Skip to main content
SpringerLink
Log in

Realization is universal

  • Published:
Mathematical systems theory Aims and scope Submit manuscript

Abstract

Behavior is left adjoint to minimal realization, as functors between certain categories of machines and behaviors. This gives a succinct characterization of minimal realization valid for discrete as well as linear machines, with or without finiteness condition. Realization theory is therefore expressed in contemporary algebra in a way which reveals its inner structure and suggests generalizations. An adjunction between regular sets and finite state acceptors follows as a corollary.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. A. Arbib,Theories of Abstract Automata, Prentice-Hall, Englewood Cliffs, New Jersey, 1969.

    Google Scholar 

  2. M. A. Arbib andH. P. Zeiger, On the relevance of abstract algebra to control theory,Automatica 5 (1969), 589–606.

    Google Scholar 

  3. Y. Give'on andY. Zalcstein, Algebraic structures in linear systems theory,J. Computer and System Sciences 4 (1970), 539–556.

    Google Scholar 

  4. J. A. Goguen, “Mathematical Representation of Hierarchically Organized Systems”, inGlobal Systems Dynamics (E. Attinger, editor), S. Karger, Basel, 1970.

    Google Scholar 

  5. J. A. Goguen, “Systems and Minimal Realization”, inProc. 1971 IEEE Conference on Decision and Control, Miami, Florida, 1972.

  6. J. A. Goguen, Minimal realization of machines in closed categories, to appear inBull. Amer. Math. Soc.

  7. R. E. Kalman, “Introduction to the Algebraic Theory of Linear Dynamical Systems”,Lecture Notes in Operations Research and Mathematical Economics, Vol. 11, Springer Verlag, Berlin, 1969.

    Google Scholar 

  8. S. MacLane,Categories for the Working Mathematician, Springer Verlag, Berlin, 1972.

    Google Scholar 

  9. S. MacLane andG. Birkhoff,Algebra, MacMillan, New York, 1967.

    Google Scholar 

  10. M. B. Mesarović, “Foundations for a General Systems Theory”, inViews on General Systems, Proceedings of the Second Systems Symposium at Case Institute of Technology, J. Wiley, New York, 1964.

    Google Scholar 

  11. B. Mitchell,Theory of Categories, Academic Press, New York, 1965.

    Google Scholar 

  12. A. Nerode, Linear automaton transformations,Proc. Amer. Mach. Soc. 9 (1958), 541–544.

    Google Scholar 

  13. Y. Rouchaleau, Finite-dimensional, constant, discrete-time, linear dynamical systems over some classes of commutative rings, Ph.D. Dissertation, Operations Research Dept., Stanford University, 1972.

Download references

Author information

Authors and Affiliations

  1. Mathematical Sciences Department, IBM T. J. Watson Research Center, 10598, Yorktown Heights, New York, USA

    J. A. Goguen

Authors
  1. J. A. Goguen
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Soli Deo gloria.

Written on leave from the Committee on Information Sciences, University of Chicago. Address during 1972–73: Department of Computer Science, UCLA, Los Angeles, Calif. 90024.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Goguen, J.A. Realization is universal. Math. Systems Theory 6, 359–374 (1972). https://doi.org/10.1007/BF01843493

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01843493

Keywords

Search

Navigation