Properties of $s\Sigma$-reducibility

We couch the definition of $s\Sigma$-reducibility on structures, describe some properties of $s\Sigma$-reducibility, and also exemplify explicitly how to use it. In particular, we consider natural expansions of structures such as Morleyization and Skolemization. Previously, a class of quasiregular structures was defined to be a class of fixed points of Morleyization with respect to $s\Sigma$-reducibility, extending the class of models of regular theories and the class of effectively model-complete structures. It was proved that an $\mathrm{HF}$-superstructure over a quasiregular structure is quasiresolvent and, consequently, has a universal $\Sigma$-function and possesses the reduction property. Here we show that an $\mathrm{HF}$-superstructure over a quasiregular structure has the $\Sigma$-uniformization property iff with respect to $s\Sigma$-reducibility this structure is a fixed point for some of its Skolemizations with an extra property, that of well-definededness. In this case an $\mathrm{HF}$-superstructure and a Moschovakis superstructure over a given structure are $s\Sigma$-equivalent.