Finitely Valued f-Modules, An Addendum

Abstract

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 $$\Gamma _{\Omega 1} (M)$$ and $$\Gamma _{\Omega _2 } (M)$$ . If R is a unital ℓ-ring, then each unital ℓ-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.