A Category-Theoretic Approach to Systems in a Fuzzy World

  • Michael A. Arbib
  • Ernest G. Manes
Part of the Boston Studies in the Philosophy of Science book series (BSPS, volume 31)


The last 30 years have seen the growth of a new branch of mathematics called Category Theory which provides a general perspective on many different branches of mathematics. Many workers (see Lawvere, 1972) have argued that it is category theory, rather than Set Theory, that provides the proper setting for the study of the Foundations of Mathematics.


Fuzzy System Category Theory Intuitionistic Logic Sequential Machine Forgetful Functor 
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  1. Arbib, M. A. and Manes, E. G.: 1974a, ‘Machines in a Category: An Expository Introduction’, SIAM Review 16, 163–192.CrossRefGoogle Scholar
  2. Arbib, M. A. and Manes, E. G.: 1974b, ‘Foundations of System Theory: Decomposable Machines’, Automatica 10, 285–302.CrossRefGoogle Scholar
  3. Arbib, M. A. and Manes, E. G.: 1975a, Arrows, Structures, and Functors: The Categorical Imperative, Academic Press, New York.Google Scholar
  4. Arbib, M. A. and Manes, E. G.: 1975b, ‘Fuzzy Machines in a Category’, Bull Aust.Math. Soc. 13, 169–210.CrossRefGoogle Scholar
  5. Athans, M. and Falb, P. L.: 1966, Optimal Control, McGraw-Hill.Google Scholar
  6. Bainbridge, E. S.: 1975, ‘Addressed Machines and Duality’, in E. G. Manes (ed.), Category Theory Applied to Computation and Control, Lecture Notes in Computer Science 25, 93–98, Springer-Verlag, Heidelberg.Google Scholar
  7. Bobrow, L. S. and Arbib, M. A.: 1974, Discrete Mathematics, Saunders, Philadelphia.Google Scholar
  8. Ehrig, H., Kiermeier, K.-D., Kreowski, M.-J, and Kuhnel, W.: 1974, Universal Theory of Automata: A Categorical Approach, Teubner.Google Scholar
  9. Goguen, J. A.: 1967, ‘L-Fuzzy Sets’, J. Math. Anal Appl. 18, 145–174.CrossRefGoogle Scholar
  10. Goguen, J. A.: 1969, ‘The Logic of Inexact Concepts’, Synthese 19, 325–373.CrossRefGoogle Scholar
  11. Goguen, J. A.: 1972, ‘Minimal Realization of Machines in Closed Categories’, Bull. Amer. Math. Soc. 78, 777 - 783.CrossRefGoogle Scholar
  12. Goguen, J. A.: 1973, ‘Realization is Universal’, Math. Sys. Th. 6, 359–374.CrossRefGoogle Scholar
  13. Goguen, J. A.: 1974, ‘Concept Representation in Natural and Artificial Languages: Axioms, Extensions and Applications for Fuzzy Sets’, Int. J. Man-Machine Studies 6, 513–561.CrossRefGoogle Scholar
  14. Goguen, J. A., Thatcher, J. W., Wagner, E. G., and Wright, J. B.: 1973, ‘A Junction Between Computer Science and Category Theory, I: Basic Concepts and Examples (Part 1)’, IBM Research Report RC 4526, T. J. Watson Research Center.Google Scholar
  15. Kalman, R. E., Falb, P. L., and Arbib, M. A.: 1969, Topics in Mathematical System Theory, Mc-Graw Hill.Google Scholar
  16. Lawvere, F. W. (ed.): 1972, Toposes, Algebraic Geometry and Logic, Lecture Notes in Mathematics 274, Springer-Verlag.Google Scholar
  17. MacLane, S.: 1971, Categories for the Working Mathematician, Springer-Verlag.Google Scholar
  18. Manes, E. G.: 1975a, Algebraic Theories, Springer-Verlag.Google Scholar
  19. Manes, E. G. (ed.): 1975b, Category Theory Applied to Computation and Control Proceedings of the First International Symposium, Lecture Notes in Computer Science 25, Springer-Verlag, Heidelberg.Google Scholar
  20. Padulo, L. and Arbib, M. A.: 1974, Systems Theory, Saunders, Philadelphia.Google Scholar
  21. Schank,R. C. and Colby, K. M.(eds.): 1973, Computer Models of Thought and Language, W. H. Freeman.Google Scholar
  22. Schützenberger, M. P.: 1962, ‘On a Theorem of R. Jungen’, Trans. Amer. Math. Soc. 13, 885–890.Google Scholar
  23. Thom, R.: 1972, Stabilité structurelle et morphogénèse, W. A. Benjamin, Inc.Google Scholar
  24. Zadeh, L.: 1965, ‘Fuzzy Sets’, Inform. Control 8, 338–353.CrossRefGoogle Scholar

Copyright information

© D. Reidel Publishing Company 1983

Authors and Affiliations

  • Michael A. Arbib
    • 1
  • Ernest G. Manes
    • 1
  1. 1.Computer and Information Science and MathematicsUniversity of MassachusettsAmherstUSA

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