Partial Maltsev Polymorphisms

• Partial Maltsev polymorphisms are partial functions on a finite set that satisfy the identities m(a, a, b)=b and m(b, a, a)=b when defined, capturing essential algebraic structure.
• Maltsev-closed quantifiers leverage these polymorphisms to create strict arity hierarchies and model comparison games, highlighting limits of logical expressiveness in finite model theory.
• This framework bridges algebraic CSP tractability and logical definability, offering a uniform tool for exploring quantifier expressiveness in generalized logics.

A partial Maltsev polymorphism is a central concept in the algebraic theory of generalized quantifiers and constraint satisfaction problems (CSPs), providing a lens to organize and stratify the expressive power of logics extended by quantifiers closed under specified equational conditions. Maltsev-closed quantifiers, and their associated arity hierarchies, yield a fine-grained algebraic and combinatorial framework within finite model theory that bridges polymorphism-based CSP tractability analysis with the logical characterization of definability.

1. Definition: Partial Maltsev Polymorphisms and Maltsev-Closed Quantifiers

Let $A$ be a finite set. A partial Maltsev polymorphism on $A$ is a partial function

$m_A : A^3 \longrightarrow A$

satisfying the identities

$$m_A(a, a, b) = b \qquad \text{and} \qquad m_A(b, a, a) = b$$

whenever these terms are defined, and $m_A(a, b, c)$ is undefined if neither $a = b$ nor $b = c$. This partiality encodes the essential behavior required for the Maltsev property while permitting undefinedness on tuples not matching the required patterns. The operation is invariant under bijections: for any $f: A \to B$,

$f\bigl(m_A(a,a,b)\bigr) = m_B(f(a),f(a),f(b))$

whenever both sides are defined.

Given a Lindström quantifier $Q_K$ specified by an isomorphism-closed class $K$ of finite $\tau$-structures, $Q_K$ is Maltsev-closed if $K$ is closed under partial Maltsev polymorphism in the sense of downward containment: whenever $\mathfrak{B} \in K$ and $\mathfrak{A}$ is a $\tau$-substructure of $\mathfrak{B} \cup m_B(\mathfrak{B})$, then $\mathfrak{A} \in K$ as well. Define $Q^M$ as the class of all Maltsev-closed quantifiers, and $Q^M_r$ the subclass of arity at most $r$ (Dawar et al., 14 Nov 2025, Dawar et al., 2023).

2. Arity Hierarchy and Separation in Maltsev-Closed Logics

Maltsev-closed quantifiers admit a strict arity hierarchy paralleling classical results for full quantifier logics. For each $r \geq 3$,

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

where $L(Q^M_r)$ denotes the logic obtained by augmenting first-order logic with all arity-${\leq}r$ Maltsev-closed quantifiers. The canonical example is provided by the template

$B_r = (\{0,1\}, R_0, R_1)$

with

$$R_i = \Bigl\{(a_1,\dots,a_r) \in \{0,1\}^r ~|~ \sum_j a_j \equiv i \pmod{2} \Bigr\}$$

for $i=0,1$. $B_r$ admits a total Maltsev polymorphism via the group operation on $(\mathbb{Z}_2, +)$, and the corresponding CSP quantifier $Q_{\mathrm{CSP}(B_r)}$ lies in $Q_r^M$. Classically, $\mathrm{CSP}(B_{r+1})$ is not definable in $L(Q^M_r)$ (Dawar et al., 14 Nov 2025).

A sharper separation is observed when bounding the number of variables: for each $k\geq3$,

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

where $L^k$ denotes the $k$-variable fragment. Thus, Maltsev-closed quantifiers of a given arity and bounded variable-width do not capture the full expressive power of general quantifiers of that arity (Dawar et al., 14 Nov 2025).

3. CFI-Style Inexpressibility and the Maltsev Quantifier Pebble Game

The arity and variable-bound separation results rely on a generalized Cai-Fürer-Immerman (CFI)-type construction, coupled with a new Maltsev quantifier pebble game. This game, developed by Dawar and Hella (2024), is adapted to the semantics of partial Maltsev closure.

Construct non-isomorphic structures $\mathfrak{A}^M(G)$ and $\widetilde{\mathfrak{A}}^M(G)$ by decorating the vertices of a carefully chosen $k$-regular near-$k$-connected graph $G$ with gadgets $A^M(v,s)$, where the gadget universe is $E(v) \times \mathbb{Z}_4$ and the relations $R_0, R_1$ are specified by parity constraints on the sum of coordinates modulo $4$ (explicitly: $R_0$ contains tuples with sum $2s$ or $2s+1 \bmod 4$, $R_1$ those with sum $2s+2$ or $2s+3$).

