Finitely valued f-modules, an addendum

• Autores: Stuart A. Steinberg
• Localización: Czechoslovak mathematical journal, ISSN 0011-4642, Vol. 51, Nº 2, 2001, págs. 387-394
• Idioma: inglés
• DOI: 10.1023/a:1013707316630
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• Resumen

  • In an ℓ-group M with an appropriate operator set Ω it is shown that the Ω-value set ΓΩ(M) can be embedded in the value set Γ(M). This embedding is an isomorphism if and only if each convex ℓ-subgroup is an Ω-subgroup. If Ω(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 Ω1 and Ω2 and the corresponding Ω-value sets ΓΩ1(M) and ΓΩ2(M). If R is a unital ℓ-ring, then each unital ℓ-module over R is an f-module and has Γ(M)=ΓR(M) exactly when R is an f-ring in which 1 is a strong order unit.