Subdirectly Irreducible and Semisimple Double Boolean Algebras

Double Boolean algebras are algebras of type (2, 2, 1, 1, 0, 0) introduced by Rudolf Wille to capture the equational theory of protoconcept algebras. A famous theorem of Birkhoff says that any variety is determined by its subdirectly irreducible members. In this work we give a construction that leads to a concrete embedding of double Boolean algebras into the protoconcept algebra. We characterize subdirectly irreducible, simple and semisimple double Boolean algebras.