Galois theory for finite algebras of operations and multioperations of rank 2

The construction of Galois theory for the algebras of operations and relations is a popular topic for investigation. It finds numerous applications in both algebra and discrete mathematics – especially for the perfect Galois connection, since if such a connection is established for the sets of all subalgebras of some algebra then the algebraic closure in this algebra coincides with the Galois closure, and this is an efficient tool for solving many algebraic problems. The perfect Galois connection is well known for clones and co-clones, as well as for some algebras of operations, for example, for clones and superclones. In all these cases, infinite algebras are considered.
In this article we study Galois theory for finite algebras of operations and multioperations with fixed rank and arbitrary dimension. We find the necessary and sufficient conditions on the dimension of algebras so that the Galois connection between the lattices of subalgebras of algebras of operations and algebras of multioperations of rank 2 is perfect. The problem of finding the necessary and sufficient conditions for the existence of a perfect Galois connection between lattices of algebras of operations and algebras of multioperations of arbitrary fixed rank is posed.