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Analysis explores various products and their relation to bialgebras in commutative rings, suggesting deeper algebraic structures.
For factorizing and decomposing the representative (or rational) series, with coefficients in a commutative ring A containing ℚ, we examine various products (such as concatenation, shuffle, quasi-shuffle) defined on the free monoid which are such that their associated bialgebras are isomorphic to the Sweedler’s dual, for A being a field K.
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Bùi et al. (2025) studied this question.
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