An R-module M is observable iff all its elements can be distinguished by observing them by means of linear morphisms from M to R. We show that free observable R-modules can be explicitly described as the cores of the final power domains with characteristic semiring R. Then, the general theory is applied to the cases of the lower and the upper semiring. All lower modules are observable, whereas there are non-observable upper modules. Accordingly, all known lower power constructions coincide, whereas there are at least three different upper power constructions. We show that they coincide for continuous ground domains, but differ on more general domains.
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Reinhold Heckmann /