Categorical Horn theories and modules

We study connection between categorical Horn theories and modules. We show that each function enrichment of any abelian group to a primitive normal structure is primitively equivalent to some module. We give a description for the categorical Horn classes of modules. We propose some sufficient conditions for a categorical Horn theory to be primitively equivalent to a theory of modules. In particular, such are the categorical Horn theories of enrichments of abelian groups with the conditions of primitive rank $\le3$ and the absence of predicate symbols of arity $\ge3$ in the language.