Partial Maltsev Polymorphisms in Logic

• Partial Maltsev polymorphisms are partial functions satisfying the Maltsev identity, central to extending first-order logic with generalized quantifiers.
• CFI-style algebraic constructions and pebble games are used to rigorously establish expressiveness hierarchies for Maltsev-closed quantifiers.
• The methodology reveals that under partial Maltsev closure, logical distinctions diminish, impacting definability in constraint satisfaction frameworks.

A partial Maltsev polymorphism is a central concept in the study of expressiveness in logics extended with generalized quantifiers closed under certain algebraic closure conditions. Within the wider context of constraint satisfaction and logic, it serves to delineate classes of quantifiers whose definability and complexity are intimately tied to the algebraic behavior of partial operations that satisfy a Maltsev identity on their domain. The recent investigation by Dawar and others has provided a formal framework for these conditions and established sharp hierarchical results regarding the expressive power of such quantifiers, particularly through the use of CFI-style algebraic constructions and pebble games (Dawar et al., 14 Nov 2025).

1. Formal Definition of Partial Maltsev Polymorphism

Given a relational structure $A$ with universe $A$, a partial function $p\colon A^r \to A$ is termed a partial polymorphism of $A$ if, for every relation $R\subseteq A^r$ part of $A$, the lifted map

$$\hat{p}\colon R^\ell \to A^r,\qquad \hat{p}(\bar a^{(1)}, \dots, \bar a^{(\ell)})_j = p(a^{(1)}_j, \dots, a^{(\ell)}_j)$$

is everywhere defined on $R^\ell$ and produces a tuple in $R$ whenever its arguments belong to $R$.

For the Maltsev setting, the partial Maltsev family $M$ comprises functions $m_A\colon A^3 \to A$ defined on those triples with either equal outer elements or equal inner elements: $m_A(a, a, b) = b,\qquad m_A(b, a, a) = b,$ undefined elsewhere. Therefore, the operation realizes the usual Maltsev identity where $m(a, a, b) = m(b, a, a) = b$ and is undefined otherwise—enforcing a strong, but partial, symmetry constraint.

2. Generalized Quantifiers Closed under Partial Maltsev Families

For a family of partial functions $P$, a generalized quantifier $Q_K$ (with $K$ a class of $\tau$-structures) is $P$-closed if, for every structure $B\in K$, all structures $A\leq B\cup p_B(B)$ (i.e., $A$ is contained in $B$ with the action of $p_B$) also satisfy $A\in K$. Specifically, $Q^M$ denotes the collection of all quantifiers closed under the partial Maltsev family $M$, and $Q_r^M = Q^M \cap Q_r$ are those quantifiers of arity at most $r$. A quantifier $Q_K$ thus lies in $Q^M_r$ precisely when the defining class $K$ is closed under $m_B$ for any $B\in K$.

3. Main Inexpressibility Theorems

The two principal theorems characterizing the limits of logic extended by Maltsev-closed quantifiers are:

$\text{\emph{Theorem (Hierarchy): For every %%%%32%%%%,}}\qquad L(Q_r^M) \lneq L(Q_{r+1}^M)$

$\text{\emph{Theorem (Separation): For every %%%%33%%%%,}}\qquad L^k(Q_{k}^M) \lneq L^k(Q_{k})$

Here, $L(Q_r^M)$ is first-order logic extended by all $r$-ary Maltsev-closed quantifiers, and $L^k(\cdot)$ denotes the $k$-variable fragment. The first theorem asserts a strict arity hierarchy for Maltsev-closed quantifiers; i.e., increasing arity yields strictly more expressiveness. The second theorem demonstrates that the $k$-variable fragment of logic with $k$-ary Maltsev-closed quantifiers is strictly weaker than with all $k$-ary quantifiers (Dawar et al., 14 Nov 2025).

