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.