On decidability and axiomatizability of finite subset algebra for binary operations

We consider algebras of finite subsets for infinite groupoids. It is proved that for linear spaces over fields of finite characteristic the theory of constructed algebras are algorithmically equivalent to elementary arithmetic. Further, this result is generalized to arbitrary infinite Abelian groups. For classes of Abelian groups, arbitrary groups, monoids, semigroups, groupoids we prove that the theory of corresponding classes of finite subsets algebras admits interpretation of elementary arithmetic. This also proves the impossibility of recursive axiomatization of such theories. Then we consider lattice of subalgebras for Abelian groups of finite exponent and classes of such lattices for groups, monoids and semigroups, we prove that theories are undecidable and don't have recursive axiomatization.