The existence or nonexistence of invariants generally says something about the complexity of certain problems. It therefore seems to be an appropriate area for research from the viewpoint of reverse mathematics. Once this is done, one can investigate the complexity of such statements in the hierarchy of axiom systems of reverse mathematics. One easy example is vector spaces.
Boolean algebras supply more interesting results.
The isomorphism relation for arbitrary Boolean algebras is Sigma11 complete so no simply defined invariants for isomorphism can exist and the existence of invariants implies Pi11-CA . However, it is a classical result of Tarski that there are number invariants for elementary equivalence of Boolean algebras. These invariants are, of course, isomorphism invariants for the class of saturated Boolean algebras. These algebras can be described algebraically and are called the dense Boolean algebras.