Simple and sub-directly irreducible double Boolean algebras

Abstract

Double Boolean algebras are algebras $\underline{D}=(D;\sqcap ,\bigsqcup ,\neg ,┘,\perp ,\top )$ 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 $\underline{D}$ contains two Boolean algebras denoted by ${\underline{D}}_{\sqcap }$ and ${\underline{D}}_{\bigsqcup }$. A double Boolean algebra $\underline{D}$ is said pure if $D=D_{\sqcap }\cup D_{\bigsqcup }$, and trivial if $\perp \bigsqcup \perp =\top \sqcap \top$. 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.

18 pages, 3 figures, 19 tables