Double Heyting Algebras: Structure & Applications

• Double Heyting algebras are distributive lattices equipped with both intuitionistic and dual implications, offering a unified framework for algebraic semantics.
• Their construction involves embedding Heyting algebras into richer structures, employing techniques like hyper–MacNeille completions and Boolean product representations.
• They support a range of logical systems and categorical models, bridging order theory, topology, and proof theory to extend intuitionistic logics.

A double Heyting algebra, also known in the literature as a bi-Heyting algebra, is a distributive lattice equipped with both the structure of a Heyting algebra (supporting intuitionistic implication, relative pseudocomplementation) and a co-Heyting algebra (supporting dual implication, relative dual pseudocomplementation), with each operation satisfying its respective residuation law. Double Heyting algebras serve as a canonical environment for the algebraic semantics of logics that incorporate both intuitionistic implication and its dual, and are the subject of intense paper, especially for their numerous connections to order theory, topology, categorical logic, and algebraic proof theory.

1. Algebraic Structure and Axiomatization

A double Heyting algebra $A$ is a distributive lattice $(A, \land, \lor, 0, 1)$ furnished with two residual operations:

• Heyting implication: $a \to b$ characterized by $c \leq (a \to b) \iff (c \land a) \leq b$
• Co-Heyting implication (sometimes called subtraction or dual implication): $a \leftarrow b$ (or $a + b$ in some texts) characterized by $c \geq (a \leftarrow b) \iff (c \lor b) \geq a$

The existence of two residuals induces pseudocomplementation ($a^* = a \to 0$) and dual pseudocomplementation ($a^+ = 0 \leftarrow a$) operations. Double Heyting algebras can equivalently be axiomatized via terms composed of these residuals, conjunction, disjunction, and bounds.

The fundamental algebraic concept is elucidated in Katriňák’s theorem (Cornejo et al., 2022), which shows that the double structure arises by constructing binary terms (notably $K(x, y)$ and $Kd(x, y)$) utilizing pseudocomplements and their duals. Conversely, in regular dually pseudocomplemented Heyting algebras, the implication operation $\to$ precisely recovers the Katriňák term, establishing a strong term-equivalence among key varieties:

• Regular double $p$-algebras (RDBLP)
• Regular double Heyting algebras (RDBLH)
• Regular dually pseudocomplemented Heyting algebras (RDPCH)
• Regular pseudocomplemented dual Heyting algebras (RPCH$^d$)

Notably, the operation $\to$ and its dual can be defined via: $$K(x, y) := (x^* \lor y^{**})^{**} \land [ (x \lor x^*)' \lor (x^* \lor (y \lor y^*)^*) ]$$

$$Kd(x, y) := (x^+ \land y^{++})^{++} \lor [ (x \lor x^+)^* \land (x^+ \lor (y \lor y^*)^+) ]$$

2. Embeddings, Extensions, and Completions

The embedding theory of Heyting algebras into richer algebras supports the construction of double Heyting algebras. Every Heyting algebra can be embedded into an “enrichable” algebra (Muravitsky, 2017) and then, via iterative expansion (using the Stone spectrum and filter constructions), extended so that both the original implication and a dual operation coexist robustly. The embedding preserves essential properties, such as countability and subdirect irreducibility, and the variety generated by the original algebra is unchanged.

For completion procedures, every Heyting algebra admits a centrally supplemented extension in the same variety (Harding et al., 2019). A centrally supplemented Heyting algebra ensures every element $a$ has a dual pseudocomplement $a^+$ that is central ($a \lor a^+ = 1$ and $a^+ \in Z(A)$, the Boolean center). The hyper–MacNeille completion is then obtained by completing this centrally supplemented extension. These completions are critical for constructing double Heyting algebras that are both complete and retain central supplement properties, allowing for Galois connections based on polarity relations of the form $s \lor t \lor (a \to b) = 1$.

Boolean product representations are significant, as the center $Z(S(A))$ in these centrally supplemented settings becomes a complete Boolean algebra, and the algebra is represented as a Boolean product over a Stone space of minimal prime filters.