The Maltsev quantifier pebble game $M_k(\mathfrak{A},\mathfrak{B})$ requires the Duplicator, on every move, to identify three tuples in $B^r$ whose partial Maltsev closure contains the challenged tuple. The winning strategy builds an invariant tracking the unique mismatch in the global bijection (between the structures), shifting this mismatch via alternating paths in $G$.

Consequently, the structures are not isomorphic, yet indistinguishable by $L^k(Q^M_k)$:

$$\mathfrak{A}^M(G) \not\cong \widetilde{\mathfrak{A}}^M(G) \qquad \text{but} \qquad \mathfrak{A}^M(G) \equiv_{L^k(Q^M_k)} \widetilde{\mathfrak{A}}^M(G)$$

establishing a proper separation between $L^k(Q^M_k)$ and $L^k(Q_k)$ (Dawar et al., 14 Nov 2025).

4. Failure of Partial Maltsev Closure: Illustrative Examples

Partial Maltsev closure imposes substantive restrictions not satisfied by many natural relational structures. A key example: for the relation

$R = \{(w,x,y,z) \in \mathrm{Sym}_3^4 ~|~ wxyz=e\}$

arising from 4-ary systems of equations in the non-abelian group $\mathrm{Sym}_3$, each 3-ary projection of $R$ is closed under the partial Maltsev operation, but $R$ as a whole fails closure: there exist three tuples such that their componentwise application of $m$ yields a tuple not in $R$. Hence, $R$ is not Maltsev-closed, and such structures fall outside $Q^M$. This demonstrates that CSPs with higher arity or over non-abelian groups may escape Maltsev closure, even when all lower-arity projections suggest the presence of a polymorphism (Dawar et al., 14 Nov 2025).

5. The Landscape of Arity-Hierarchies and Polymorphism-Closed Quantifiers

The study of quantifiers closed under partial polymorphisms, including the Maltsev case, situates Maltsev-closed logics as an intermediate complexity layer in the arity-hierarchy landscape:

• Near-unanimity closed quantifiers ($N^\ell$) are associated with a strict, infinite arity hierarchy, where $L(Q_{\ell-1}) < L(Q^{N^\ell}_{\ell+1}) < L(Q_\ell)$ for every $\ell \geq 3$; these quantifiers capture all “bounded-width” CSP quantifiers, including instances without a genuine near-unanimity polymorphism.
• Maltsev-closed quantifiers ($M$) similarly enjoy an infinite strict hierarchy $L(Q^M_r)<L(Q^M_{r+1})$, but in bounded-variable fragments are strictly weaker than the full arity class. The CSPs definable by a single Maltsev-closed quantifier are precisely those whose template admits a Maltsev polymorphism, notably including systems of equations mod $2$ (e.g., $C_\ell$).
• Counting, group-theoretic, and linear-algebraic quantifiers constitute further distinct logics, each with their model-theoretic game characterizations.

Maltsev-closed logics admit proper model comparison games, are richer than plain FO, and occupy an essential role in identifying the boundaries of tractable and inexpressible CSPs from a descriptive complexity perspective (Dawar et al., 14 Nov 2025, Dawar et al., 2023).

6. Expressiveness, Pebble Game Characterization, and Open Questions

All CSPs whose template has a Maltsev polymorphism are definable by a Maltsev-closed quantifier. Within $L_{\infty\omega}(Q_M)$, such CSP quantifiers are available, and the k-pebble Maltsev game $PG^M_k$ precisely captures the distinguishing power of $L^k_{\infty\omega}(Q_M)$ (Dawar et al., 2023).

Unlike the case with near-unanimity closures (where solvability mod $2$ is not definable in the corresponding logics), no nontrivial inexpressibility result is known for Maltsev-closed quantifiers beyond k-variable counting lower bounds. The extent to which tractable CSPs such as solvability mod $p$ (for prime $p$) are fully captured by $L_{\infty\omega}(Q_M)$ remains open. No arity hierarchy strictness has yet been confirmed for $Q_M$ on the infinitary (unbounded-variable) level in (Dawar et al., 2023), but (Dawar et al., 14 Nov 2025) establishes strictness in the first-order and finite-variable fragments.

The general methodology of partial polymorphism closure—specialized to the Maltsev case—offers a uniform algebraic and logical tool for delineating the expressive boundaries of logics with generalized quantifiers, via their closure conditions and associated pebble games. This research program motivates the exploration of other polymorphism-based stratifications, such as weak near-unanimity and mixture polymorphisms, in the pursuit of even finer logical hierarchies and descriptive complexity classifications.