Partial Maltsev Polymorphism

• Partial Maltsev polymorphism is a concept that applies a partial function satisfying limited Maltsev identities to restrict generalized quantifiers in algebra and logic.
• It underpins key results such as arity hierarchies and fixed-variable separations, delineating expressive boundaries in logical systems.
• CFI-style algebraic constructions provide concrete counterexamples, demonstrating practical limitations in constraint satisfaction and logical definability.

A partial Maltsev polymorphism is a fundamental concept in the algebraic and logical analysis of generalized quantifiers, closely tied to the structural theory of constraint satisfaction problems (CSPs) and the expressiveness of finite-variable logics. Partial Maltsev polymorphisms restrict the set of generalized quantifiers to those invariant under specified algebraic operations, and recent results crystallize the relationship between such quantifiers, arity hierarchies, and the limits of logical definability—most sharply through inexpressibility theorems and Cai–Fürer–Immerman–type (CFI-type) algebraic counterexamples (Dawar et al., 14 Nov 2025).

1. Definition and Properties of Partial Maltsev Polymorphisms

Given a relational structure $A$ with universe $A$, a partial polymorphism is a partial function $p: A^r \to A$ such that for any relation $R \subseteq A^r$ of $A$, the lifted map

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

is everywhere defined on $R^\ell$ and, whenever each $\bar a^{(i)} \in R$, yields that $\hat p(\bar a^{(1)},\ldots, \bar a^{(\ell)}) \in R$.

The partial Maltsev family $M$ consists, for every finite set $A$, of the partial function $m_A: A^3 \to A$ given by

$m_A(a,a,b) = b, \qquad m_A(b,a,a) = b,$

undefined when neither $x=y$ nor $y=z$ for the triple $(x,y,z)$. Thus, $m_A$ implements the Maltsev identities only partially, admitting inputs with repeated elements in the outer or inner position, but is not total.

2. Maltsev-Closed Generalized Quantifiers

For a family of partial operations $P$ (here, $M$), a generalized quantifier $Q_K$ is $P$-closed if, for any $B \in K$ and $A \leq B \cup p_B(B)$, then $A \in K$. Here, $A \leq B \cup p_B(B)$ means $A$ is a substructure of the structure obtained by adjoining the image of $p_B(B)$.

Let:

• $Q^M$ denote all generalized quantifiers closed under the partial Maltsev family,
• $Q^M_r$ denote those of arity at most $r$ (i.e., $Q^M \cap Q_r$, with $Q_r$ the class of Lindström quantifiers using relations of arity $\leq r$).

An $r$-ary quantifier $Q_K$ lies in $Q^M_r$ if and only if $K$ is closed under the Maltsev operation $m_B$ on any $B \in K$. This yields a robust family of constraints that restrict Lindström extensions, underlying the subsequent inexpressibility hierarchy.

3. Inexpressibility Theorems for Partial Maltsev Closure

Two central theorems establish strict inexpressivity of logics augmented by Maltsev-closed quantifiers:

Theorem (Arity Hierarchy)

For every $r \geq 3$,

$L(Q_r^M) \lneq L(Q_{r+1}^M).$

Here, $L(Q_r^M)$ denotes first-order logic extended by all $r$-ary quantifiers closed under $M$, and $\lneq$ denotes strict containment.

Theorem (Fixed-Variable Separation)

For every $k \geq 3$,

$L^k(Q_{k}^M)\;\lneq\;L^k(Q_{k}),$

where $L^k(\cdot)$ is the $k$-variable fragment. Thus, the expressive power of $k$-variable logic with all $k$-ary Maltsev-closed quantifiers is strictly less than with all $k$-ary quantifiers (Dawar et al., 14 Nov 2025).

4. CFI-Style Algebraic Constructions for Lower Bounds

The key separation results are witnessed by modifications of the classic Cai–Fürer–Immerman (CFI) gadgets tailored to the partial Maltsev context.

