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arXiv · 2607.19093

Various Bialgebras Of Representative Functions On Free Monoids

Vincel Hoang Ngoc Minh
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Abstract

Factorization and decomposition of representative functions on a free monoid X * (generated by an alphabet X ) and with values in a ring A containing Q are equivalent to factorization and decomposition of their graphs (within the A-algebra of rational noncommutative series over X ) admitting linear representations (thanks to the Kleene-Sch{\"u}tzenberger theorem). To factorize and to decompose effectively these graphs, we examine various products of noncommutative series (as concatenation, shuffle and its $\phi$deformations) and co-products such that, for A is a field K, their associated non graded commutative and co-noncommutative bialgebras of series are isomorphic to the Sweedler's dual of the graded noncommutative co-commutative bialgebras of polynomials having, for the concatenation, only Kleene stars of the planes as characters, or equivalently, only the planes are infinitesimal characters (thanks to a Ree's theorem like).

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Vincel Hoang Ngoc Minh. 2026-07-21. Various Bialgebras Of Representative Functions On Free Monoids. https://arxiv.org/abs/2607.19093

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