Simple and sub-directly irreducible double Boolean algebras

Double Boolean algebras are algebras $\underset{‾}{D}=(D;\sqcap ,\bigsqcup ,\mathrm{\neg },,\mathrm{\perp },\mathrm{\top })$ 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 denoted by ${\underset{‾}{D}}_{\sqcap }$ and ${\underset{‾}{D}}_{\bigsqcup }$. A double Boolean algebra $\underset{‾}{D}$ is said pure if $D=D_{\sqcap }\cup D_{\bigsqcup }$, and trivial if $\mathrm{\perp }\bigsqcup \mathrm{\perp }=\mathrm{\top }\sqcap \mathrm{\top }$. In this work, we first show that a double Boolean algebra is pure and trivial if and only if it is a glued sum of two Boolean algebras; secondly, we characterize simple double Boolean algebras; and finally, we determine up to isomorphism all sub-directly irreducible algebras of some sub-classes of the variety of double Boolean algebras.