Rigid Boolean Functions & Polynomial Invariants

• Rigid boolean functions are defined on power sets (vanishing at the empty set) and exhibit structural decompositions through genuine additive splitting.
• The unique polynomial invariant generalizes the chromatic polynomial by encoding deep bialgebraic symmetries present in combinatorial and algebraic frameworks.
• This framework applies to hypergraphs, graphs, and matroids, providing deletion–contraction recurrences that mirror classical algorithms for computing polynomial invariants.

A polynomial invariant on rigid boolean functions is a uniquely defined polynomial arising from the double bialgebra structure imposed on the subspecies of rigid boolean functions. Boolean functions in this context are functions $f: P(X) \to \mathbb{Z}$ for a finite set $X$, vanishing at the empty set, and encompassing important combinatorial constructs such as the indicator function of hypergraphs and the rank function of matroids. The construction of the polynomial invariant generalizes the chromatic polynomial of graphs and encodes deep bialgebraic symmetries present in combinatorics and algebraic structures (Foissy, 20 Jan 2026).

1. Boolean Functions, Bialgebraic Structures, and Products

Let $X$ be a finite set. The species $\mathrm{Bool}(X)$ consists of all functions $f: P(X) \to \mathbb{Z}$ with $f(\emptyset) = 0$. This species supports a two-parameter family of associative products defined as follows: given $f \in \mathrm{Bool}(X)$, $g \in \mathrm{Bool}(Y)$, and $X$ disjoint from $Y$,

$$(f *_{q_1, q_2} g)(A) = q_1 f(A \cap X) + q_2 g(A \cap Y), \quad A \subseteq X \cup Y$$

This product is commutative if and only if $q_1 = q_2$. The canonical (commutative) instance is $q_1 = q_2 = 1$.

The restriction-coproduct is given by

$$\Delta(f) = \sum_{X = I \sqcup J} f|_I \otimes f|_J$$

with $f|_I(A) = f(A \cap I)$. The triple $(\mathrm{Bool}, *_{q_1,q_2}, \Delta)$ defines a twisted (set-graded) bialgebra.

Applying the Fock functor $F$, one obtains a genuine commutative bialgebra $H_{\mathrm{Bool}} = F(\mathrm{Bool})$, with $*$ for $*_{1,1}$.

2. Contraction, Restriction, and the Maximal Subspecies

Attempting to define a second coproduct, namely a contraction–restriction coproduct, necessitates additional structure: $$\delta(f) = \sum_{\sim \in \mathcal{E}(f)} f/\!\sim \otimes f|_{\sim}$$ For any equivalence relation $\sim$ on $X$, define: $$f/\!\sim(A) = f(\pi^{-1}(A)),\quad f|_{\sim}(A) = \sum_{Y \in X/\!\sim} f(A \cap Y)$$ where $\pi: X \to X/\!\sim$ is the canonical quotient. Imposing bialgebraic compatibility axioms, it is established that no rule for $\mathcal{E}(f)$ works globally for all boolean functions. Thus, one is forced to work inside a maximal subspecies $\mathrm{Bool}_{\max} \subset \mathrm{Bool}$ where weak and strong equivalence families coincide.

3. Rigid Boolean Functions and Double Bialgebra Structure

A boolean function $f \in \mathrm{Bool}(X)$ is called indecomposable if it cannot be written nontrivially as $f = g * h$, for a partition $X = Y \sqcup Z$. The unique maximal factorization

$f = f_{X_1} * \cdots * f_{X_k}$

yields equivalence classes defining the indecomposable blocks.

A function $f$ is called rigid if any additive splitting over disjoint subsets $A, B \subseteq X$,

$f(A \cup B) = f(A) + f(B)$

implies a structural decomposition $f|_{A \cup B} = f|_A * f|_B$. Thus, rigid functions account only for "genuine" additive splitting reflecting actual disconnected blocks in $X$. Rigid boolean functions form a maximal subspecies $\mathrm{Bool}_r$ stable under $*$, $\Delta$, and the contraction–restriction coproduct.

On $\mathrm{Bool}_r$, the weak and strong equivalences coincide, and a bona fide second coproduct $\delta$ exists,

$$\delta: \mathrm{Bool}_r \longrightarrow \bigoplus_{X = I \sqcup J} \mathrm{Bool}_r(I) \otimes \mathrm{Bool}_r(J)$$

After linearization using the Fock functor, one obtains a connected double bialgebra: $(H_{\mathrm{Bool}_r}, *, \Delta, \delta)$

4. Second Coproduct and Polynomial Invariant

The second coproduct for $f \in \mathrm{Bool}_r(X)$ is defined as: $$\delta(f) = \sum_{\sim \in \mathcal{E}(f)} f/\!\sim \otimes f|_{\sim}$$ where equivalence relations respect indecomposable components. The map $\delta$ is coassociative and compatible with both $*$ and $\Delta$, and admits a two-sided counit $\varepsilon_\delta$, with $\varepsilon_\delta(f) = 1$ iff $f$ is modular (splitting as a pure sum of singleton restrictions).

