Double Boolean algebras: Constructions, sub-structures and morphisms

Double Boolean algebras are algebras $\underset{\_}{D}:=(D;\sqcap ,\bigsqcup ,\neg ,⌟,\perp ,\top )$ 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 $\underset{\_}{D}$ contains two Boolean algebras: ${\underset{\_}{D}}_{\sqcap }$ and ${\underset{\_}{D}}_{\bigsqcup }$. Three main goals are achieved in this paper. First we characterize sub-algebras of a double Boolean algebra $\underset{\_}{D}$ as join sets of sub-algebras of the Boolean algebras ${\underset{\_}{D}}_{\sqcap }$ and ${\underset{\_}{D}}_{\bigsqcup }$ and a subset of $D﹨D_p$ (where $D_p=D_{\sqcap }\cup D_{\bigsqcup }$) satisfying certain conditions. Second, we characterize homomorphisms between two double Boolean algebras $\underset{\_}{D}$ and $\underset{\_}{E}$ by homomorphisms between the Boolean algebras ${\underset{\_}{D}}_{\sqcap }$ and ${\underset{\_}{E}}_{\sqcap }$, ${\underset{\_}{D}}_{\bigsqcup }$ and ${\underset{\_}{E}}_{\bigsqcup }$ 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.