Bialgebraic structures on boolean functions

TL;DR

The paper develops a bialgebraic framework for boolean functions on finite sets, introducing a two-parameter family of products ${\star_{q_1,q_2}}$ and a restriction coproduct to form twisted bialgebras on Bool. A bosonic Fock functor then yields a two-parameter family of bialgebras on isomorphism classes, with a morphism $\Phi$ to ${\mathbb K}[T]$ that encodes modular colorings; in particular, hypergraph indicators recover the chromatic polynomial, and rank functions of graphical matroids yield forest-coloring polynomials. The central challenge is to define a second coproduct via contractions to obtain a double bialgebra on all of Bool; this is impossible, but a maximal convenient subspecies ${\rm Bool}_{\rm max}$ (and its rigid/hyper-rigid variants) does admit a second coproduct compatible with the product and the original coproduct, enabling a unique double bialgebra morphism to ${\mathbb K}[T]$. The rigidity framework unifies hypergraphs and matroids under contraction–restriction, and yields a polynomial invariant with combinatorial interpretations in terms of colorings and forests. An appendix establishes that matroid rank functions are rigid boolean functions, strengthening the connection between matroids and the twisted double bialgebra structure explored. The work thus deepens links between combinatorial Hopf algebras, contraction/restriction operations, and classical invariants such as the chromatic polynomial.

Abstract

We study several bialgebraic structures on boolean functions, that is to say maps defined on the set of subsets of a finite set $X$, taking the value $0$ on $\emptyset$. Examples of boolean functions are given by the indicator function of the hyperedges of a given hypergraph, or the rank function of a matroid. We give the species of boolean functions a two-parameters family of products and a coproduct, and this defines a two-parameters family of twisted bialgebras. We then try to define a second coproduct on boolean functions, based on contractions, in order to obtain a double bialgebra. We show that this is not possible on the whole species of boolean functions, but that there exists a maximal subspecies where this is possible. This subspecies being rather mysterious, we introduce rigid boolean functions and show that this subspecies has indeed a second coproduct, as wished, and that it contains rank functions of matroids and indicator functions associated to hypergraphs. As a consequence, we obtain a unique polynomial invariant on rigid boolean functions, which is a generalization of the chromatic polynomial of graphs.

Theorems & Definitions (188)

• Definition 1.1
• Example 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 1.4
• Definition 1.5
• Definition 1.6
• Definition 1.7
• Definition 1.8
• Example 1.2