This is a summary of some of the main results in the monograph Faithfully Ordered Rings (Mem. Amer. Math. Soc. 2015), presented by the first author at the ALANT conference, Będlewo, Poland, June 8-13, 2014. The notions involved and the results are stated in detail, the techniques employed briefly outlined, but proofs are omitted. We focus on those aspects of the cited monograph concerning (diagonal) quadratic forms over preordered rings.

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...