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

TL;DR

This paper introduces a simplified axiom system for double Boolean algebras, improving their structural understanding and providing a topological representation, thus advancing the theoretical foundation of these algebraic structures.

Contribution

It presents a new, simplified axiom system for double Boolean algebras, along with refined representation theorems and topological characterizations, enhancing their theoretical framework.

Findings

Simplified axiom system for double Boolean algebras

Refined Boolean representation theorem with fewer conditions

Stone-type topological representation of dBas

Abstract

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.