3. Topological and Categorical Representations

The natural setting for double Heyting algebraic logic is found in the internal logic of Grothendieck toposes (Caramello, 2012). The subobject classifier $\Omega$ in a topos is an internal Heyting algebra; by introducing intermediate logic axioms via appropriate local operators (e.g., the double-negation or De Morgan topology), one can “tune” the topos so its internal logic closely mirrors a double Heyting algebra structure. The essential operation involves stabilizing both the Heyting implication and pseudocomplementation under sheafification, ensuring their preservation by commutative diagrams of the form: $$\begin{array}{ccc} \Omega & \xrightarrow{\neg} & \Omega \ \downarrow^{e_j} && \downarrow^{e_j} \ \Omega_j & \xrightarrow{\neg_j} & \Omega_j \end{array}$$ This guarantees that the passage to sheaves does not disturb key logical operations, leading to subtoposes whose internal logics admit both implication and its dual.

Duality theory for double Heyting algebras is naturally extended through $\nabla$-spaces (Tabatabai et al., 16 Sep 2024), which generalize Priestley and Esakia spaces by equipping a distributive lattice with a binary relation $R$ dualizing the extra modal (or dual) operator. The corresponding dual equivalence relates algebraic objects to geometric/topological structures, where co-Heyting implications are modeled via upward- and downward-closed sets, or by modalities induced by dynamic transitions.

4. Logical Systems and Varieties

Double Heyting algebras furnish semantics for a multitude of systems:

• BP-algebraizable systems such as RDPCH, RPCH$^d$, and RDMH (Cornejo et al., 2022), which correspond via algebraic semantics to varieties of double Heyting and related algebras.
• $\nabla$-algebra logics (Tabatabai et al., 18 May 2024, Tabatabai et al., 16 Sep 2024), using sequent calculi with extended modalities and rules capturing normality, faithfulness, fullness, and interaction between $\to$ and its dual, with semantics spanning algebraic, Kripke, topological, and ring-theoretic perspectives.

These varieties exhibit extremely rich structure: subvariety lattices and extension lattices have cardinality $2^{\aleph_0}$ (Cornejo et al., 2022), reflecting the profusion of intermediate logics and algebraic behaviors supported by the double structure.

5. Applications and Interconnections

Double Heyting algebras appear naturally in the semantics of intermediate logics, dynamic and temporal logic, duality theory, proof theory (especially in understanding assertoric equipollence, interpolation, and amalgamation properties), and topological models (including dynamic topological systems and bitopological dualities (Das et al., 2023)).

The underlying algebraic and categorical mechanisms permit:

• Canonical construction and Kripke representation, connecting prime filters and relations to operational semantics,
• Deductive interpolation, which guarantees modularity and amalgamation for proof-theoretic investigations,
• Closure under Dedekind-MacNeille and hyper–MacNeille completions (Tabatabai et al., 18 May 2024, Harding et al., 2019), ensuring robustness under extension.

6. Term-Equivalence and De Morgan Variants

The landscape of double Heyting algebra varieties features extensive term-equivalence between regular double $p$-algebras, regular double Heyting, and regular De Morgan variants (Cornejo et al., 2022). This equivalence implies isomorphic lattices of subvarieties, congruence structures, and algebraizations of logical systems. Algebraic identities such as the regularity condition $$(x \wedge x^+) \lor (y \lor y^*) \approx y \lor y^*$$ play central roles in distinguishing elements via pseudocomplements and their duals.

7. Methodological Significance and Future Directions

The paper of double Heyting algebras demonstrates that extending intuitionistic frameworks to accommodate dual implications is both conservative (does not alter the equational theory or assertoric content) and generative (produces algebraic semantics with enhanced expressivity and representation). The interplay of categorical logic, topology, algebraic embedding, and completion theories underpins the systematic generalization of logic from classical and intuitionistic to richer non-classical settings. The incorporation of modalities (as in $\nabla$-algebras), dynamic and temporal operators, and coalgebraic dualities suggests ongoing avenues for analysis and generalization.

In conclusion, double Heyting algebras are a fundamental and unifying algebraic structure, characterized by the interaction of intuitionistic and dual implication operations, with deep connections across algebra, topology, logic, and category theory. Their term-equivalent varieties, categorical representations, and logical semantics provide the backbone for diverse domains, including intermediate logics, dynamic systems, algebraic proof theory, and duality-driven algebraic logic.