Equational theories of profinite structures

 I will present a simple way of defining profinite structures in a very general framework. The applications include such objects as words, trees and generic finite structures. The procedure gives interesting results in the context of FO logic and relational models.

The paper ,,Duality and equational theory of regular languages'' by M. Gehrke, S. Grigorie and J. Pin presents the following theorem: A family of regular languages is a lattice if and only if it is definable by a set of equations on profinite words.

I will provide a simple topological proof of the theorem in a more general setting, including all frameworks mentioned above.