Towards a Simplified Theory of Double Boolean Algebras: Axioms and Topological Representation

Double Boolean algebras (dBas), introduced by Wille, are based on twenty-three identities. We present a simplified axiom system, the D-core algebra, and prove it is equivalent to Wille's original definition. This reduction allows improved structural results, including a refined Boolean representation theorem showing fewer conditions suffice to represent a dBa as a pair of Boolean algebras linked by adjoint maps. We generalize the glued-sum construction to possibly overlapping Boolean algebras, characterize them via a generalized order, and establish a Stone-type topological representation: every dBa is quasi-isomorphic to a dBa of clopen subsets of a Stone space. Simplified logical systems for contextual and pure dBas are developed with soundness and completeness.