator set \(\Omega\) it is shown that the \(\Omega\)-value set \(\Gamma_{\Omega}(M)\) can be embedded in the value set \(\Gamma(M)\). This embedding is an isomorphism if and only if each convex \(\ell\)-subgroup is an \(\Omega\)-subgroup. If \(\Gamma(M)\) has a.c.c. and \(M\) is either representable or finitely valued, then the two value sets are identical. More generally, these results hold for two related operator sets \(\Omega_1\) and \(\Omega_2\) and the corresponding \(\Omega\)-value sets \(\Gamma_{\Omega_1}(M)\) and \(\Gamma_{\Omega_2}(M)\). If \(R\) is a unital \(\ell \)-ring, then each unital \(\ell\)-module over \(R\) is an \(f\)-module and has \(\Gamma(M) = \Gamma_R(M)\) exactly when \(R\) is an \(f\)-ring in which \(1\) is a strong order unit.
The title of this article refers to the author’s paper in Pac. J. Math. 40, 723-737 (1972; Zbl 0218.16008)].
MSC:
06F15 Ordered groups
06F25 Ordered rings, algebras, modules
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References:
[1] M. Anderson and T. Feil: Lattice-Ordered Groups. D. Reidel, Dordrecht, 1988. · Zbl 0636.06008
[2] A. Bigard and K. Keimel: Sur les endomorphismes conservant les polaires d’ un groupe réticulé archimédien. Bull. Soc. Math. France 97 (1970), 81-96. · Zbl 0215.34203
[3] A. Bigard, K. Keimel, S. Wolfenstein: Groupes Et Anneaux Réticulés. Springer-Verlag, Berlin, 1977.
[4] G. Birkhoff and R. S. Pierce: Lattice-ordered rings. Am. Acad. Brasil. Ci. 28 (1956), 41-69. · Zbl 0070.26602
[5] P. Conrad: The lattice of all convex \(\ell \)-subgroups of a lattice-ordered group. Czechoslovak Math. J. 15 (1965), 101-123. · Zbl 0135.06301
[6] P. Conrad: Lattice-Ordered Groups. Tulane Lecture Notes, New Orleans, 1970. · Zbl 0258.06011
[7] P. Conrad and J. Diem: The ring of polar preserving endomorphisms of an abelian lattice-ordered group. Illinois J. Math. 15 (1971), 222-240. · Zbl 0213.04002
[8] P. Conrad, J. Harvey and C. Holland: The Hahn embedding theorem for lattice-ordered groups. Trans. Amer. Math. Soc. 108 (1963), 143-169. · Zbl 0126.05002
[9] P. Conrad and P. McCarthy: The structure of \(f\)-algebras. Math. Nachr. 58 (1973), 169-191. · Zbl 0276.06016
[10] S. A. Steinberg: Finitely-valued \(f\)-modules. Pacific J. Math. 40 (1972), 723-737. · Zbl 0218.16008
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