Maltsev-Closed Generalized Quantifiers

• Maltsev-closed generalized quantifiers are Lindström quantifiers defined via closure under partial Maltsev polymorphisms, linking algebraic invariants with logical expressiveness.
• They exhibit a strict arity hierarchy and are semantically characterized by a two-player pebble game that delineates the limits of expressibility in expanded logics.
• Their framework not only refines logical hierarchies in CSP definability but also offers an algebraic approach for exploring extensions in model theory and algebraic logic.

Maltsev-closed generalized quantifiers, a key concept at the intersection of model theory, algebra, and the theory of constraint satisfaction problems (CSPs), are Lindström quantifiers characterized by closure under partial Maltsev polymorphisms. They provide an important bridge between algebraic invariants and the expressiveness of expansive logics, including applications in CSP-definability and the fine structure of logical hierarchies beyond first-order logic (Dawar et al., 14 Nov 2025, Dawar et al., 2023).

1. Definition and Algebraic Foundation

A partial polymorphism for a finite $\tau$-structure $B$ with universe $B$ is a partial function $m_B: B^3 \rightharpoonup B$ such that the induced coordinate-wise operation $\hat m_B$ satisfies $\hat m_B(R^B) \subseteq R^B$ for every relation symbol $R \in \tau$. That is, for any triple of tuples $\bar b_1,\bar b_2,\bar b_3 \in R^B$ on which $\hat m_B$ is defined, their image under $\hat m_B$ again belongs to $R^B$ (Dawar et al., 14 Nov 2025).

A partial Maltsev polymorphism is a partial operation $m_B: B^3 \rightharpoonup B$ defined by:

• $m_B(a, a, b) = m_B(b, a, a) = b$
• $m_B(a, b, c)$ undefined if $a \neq b$ and $b \neq c$

A Maltsev-closed quantifier is a Lindström quantifier $Q_K$ (for an isomorphism-closed class $K$ of $\tau$-structures) such that, for every $B \in K$ with a partial Maltsev polymorphism $m_B$ and every substructure $A \leq B \cup m_B(B)$, the structure $A$ also lies in $K$. Here, $m_B(B)$ is constructed by applying $m_B$ column-wise to all eligible tuples in $B$ (Dawar et al., 14 Nov 2025, Dawar et al., 2023).

Notation:

• $Q^M$: class of all Maltsev-closed quantifiers
• $Q^M_r$: those of arity $\leq r$

This Maltsev closure property is strictly algebraic yet directly informs logical definability, connecting the structure of polymorphisms with the expressiveness of generalized quantifier logics (Dawar et al., 2023).

2. Expressive Power and Arity Hierarchies

The expressiveness of logics extended with Maltsev-closed quantifiers admits a strict arity hierarchy. For each $r \geq 3$,

$L(Q^M_r) \subsetneq L(Q^M_{r+1})$

The construction uses the structure $B_{r+1} = (\{0,1\}, R_0, R_1)$, with $R_i$ comprising $(r+1)$-tuples of $\{0,1\}$ summing to $i$ modulo $2$. Although $B_{r+1}$ admits a total Maltsev polymorphism, the associated CSP is not definable in $L(Q^M_r)$. Therefore, $Q_{CSP(B_{r+1})} \in Q^M_{r+1}$ strictly separates the expressive power at consecutive arities (Dawar et al., 14 Nov 2025).

This arity hierarchy is strictly finer than the classical quantifier arity hierarchy and demonstrates that the closure properties encoded by Maltsev operations delineate logical power with precision.

3. Semantic Characterization via Pebble Games

Distinguishability in logics with Maltsev-closed quantifiers is characterized by a two-player pebble game, the Maltsev-pebble game $PG^M_k(A, B, \alpha, \beta)$, parameterized by a $k$-variable context (Dawar et al., 14 Nov 2025, Dawar et al., 2023). Key features include:

• Positions are partial assignments $\alpha: X_k \to A$ and $\beta: X_k \to B$.
• Rounds alternate between left and right Maltsev-moves, where Spoiler selects $r \leq k$ variables, Duplicator provides bijections, and Spoiler poses tuples; Duplicator must respond with sets from which the Spoiler's tuple can be retrieved via a finite chain of Maltsev operations.
• Duplicator wins by maintaining partial isomorphism indefinitely, thereby demonstrating indistinguishability in $L^k(Q^M_k)$.

