Automorphisms of Boolean algebras definable by fixed elements

Enriched Boolean algebras are studied. We give an answer to the question asking under which conditions, given a subalgebra of a Boolean algebra, we can uniquely reconstruct an automorphism for which the given subalgebra is a subalgebra of fixed elements. Also we furnish a complete description of subalgebras of Boolean algebras that are fixed subalgebras of automorphisms definable by fixed elements. It is proved that an automorphism of a Boolean algebra is defined by fixed elements iff it is an involution. Subalgebras of fixed elements of automorphisms of atomic and superatomic Boolean algebras are examined. It is shown that an automorphism of a distributive lattice is defined by fixed elements iff it is an involution, and that this is untrue of finite modular lattices.