Double Boolean algebras: Constructions, sub-structures and morphisms

Double Boolean algebras are algebras D _ : = ( D ; ⊓ , ⊔ , ¬ , ⌟ , ⊥ , ⊤ ) of type ( 2 , 2 , 1 , 1 , 0 , 0 ) introduced by Rudolf Wille to capture the equational theory of the algebra of protoconcepts. Every double Boolean algebra D _ contains two Boolean algebras: D _ ⊓ and D _ ⊔ . Three main goals are achieved in this paper. First we characterize sub-algebras of a double Boolean algebra D _ as join sets of sub-algebras of the Boolean algebras D _ ⊓ and D _ ⊔ and a subset of D ﹨ D p (where D p = D ⊓ ∪ D ⊔ ) satisfying certain conditions. Second, we characterize homomorphisms between two double Boolean algebras D _ and E _ by homomorphisms between the Boolean algebras D _ ⊓ and E _ ⊓ , D _ ⊔ and E _ ⊔ and maps between D ﹨ D p and E satisfying certain conditions. Third, we give some tools to construct some classes of pure double Boolean algebras. • Characterize sub-algebras of double Boolean algebras as well as morphims between them. • Describe the lattice of sub-algebras of trivial double Boolean algebras. • Provide a method to construct pure double Boolean algebras.