The winning condition for Duplicator precisely matches indistinguishability in the associated infinitary logic with $P$-closed quantifiers. This game-based approach extends the utility of the pebble game paradigm to partial polymorphism-closed logics (Dawar et al., 2023).

4. CFI-style Constructions and Inexpressibility

Maltsev-closed quantifiers delimit expressiveness strictly below that of arbitrary $k$-ary quantifiers. For each $k \geq 3$, explicit CFI-style structures $A^M(G)$ and $\tilde{A}^M(G)$ can be constructed, based on $k$-regular bipartite graphs, with the following properties:

• $A^M(G) \not\cong \tilde{A}^M(G)$ (non-isomorphic)
• Duplicator wins $M_k(A^M(G), \tilde{A}^M(G))$, so they are not separated by any formula in $L^k(Q^M_k)$
• There exists a $k$-ary quantifier in $L^k(Q_k)$ that separates them

The construction is based on parity conditions in cyclic groups, combined with gadget-based decomposition and leveraging the Maltsev-closure properties to propagate local consistency. The separation shows:

$L^k(Q^M_k) \subsetneq L^k(Q_k)$

indicating that Maltsev-closure is a genuine restriction in expressive power (Dawar et al., 14 Nov 2025).

5. Connections with CSPs and Other Polymorphism-Based Quantifiers

Maltsev-closed quantifiers generalize quantifiers arising from CSPs whose polymorphism algebras satisfy the Maltsev identities. For a finite structure $C$ with a total Maltsev polymorphism $p: C^3 \to C$, the CSP quantifier $Q_{CSP(C)}$ (class of all homomorphic pre-images of $C$) is always Maltsev-closed (Dawar et al., 2023).

Maltsev-closed quantifiers are downward monotone—closed under taking substructures. The class is also strongly invariant under injections between universes. This property distinguishes Maltsev-closed quantifiers from those closed under near-unanimity operations (which enforce bounded width and stronger algebraic closure).

A representative comparison:

Family	Closure Property	Expressiveness
Maltsev ($M$)	Partial, idempotent choice	Tractable, unbounded width
Majority ($MJ$)	Total, symmetric choice	Implies $M$-closure
Near-unanimity ($N_\ell$)	Total, higher arity	Bounded width, stronger closure

6. Expressiveness Limitations and Implications

A principal limitation is that Maltsev-closed quantifiers do not suffice to express certain algebraic properties, such as the solvability of systems of linear equations over $\mathbb{Z}_2$ (parity problems), even though such CSPs are tractable and admit Maltsev polymorphisms. That is, such languages are not definable in $L^\omega_{\infty\omega}(Q_M)$, highlighting the expressive gap between closure under Maltsev operations and full linear-algebraic logics (Dawar et al., 2023).

The theoretical framework for Maltsev-closed quantifiers demonstrates a systematic method for classifying Lindström quantifiers through partial polymorphism families. This approach unifies and extends prior results from CSP theory, algebraic logic, and descriptive complexity, offering tools both for proving inexpressibility and for refining logical hierarchies (Dawar et al., 14 Nov 2025).

7. Extensions, Open Problems, and Future Directions

The current paradigm provides a natural pebble-game characterization and clean algebraic semantics for Maltsev-closed quantifiers, supporting extensions to other families of partial polymorphisms. Open questions include:

• Separation of higher-variable fragments in quantifiers closed under more general partial identities (e.g., weak near-unanimity), for which no pebble-game characterization is yet known.
• Full formalization of inexpressibility for linear-algebraic phenomena using adaptations of CFI-style constructions.
• Further systematic study of quantifier hierarchies based on other partial polymorphism conditions.

A plausible implication is that the extension of these techniques to wider classes of polymorphisms will continue to refine the map between algebraic closure properties, pebble-game logics, and the spectrum of definability for CSPs and related computational constructs (Dawar et al., 14 Nov 2025, Dawar et al., 2023).