On subset algebras and subalgebra lattices

We consider algebras of subsets of other algebras. Such construction is very natural. For example, in such a way we can go from the algebra of words to the algebra of languages, from the algebra of vectors to the algebra of subspaces, and so on. We start from the algebra of words, then we consider to some wider classes of monoids, and at the end we research the classes of all abelian groups, all groups, all monoids, all semigroup, and all groupoids. Our main result is that all such algebras have the theory that admits an interpretation of the elementary arithmetic; hence, they are algorithmically undecidable and don't have a recursive axiomatization. Using this result, we can investigate lattices of corresponding subalgebras and prove analogous results for them.