Problems of algorithmic decidability and axiomatizability of finite subset algebra for binary operations

We consider algebras of finite subsets under the assumption that the original algebra is an infinite groupoid. For linear spaces over fields of finite characteristic, we prove that the finite subsets algebra is algorithmically equivalent to the first-order arithmetic. We also generalize this result to arbitrary infinite Abelian groups. As a corollary, for many classes of original algebras such as Abelian groups, arbitrary groups, monoids, semigroups, and general groupoids, if we consider the corresponding class of all algebras of finite subsets, it is shown that the theory of the last class has degree of undecidability not smaller than the first-order arithmetic. This also proves that the theories of such classes have no recursive axiomatization. For Abelian groups of finite exponent, we show that the theories of the subalgebra lattices are algorithmically undecidable and have no recursive axiomatization; also, for the class of such lattices for groups, monoids, or semigroups, we show that the theory of this class is also undecidable and has no recursive axiomatization.