  In this paper we consider a general way of constructing profinite struc-
tures based on a given framework - a countable family of objects and a
countable family of recognisers (e.g. formulas). The main theorem states:
  A subset of a family of recognisable sets is a lattice if and only if it is
definable by a family of profinite equations.
  This result extends Theorem 5.2 from [GGEP08] expressed only for finite words
and morphisms to finite monoids. One of the applications of our theorem is the
situation where objects are finite relational structures and recognisers are
first order sentences. In that setting a simple characterisation of lattices of
first order formulas arise.