By general results on double bialgebras, there exists a unique morphism

$\Phi: H_{\mathrm{Bool}_r} \to \mathbb{K}[T]$

to the binomial Hopf algebra, characterized by

$$\Phi(f) \in \mathbb{K}[T], \quad \Phi(f)(1) = \varepsilon_\delta(f)$$

This morphism gives rise to the fundamental polynomial invariant $P_f(T) = \Phi_f(T)$ for rigid boolean functions.

The morphism satisfies the following recursion: $$\Phi_f(T + S) = \sum_{\sim \in \mathcal{E}(f)} \Phi_{f/\!\sim}(T) \cdot \Phi_{f|_{\sim}}(S)$$ and a Taylor-style expansion: $$\Phi_f(T) = \sum_{k \geq 1} \frac{1}{k!} \big((\Phi-\varepsilon_\delta)*\cdots*(\Phi-\varepsilon_\delta)\big)(f) \,\, T(T-1)\cdots (T-k+1)$$ Combinatorially,

$$\Phi_f(n) = \#\left\{ c: X \to \{1, \dots, n\} \,\Bigg|\, \forall i,\, f(c^{-1}(i))\;\text{is modular} \right\}$$

for $n > 0$. $P_f(T)$ is thus a polynomial of degree $|X|$ with integer coefficients.

5. Chromatic Polynomial, Matroids, and Hypergraph Examples

For a hypergraph $H$ with vertex set $X$ and hyperedge set $E(H)$, the indicator function

$y(H)(A) = \#\{e \in E(H) \mid e \subseteq A\}$

is rigid, and $\Phi_{y(H)}(n)$ yields the chromatic polynomial of the hypergraph (i.e., the number of proper colorings avoiding monochromatic hyperedges). For graphs $G$ (as $2$-uniform hypergraphs), this reduces to the classical chromatic polynomial.

For matroids:

• If $f = \operatorname{rk}_G$ is the rank-function of the graphic matroid on the edges $E(G)$ of a graph $G$, then

$$\Phi_{\operatorname{rk}_G}(n) = \#\left\{c: E(G) \to [n] \mid \forall i,\, G[c^{-1}(i)] \text{ is a forest} \right\}$$

• For a linear matroid, $f(A) = \operatorname{rank}(\{v_x\}_{x \in A})$, then

$$\Phi_f(n) = \#\left\{ c: X \to [n] \mid \forall i,\, \{v_x : c(x) = i\} \text{ is independent} \right\}$$

The table below summarizes key rigid boolean function classes and their associated polynomial invariants:

Structure	Boolean Function	Polynomial Invariant
Hypergraph	Indicator $y(H)$	Chromatic polynomial of $H$
Graph	Indicator of $2$-uniform	Classical chromatic polynomial
Graphic Matroid	Rank function $\operatorname{rk}_G$	Coloring forests per edge-set
Linear Matroid	Rank of vectors	Coloring independent sets

6. Universality and Relation to the Tutte Polynomial

The construction $f \mapsto P_f(T)$ is universal among invariants that factor through the double bialgebra $(H_{\mathrm{Bool}_r}, *, \Delta, \delta)$ into the binomial Hopf algebra. Special cases and parameter specializations recover the classical graph-chromatic, hypergraph-chromatic, and via bivariate refinements, the Tutte polynomial of a matroid.

A plausible implication is that this framework systematically subsumes and generalizes major combinatorial polynomial invariants, providing a unified platform for their algebraic and structural analysis.

7. Computational Aspects and Applications

Evaluation of $\Phi_f(n)$ is $\#P$-hard even for graph-theoretic instances. Nonetheless, the bialgebraic approach yields deletion–contraction recurrences and broken-circuit expansions that mirror efficient algorithms for computing chromatic and Tutte polynomials, emphasizing both theoretical clarity and computational tractability where possible.

Potential applications include graph and hypergraph coloring, reliability of networks, moment–cumulant relations in algebraic probability (through duality in species theory), and the combinatorics underlying regularity structures in stochastic PDEs (Foissy, 20 Jan 2026).