Finitely Valued f-Modules, An Addendum

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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 Γ(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.

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