Bialgebraic structures on boolean functions

TL;DR

This paper explores bialgebraic structures on boolean functions, introducing a two-parameter family of products and coproducts, and identifies a special subspecies with a unique polynomial invariant generalizing the chromatic polynomial.

Contribution

It defines a new family of twisted bialgebras on boolean functions and introduces rigid boolean functions with a second coproduct, expanding algebraic understanding of combinatorial invariants.

Findings

Established a two-parameter family of twisted bialgebras on boolean functions.

Identified a maximal subspecies where a second coproduct exists, called rigid boolean functions.

Derived a polynomial invariant generalizing 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.