On properties of subset algebras

Any operation over any domain can be generalized on arbitrary subsets of the domain. So for any universe we can consider algebras of subsets with the same operations. It can be algebra of all subsets or some subsets, for example, finite subsets. We investigate subsets algebras for various original universes. We have established results on elementary equivalence, algorithmic decidability, definability, and other properties.
We pay special attention to semigroups. The free semigroup is the algebra of all words with concatenation. So the subsets algebras is the corresponding algebras of languages. Another examples are subsets of natural numbers or unity-coefficient polynomials over any idempotent semiring with unity.
Another universes we consider are unars with an injective function. Then the subsets algebras are of the same kind. We have established structure of its theory.