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| Format: | Preprint |
| Veröffentlicht: |
2026
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| Online-Zugang: | https://arxiv.org/abs/2601.13773 |
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| _version_ | 1866909995646320640 |
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| author | Foissy, Loïc |
| author_facet | Foissy, Loïc |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_13773 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Bialgebraic structures on boolean functions Foissy, Loïc Combinatorics 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. |
| title | Bialgebraic structures on boolean functions |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2601.13773 |