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...

In this paper we have given the construction of free $\ell$-groups generated by a po-group and the construction of free products in any sub-product class $𝒰$ of $i\ell$-groups. We have proved that the $𝒰$-free products satisfy the weak subalgebra property.