Jónsson–Tarski Representation in Algebraic Logic

• Jónsson–Tarski representation is a framework that links modal algebras with relational and combinatorial structures, underpinning canonical model constructions and completeness theorems.
• It extends classical representation by incorporating neighborhood frames and Q-filters to manage non-normal modalities and infinitary operations, ensuring versatile applications across modal logics.
• The framework establishes an isomorphism with endomorphism monoids of free Jónsson–Tarski algebras, providing finite presentability and deep insights into algebraic logic and transformation monoids.

The Jónsson–Tarski representation comprises a collection of algebraic and duality principles connecting modal and similar algebraic structures to relational or combinatorial objects, enabling canonical model constructions, structural analyses, and finite presentability results. Its multiple incarnations include the classical representation of modal algebras (both normal and non-normal), the realization of certain transformation monoids as endomorphism monoids of free Jónsson–Tarski algebras, and extensions that uniformly treat infinitary modal operations via Q-filters and neighborhood frames. The framework underlies deep completeness and presentation results for modal logics and their algebraic counterparts, structuring both foundational and computational approaches in algebraic logic and group theory (Witt et al., 2023, Tanaka, 2021).

1. The Classical Jónsson–Tarski Representation in Modal Algebra

For a normal modal algebra $A = \langle A;\vee,\wedge,¬,\Box,0,1 \rangle$, where $(A;\vee,\wedge,¬,0,1)$ is a Boolean algebra and $\Box$ is a unary modality satisfying $\Box1 = 1$ and $\Box(x \wedge y) = \Box x \wedge \Box y$, the Jónsson–Tarski theorem asserts that $A$ admits a canonical embedding into a complex algebra derived from a Kripke frame constructed via its set of prime filters. Specifically, the associated Kripke frame $\mathcal{F} = (\Pr(A), R)$, with $F\ R\ G$ iff $\forall a$ $(\Box a \in F \Longrightarrow a \in G)$ and the embedding $e(a) = \{ F \in \Pr(A) \mid a \in F \}$, is an injective modal algebra homomorphism. This result enables the canonical model construction principle and completeness theorems for a broad class of modal logics (Tanaka, 2021).

2. Neighborhood Frames and the Extension to Non-Normal Modalities

To generalize beyond normal modal algebras and accommodate infinitary meets and joins, the framework transitions from Kripke frames to neighborhood frames $(C, \nu)$, where $\nu: C \to \mathcal{P}(\mathcal{P}(C))$ assigns neighborhoods to points with monotonicity, topping, and cufi (closure under finite intersections) capturing variations in logical strength. Dual algebras $K(Z)$ for a neighborhood frame $Z$ host the generalized relational semantics. Central to the infinitary extension is the Q-filter: for a Boolean algebra $A$ and countable $S \subseteq \mathcal{P}(A)$, a Q-filter $F$ ensures that whenever $X \in S$ with $X \subseteq F$ and $\bigwedge X$ exists, then $\bigwedge X \in F$. The Rasiowa–Sikorski lemma ensures richness of Q-filters and injectivity in the representation map (Tanaka, 2021).

3. The Extended Representation Theorem and Preservation of Infinitary Operations

Given a modal algebra $A$ (possibly non-normal) and a countable family $S$ of subsets of $A$, the extended Jónsson–Tarski representation constructs a neighborhood frame $J_S(A) = (\QFilt_S(A), V_A)$, with $V_A(F) = \{ \{ G \mid x \in G \} \mid x \in F \}$. The canonical map $f(a) = \{ F \in \QFilt_S(A) \mid a \in F \}$ embeds $A$ as a Boolean subalgebra of $K(J_S(A))$, preserving the modal operator and, crucially, any infinitary meets and joins prescribed by $S$. This mechanism underpins completeness results for both finitary and infinitary predicate modal logics (normal or not), whenever appropriate Barcan-style closure conditions or countability constraints on $S$ hold (Tanaka, 2021).

4. Jónsson–Tarski Representation for Endomorphism Monoids and Finite Presentability

For fixed integers $n \ge 1$, $k \ge 2$, the free $n$-dimensional $k$-ary Jónsson–Tarski algebra $J_n = F_{n,k,1}$ (one generator) possesses a canonical endomorphism structure closely tied to the Brin–Higman–Thompson transformation monoids $\mathrm{totnM}_{k,1}$. The construction identifies $\mathrm{totnM}_{k,1}$, the monoid of all continuous self-maps of one-rooted $n$-dimensional $k$-ary Cantor space, with the endomorphism monoid $\operatorname{End}(J_n)$ by mapping each partial combinatorial pseudotree transformation to a unique endomorphism of $J_n$. This isomorphism is established via bases corresponding to shrubberies (complete prefix codes) and universality properties of $J_n$ (Witt et al., 2023).

5. Structural Properties, Key Lemmas, and Combinatorial Features

The central combinatorial insight is the correspondence between expansions of prefix codes in Cantor space and the generation of new free bases in $J_n$. Key structural results include:

• A finite set $A$ of shrubberies partitions Cantor space if and only if $\{\mathrm{tree}(a): a \in A\}$ freely generates $J_n$.
• Refinements of pseudotrees correspond bijectively to expansions of generating sets.
• Every endomorphism of $J_n$ is induced by a pseudotree map.

These properties guarantee the well-definiteness and multiplicativity of the isomorphism $$\varphi: \mathrm{totnM}_{k,1} \to \operatorname{End}(J_n)$$ and its inverse (Witt et al., 2023).

6. Finite Presentation and Algebraic Consequences

Exploiting the structural clarity of the Jónsson–Tarski representation, finite presentability of $\mathrm{totnM}_{k,1}$ (and thus $\operatorname{End}(J_n)$) is established by adapting depth-filtration arguments. Every endomorphism can be written as a composition from a finite generating set $A = \{a_1, \dots, a_m\}$ of transformations corresponding to pseudotrees of bounded depth and a finite list $R$ of defining relations (covering group-of-units relations, λ–α cancellations, and finite-depth combinatorics). Thus:

$$\mathrm{totnM}_{k,1} \cong \langle a_1, \dots, a_m \mid r_1, \dots, r_p \rangle$$

with both $m$ and $p$ finite, realizing a finite presentation motivated entirely by the Jónsson–Tarski framework (Witt et al., 2023).

7. Implications, Generalizations, and Logical Corollaries

The Jónsson–Tarski representation yields several notable consequences:

• Predicate–modal completeness: Any (possibly non-normal) predicate modal logic admits canonical neighborhood models manifesting the completeness and preservation of infinitary meets/joins captured in the algebraic $S$ (Tanaka, 2021).
• Infinitary logics: The completeness of infinitary modal logics is maintained, provided countability and Barcan-style axioms are met.
• Uniform treatment of normal and non-normal cases: Differing only in restrictions on the frame or neighborhood structure, the JT representation accommodates both settings using a shared algebraic formalism.
• Recovery of classical results: Selecting $S = \varnothing$ or restricting to finite $S$ recovers the original representation via Kripke frames, while richer $S$ captures corresponding infinitary behavior.
• Explicit model construction: The method provides explicit constructions for canonical models and presentations in both logic and algebra, with direct computational applications through finite presentability and basis expansion mechanisms (Witt et al., 2023, Tanaka, 2021).

These developments emphasize the integral role of the Jónsson–Tarski representation across algebraic logic, modal completeness theory, and the structure theory of transformation monoids and infinite groups.