The shuffle bialgebra

  • David B. Benson
Part VI New Directions
Part of the Lecture Notes in Computer Science book series (LNCS, volume 298)


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.


Formal Language Monoidal Category Countable Product Algebra Morphism Identity Morphism 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. A. Arbib and E. B. Manes, Adjoint Machines, State-Behavior Machines, and Duality, J. Pure Appl. Alg. 6(1975), 313–344.CrossRefGoogle Scholar
  2. 2.
    M. Broy, Semantics of Communicating Processes, Inform. & Control 61(1984), 202–241.Google Scholar
  3. 3.
    S. Eilenberg and G. M. Kelly, Closed Categories in Proc. Conf. Categorical Alg., La Jolla 1965, Springer-Verlag, New York, 1966, pp. 421–562.Google Scholar
  4. 4.
    S. Eilenberg & S. MacLane, On the Groups H (п, n). I., Ann. of Math 58(1953), 55–106.Google Scholar
  5. 5.
    N. Francez, Fairness, Springer-Verlag, New York, 1986.Google Scholar
  6. 6.
    H. Gaifman and V. Pratt, Partial Order Models of Concurrency and the Computation of Functions, Proc. IEEE Symp. Logic in Comput. Sci., Ithaca, NY, 1987.Google Scholar
  7. 7.
    S. Ginsburg, The Mathematical Theory of Context-Free Languages, McGraw-Hill, 1966.Google Scholar
  8. 8.
    S. Ginsburg and E. H. Spanier, Mapping of Languages by Two-tape Devices, J. Assoc. Comput. Mach. 12(1965), 423–434.Google Scholar
  9. 9.
    J. L. Gischer, Partial Orders and the Axiomatic Theory of Shuffle, Stanford Univ. Report STAN-CS-84-1033.Google Scholar
  10. 10.
    H. Herrlich & G. Strecker, Category Theory, Heldermann-Verlag, Berlin, 1979.Google Scholar
  11. 11.
    R. J. Lorentz & D. B. Benson, Deterministic and Nondeterministic Flowchart Interpretations, J. Comput. Sys. Sci. 27(1983), 400–433.CrossRefGoogle Scholar
  12. 12.
    S. MacLane, Categories for the Working Mathematician, Springer-Verlag, New York, 1971.Google Scholar
  13. 13.
    S. MacLane, Homology, Academic Press, New York, 1963.Google Scholar
  14. 14.
    M. Main and D. B. Benson, Functional Behavior of Nondeterministic and Concurrent Programs, Inform. & Control 62(1984), 144–189.Google Scholar
  15. 15.
    E. G. Manes, Algebraic Theories, Springer-Verlag, New York, 1976.Google Scholar
  16. 16.
    E. G. Manes, Additive Domains, Springer-Verlag LNCS 239, 1986, 184–195.Google Scholar
  17. 17.
    E. G. Manes, Weakest Preconditions: Categorical Insights, Springer-Verlag LNCS 240, 1986, 182–197.Google Scholar
  18. 18.
    E. G. Manes, Assertional Categories, Third Workshop on Math. Found. Program. Semantics, Tulane, April 1987, These Proceedings.Google Scholar
  19. 19.
    M. Pfender, Universal Algebra in S-Monoidal Categories, Bericht Nr. 20, Mathematisches Institut, Univ. München, 1974.Google Scholar
  20. 20.
    V. Pratt, Modeling Concurrency with Partial Orders, International J. Parallel Programming, 15(1986), 33–72.CrossRefGoogle Scholar
  21. 21.
    L. Redei, The Theory of Finitely Generated Commutative Semigroups, Pergammon Press, 1965.Google Scholar
  22. 22.
    W. E. Riddle, Modelling and Analysis of Supervisor Systems, Ph.D. Thesis, Stanford Univ., 1972.Google Scholar
  23. 23.
    W. E. Riddle, An Approach to Software System Behavior Description, Computer Languages 4(1979), 29–47.CrossRefGoogle Scholar
  24. 24.
    A. Salomaa and M. Soittola, Automata-Theoretic Aspects of Formal Power Series, Springer-Verlag, New York, 1978.Google Scholar
  25. 25.
    E. H. Spanier, Algebraic Topology, McGraw-Hill Book Co., New York, 1966.Google Scholar
  26. 26.
    M. Sweedler, Hopf Algebras, W. A. Benjamin, New York, 1969.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • David B. Benson

There are no affiliations available

Personalised recommendations