Inclusive (universal positive) theory of Abelian groups

The model theory of Abelian groups has been extensively studied in the literature, also in recent years.  An identical inclusion is a formula that can be expressed as a (possibly infinitary) disjunctive identity
(u = v_1) V (u = v_2) V (u = v_3) V …
or, equivalently, as a universally closed identical equality of subsets of words (terms).  For groups and rings, the classes defined by identical inclusions and by infinitary disjunctive identities coincide, but for semigroups they do not.  A class of algebras defined by a set of identical inclusions is called an inclusive variety.  An inclusive variety that cannot be defined by first-order formulas is called a non-elementary inclusive variety.  An inclusive variety defined by a system of identical inclusions — each depending on a finite set of variables — is called a quasi-elementary inclusive variety.

We describe elementary, non-elementary, and quasi-elementary inclusive varieties of Abelian groups.  There exist continuum-many inclusive varieties of each of these kinds.  We also determine Abelian groups defined by identical inclusions up to isomorphism and classify Abelian groups up to inclusive equivalence.