Double Boolean Algebras (dBas)

• Double Boolean algebras (dBas) are algebraic structures that combine two mutually interlocking Boolean systems with distinct operations and negations.
• They feature simplified axioms and representation theorems that unify logical calculi with topological and duality frameworks in formal concept analysis.
• Applications span modeling complex logical systems, protoconcept algebras, and bitopological dualities, providing insights into both pure and contextual algebraic structures.

A double Boolean algebra (dBa) is an algebraic structure $$\underline{D} = (D; \sqcap, \sqcup, \neg, \lrcorner, \bot, \top)$$ of type $(2,2,1,1,0,0)$, originally introduced by Rudolf Wille to model the equational theory of protoconcept algebras arising in Formal Concept Analysis. Double Boolean algebras encapsulate the interaction of two mutually interlocking Boolean-algebra-like structures—each with its own meet, join, and negation—on the same domain. Recent work shows their full algebraic, logical, and topological richness: they admit simplified axiomatics, concrete representation theorems, generalized constructions, and dualities connecting pure, contextual, and topological variants with categories of spaces and logical calculi (Kembang et al., 2023, Howlader et al., 4 Jan 2026, Howlader et al., 2021, Yang et al., 23 May 2025, Howlader et al., 2022, Howlader et al., 2022, Ledda, 2018, Clouse, 2018).

1. Algebraic Definition and Equational Theory

A double Boolean algebra $\underline{D}$ consists of a non-empty set $D$ with two binary operations (denoted $\sqcap$ and $\sqcup$), two unary operations ($\neg$ and $\lrcorner$), and two constants ($\bot$, $\top$), satisfying the paired axioms:

• Associativity, commutativity, and idempotence for both $\sqcap$ and $\sqcup$.
• Dual De Morgan laws:

$$\neg(x\sqcap y) = \neg x \sqcap \neg y,\quad \lrcorner(x\sqcup y) = \lrcorner x \sqcup \lrcorner y$$

• Absorption and distributivity:

$$x\sqcap (x \sqcup y) = x\sqcap x,\quad x\sqcup (x \sqcap y) = x\sqcup x$$

$$x\sqcap(y \vee z) = (x\sqcap y)\vee(x\sqcap z),\quad x\sqcup(y \wedge z) = (x\sqcup y)\wedge(x\sqcup z)$$

where $\vee$, $\wedge$ are secondary operations as defined by:

$$x \vee y := \neg(\neg x \sqcap \neg y),\quad x \wedge y := \lrcorner(\lrcorner x \sqcup \lrcorner y)$$

• Boolean negations:

$$\neg(x\sqcap x) = x,\quad \lrcorner(x\sqcup x) = x,\quad x\sqcup\neg x = \top,\quad x\sqcap\lrcorner x = \bot$$

• Boundary conditions:

$\neg\bot = \top,\quad \lrcorner\top = \bot$

The classic axiom system comprises up to 23 identities (Howlader et al., 4 Jan 2026), but recent refinement demonstrates that a minimal D-core of only seven schema suffices to recover full structure, streamlining both algebraic verification and representation results.

2. Boolean Reducts and Subalgebra Structure

Each dBa admits two canonically defined Boolean subalgebras:

Subalgebra	Universe	Operations	Boolean Constants
$D_{\sqcap}$	$\{x \in D \mid x\sqcap x = x\}$	$(\sqcap, \vee, \neg)$	$(\bot, \neg \bot)$
$D_{\sqcup}$	$\{x \in D \mid x\sqcup x = x\}$	$(\wedge, \sqcup, \lrcorner)$	$(\lrcorner \top, \top)$

The partial orders defined via $\sqcap$ and $\sqcup$ coincide on common elements, and the equational theory of any dBa syntactically encodes, via idempotency and dual absorption, a pair of intertwined Boolean algebras (Kembang et al., 2023, Howlader et al., 4 Jan 2026). The full algebra $D$ may not be a direct product—rather, interaction laws and shared domains induce a richer structure class.

3. Purity, Triviality, and Glued-Sum Representation

A double Boolean algebra is pure if $D = D_{\sqcap} \cup D_{\sqcup}$ (every element is idempotent for one of the two operations), and trivial if $\bot\sqcup\bot = \top$ (equivalently, $\top\sqcap\top = \bot$). The main representation theorem:

• A dBa is pure and trivial iff it is a “glued sum” of two Boolean algebras: explicitly, forming a pair $P$, $Q$ (possibly overlapping at a single point) and identifying $\top_P = \bot_Q$ (Kembang et al., 2023, Howlader et al., 4 Jan 2026).
• The operations are piecewise defined—$\sqcap$ and $\sqcup$ act as usual within each subalgebra, and “jump” outside, preserving the glue point.