4. CFI-Style Algebraic Construction for Lower Bounds

The essential lower bounds are established through gadgets modeled after the Cai–Fürer–Immerman framework, but adapted for the polymorphism context:

• Vertex Gadget Construction: For each vertex $v$ in a $k$-regular graph $G = (V, E)$ and "charge" $s \in \{0,1\}$, the gadget $A^M(v, s)$ has universe $E(v) \times \mathbb{Z}_4$, with $k$-ary relations defined so that

$$R_0^{A^M(v,s)} = \left\{ ((e_1, a_1), \dots, (e_k, a_k)) \:\Big|\: \sum_{i=1}^k a_i - 2s \equiv 0 \text{ or } 1 \pmod{4} \right\}$$

and similarly $R_1^{A^M(v,s)}$ for congruence $2$ or $3 \pmod{4}$.

• Assembly: The full structure $A^M(G, U)$ for $U \subseteq V$ unites gadgets, assigning charge $1$ to $v \in U$, $0$ otherwise.
• Distinguished Instances: $A^M(G)$ corresponds to $U = \emptyset$, whereas $\tilde{A}^M(G)$ is defined by toggling the first vertex's charge.
• Algebraic Property: There is no homomorphism from $\tilde{A}^M(G)$ to $A^M(G)$, as witnessed by the unsolvability of a linear system over $\mathbb{Z}_4$; but $A^M(G) \to A^M(G)$ does have homomorphisms.

These constructions, particularly over graphs like $K_{d, d}$ or $K_{k+1, k+1}$ with a perfect matching removed, constitute explicit CFI witnesses for the inexpressibility results.

5. Maltsev-Quantifier Pebble Game

The Maltsev-quantifier pebble game formulated by Dawar and Hella characterizes $L^k(Q_k^M)$ equivalence. The game $M_k(A,B,\alpha,\beta)$ progresses from a partial isomorphism, with each round giving Spoiler a choice between "Left" and "Right" Maltsev moves. Spoiler challenges a Maltsev-closed relation on one structure, prompting Duplicator to select a triple whose closure under $m_A$ intersects a specified image. The key property is: Duplicator wins if and only if the two structures are indistinguishable in $L^k(Q_k^M)$, aligning the combinatorial game's outcome with logic expressiveness.

6. Proof Strategy for $k$-Variable Maltsev Separation

To establish $A^M(G^k) \equiv_{L^k(Q_k^M)} \tilde{A}^M(G^k)$ but $A^M(G^k) \not\equiv_{L^k(Q_k)} \tilde{A}^M(G^k)$, the argument uses the CFI construction's combinatorial flexibility:

• The Duplicator maintains an invariant where all pebbles are off a single "designated" gadget, and a bijection with a controlled cyclic-shift error is preserved.
• On each Spoiler challenge, the Duplicator either answers within the safe region or "moves" the mismatch along escape paths, leveraging the high connectivity of $G^k$.
• The argument uses the partial Maltsev structure to prevent Spoiler from distinguishing the two structures within $L^k(Q_k^M)$, demonstrating that quantifier expressiveness collapses under Maltsev closure for the $k$-variable case.

7. Explicit CFI-Type Counterexamples

The graph pairs $(A^M(G^k), \tilde{A}^M(G^k))$ on graphs $G^k = K_{k+1, k+1} \setminus M$ serve as canonical witnesses:

• They are non-isomorphic and separable by $k$-ary CSP quantifiers, so are not equivalent in $L^k(Q_k)$.
• However, they are indistinguishable in $L^k(Q_k^M)$ via the aforementioned Duplicator strategy.

No simpler counterexample is given, establishing these CFI-constructions as the critical examples showing that partial Maltsev closure limits but does not trivialize the expressive capacity of quantifiers at arity $k$.

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The study of partial Maltsev polymorphisms delineates a rich hierarchy in the expressivity of extended first-order logics. The connection of closure properties, CFI algebraic frameworks, and logical games offers deep insights into where definability collapses and how algebraic symmetries restrict quantifier power (Dawar et al., 14 Nov 2025).