Model theory of Abelian groups is 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 = v1 ∨ u = v2 ∨ u = v3 ∨ . . . ,
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 are coincide, for semigroups they do not coincide. A class of algebras defined
by a set of identical inclusions is called an inclusive variety. An inclusive variety that
can not be defined by first order formulas is called a nonelementary inclusive variety. An
inclusive variety defined by a system of identical inclusions - each depending on a finite set
of variables - is called a quasielementary inclusive variety.
We describe elementary, nonelementary and quasielementary 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.