This theorem generalizes to overlapping subalgebras via the generalized linear sum construction $P\oplus_g Q$, under weakened intersection conditions, as characterized by abstract retraction-embedding pairs (Howlader et al., 4 Jan 2026).

4. Simple, Subdirectly Irreducible, and Atomic Structures

A dBa is simple if its congruence lattice has only the trivial and universal congruence. Classification results (Kembang et al., 2023):

• Non-pure simple dBas exist only in size 2.
• Pure simple dBas correspond to cases where one Boolean reduct is “thin”: e.g., $|D_{\sqcap}|=1$ and $|D_{\sqcup}|\leq 2$, or vice versa.
• Explicit families: two-element chains (Type I/II), three-element glued sums, certain four-element diamonds, and atomic representatives.

Subdirectly irreducible dBas are classified into five types (I-V) depending on the configuration and interaction of their Boolean reducts. With the exception of Type III (the pure nontrivial case), all nontrivial subdirectly irreducible dBas are finite and simple; every double Boolean algebra is a subdirect product of these finite "atomic" components (Kembang et al., 2023).

5. Stone-Type Topological Representation and Duality

Every double Boolean algebra admits a canonical topological representation as an algebra of clopen “rectangles” in a zero-dimensional compact Hausdorff space:

• The Stone context $(G, \tau_G), (M, \tau_M), R$ associates primary filters and ideals with Stone topologies; the relation $V$ defined by $F \cap I = \emptyset$ is continuous.
• For contextual and pure dBas, the quasi-embedding into $\mathcal{R}_T(K_T(D))$ (protoconcept algebra) is either injective or bijective; for pure dBas, an isomorphism is obtained with the semiconcept algebra of the Stone context (Howlader et al., 2021, Howlader et al., 4 Jan 2026).
• Categorical duality theorems demonstrate equivalence between the categories of pure and fully contextual dBas with Stone contexts, and between contextual dBas and their protoconcept varieties. For example:

$$\textbf{FCDBA} \simeq \textbf{PDBA} \simeq \textbf{Scxt}^{op}$$

These results unify and extend classical Stone duality to the double Boolean setting.

6. Logical Calculi: Sequent and Hypersequent Systems

The logic of double Boolean algebras is captured by sequent calculi in the contextual (L, CDBL) and pure (HL, PDBL) cases:

• HL and L are sound and complete for pure and contextual D-core algebras, respectively (Howlader et al., 4 Jan 2026, Howlader et al., 2022, Howlader et al., 2022).
• Hypersequent calculi (pipes of sequents) allow fine-grained deduction rules reflecting the dual Boolean structure and cut-free proofs.
• Modal and topological extensions (MCDBL, MPDBL) incorporate approximation operators (arising from Kripke contexts or rough set theory) with additional contraction and symmetry rules.
• Protoconcept-based semantics link formulae to concept pairs in formal contexts, ensuring full representational completeness.

7. Generalizations and Bitopological Dualities

The theory extends to d-Boolean algebras (as in bitopological Stone spaces and their duality categories), which generalize dBas by considering two distributive lattices and an order-reversing isomorphism (Yang et al., 23 May 2025). This yields:

• Coreflective and adjoint functors between the algebraic, frame, and topological categories.
• Characterizations of d-Boolean algebras via spectrum and bitopological space, paralleling classical duality.
• Synthesis with other doubly-structured algebras (e.g., Core Regular Double Stone Algebras, distributive bisemilattices), which exhibit "nearly Boolean" properties, admit embedding into standard products, and have further categorical dualities (Clouse, 2018, Ledda, 2018).

• "Simple and sub-directly irreducible double Boolean algebras" (Kembang et al., 2023)
• "Towards a Simplified Theory of Double Boolean Algebras: Axioms and Topological Representation" (Howlader et al., 4 Jan 2026)
• "Topological Representation of Double Boolean Algebras" (Howlader et al., 2021)
• "d-Boolean algebras and their bitopological representation" (Yang et al., 23 May 2025)
• "Kripke Contexts, Double Boolean Algebras with Operators and Corresponding Modal Systems" (Howlader et al., 2022)
• "A non-distributive logic for semiconcepts of a context and its modal extension with semantics based on Kripke contexts" (Howlader et al., 2022)
• "Stone-type representations and dualities for varieties of bisemilattices" (Ledda, 2018)
• "The Nearly Boolean Nature of Core Regular Double Sone Algebras, CRDSA..." (Clouse, 2018)