Vertex Gadget Construction:

• For a $k$-regular graph $G = (V, E)$ and $s \in \{0,1\}$, construct a gadget $A^M(v, s)$ with universe $E(v) \times \mathbb{Z}_4$ (one $\mathbb{Z}_4$ copy for each incident edge).
• Relations $R^{A^M(v, s)}_0$, $R^{A^M(v, s)}_1$ defined by congruence conditions:

$$R_0: \sum a_i - 2s \equiv 0,1 \pmod{4} \quad R_1: \sum a_i - 2s \equiv 2,3 \pmod{4}$$

CFI Assembly:

Structures $A^M(G, U)$ are formed by combining gadgets according to which vertices $U \subseteq V$ have “charge 1”; the default instance $A^M(G) = A^M(G, \emptyset)$, and $\tilde A^M(G) = A^M(G, \{\tilde v\})$ for a distinguished vertex $\tilde v$.

Algebraic Properties:

• No homomorphism exists from $\tilde A^M(G) \to A^M(G)$—the associated $\mathbb{Z}_4$-equations have no solution, separating these templates.
• Homomorphisms from $A^M(G) \to A^M(G)$ are possible (e.g., all-zero assignments), so the collection is not closed under isomorphism with the charged case.

5. The Maltsev-Quantifier Pebble Game

The logic $L^k(Q_k^M)$ is characterized by a model-comparison game $M_k(A, B, \alpha, \beta)$ (due to Dawar–Hella) that generalizes the usual Ehrenfeucht–Fraïssé game with maneuvers reflecting Maltsev closure:

Game Procedure:

1. If the current assignment is not a partial isomorphism, Spoiler wins.
2. Otherwise, Spoiler selects $r \leq k$ and an $r$-tuple of variables.
   • In the Left move, Duplicator picks a bijection $f:B\to A$; Spoiler chooses $\bar b\in B^r$, Duplicator presents a triple $P \subseteq A^r$ satisfying $f(\bar b) \in m_A(P)$; Spoiler chooses one $\bar a_i$ to continue.
   • Right move reverses roles.
3. If Spoiler cannot break the isomorphism in finitely many rounds, Duplicator wins.

The main result: Duplicator wins $M_k(A, B, \emptyset, \emptyset)$ if and only if $A \equiv_{L^k(Q_k^M)} B$ (Dawar et al., 14 Nov 2025).

6. High-Level Proof Strategy for the Fixed-Variable Separation

To demonstrate inequivalence in $L^k(Q_k)$ but equivalence in $L^k(Q_k^M)$ for CFI-pairs $(A^M(G^k), \tilde A^M(G^k))$, the proof hinges on Duplicator maintaining an invariant:

• All pebbled elements avoid one designated gadget $A^M(w)$.
• A global bijection $f:A^M(G)\to \tilde A^M(G)$ is a local isomorphism off $A^M(w)$. The “error” (failure of isomorphism) is concentrated and tracked by a $\mathbb{Z}_4$ parameter.
• Spoiler's moves either permit Duplicator to respond safely within the unbroken region or, if the defect is pressed, Duplicator uses Maltsev moves to shift the error elsewhere along a long enough path in $G^k$.

Because the expunged perfect matching in $G^k=K_{k+1,k+1}\setminus M$ leaves the graph highly connected, this maneuver always succeeds, ensuring indistinguishability in $L^k(Q_k^M)$.

7. Explicit Counterexamples and the Scope of Maltsev Closure

The CFI structures $A^M(G^k), \tilde A^M(G^k)$ over $G^k=K_{k+1,k+1}\setminus M$ serve as explicit, minimal counterexamples:

Structure	$L^k(Q_k^M)$ Equivalence	$L^k(Q_k)$ Equivalence
$A^M(G^k)$ vs $\tilde A^M(G^k)$	Equivalent	Not Equivalent

These instances are not isomorphic but cannot be separated by any $k$-variable logic with only Maltsev-closed quantifiers; full $k$-ary CSP quantifiers are necessary for separation. This demonstrates that partial Maltsev closure imposes substantive constraints: it is strong enough to collapse significant expressivity but still leaves a strict hierarchy below the full spectrum of $k$-ary definability. No simpler counterexamples are cited; all witnesses are constructed by this CFI mechanism (Dawar et al., 14 Nov 2025).