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Using lattice-ordered algebras it is shown that a totally ordered field which has a unique total order and is dense in its real closure has the property that each of its positive semidefinite rational functions is a sum of squares.

In an $\ell$-group $M$ with an appropriate operator set $\Omega$ it is shown that the $\Omega$-value set ${\Gamma }_{\Omega }\left(M\right)$ can be embedded in the value set $\Gamma \left(M\right)$. This embedding is an isomorphism if and only if each convex $\ell$-subgroup is an $\Omega$-subgroup. If $\Gamma \left(M\right)$ 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 ${\Gamma }_{{\Omega }_1}\left(M\right)$ and ${\Gamma }_{{\Omega }_2}\left(M\right)$. If $R$ is a unital $\ell$-ring, then each unital $\ell$-module over $R$ is an $f$-module...