Abstract
The shuffle multiplication and the cut comultiplication, a generalized car-cdr pairing, form a bialgebra. The concatenation multiplication, sometimes called tensor product, and the spray comultiplication form another bialgebra. The concatenation-spray bialgebras are the free bialgebras in the category of precise, graded bialgebras over a semiadditive symmetric monoidal category. The shuffle-cut bialgebras are the cofree bialgebras in the same category of bialgebras. These categories include many of the settings of interest in the theories of formal languages and the theories of distributed, concurrent and parallel computation. We analyze the marked shuffle, of interest in theories of distributed computing, in terms of its resolutions into the cofree shuffle-cut bialgebra.
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Benson, D.B. (1988). The shuffle bialgebra. In: Main, M., Melton, A., Mislove, M., Schmidt, D. (eds) Mathematical Foundations of Programming Language Semantics. MFPS 1987. Lecture Notes in Computer Science, vol 298. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-19020-1_32
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DOI: https://doi.org/10.1007/3-540-19020-1_32
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