Subdirectly Irreducible and Semisimple Double Boolean Algebras

Double Boolean algebras are algebras \(\underline{D}= (D, \sqcap , \sqcup , \lnot , \lrcorner , \bot , \top )\) 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.