Equation with residuated functions

Ray A. Cuninghame-Green; Karel Zimmermann

Commentationes Mathematicae Universitatis Carolinae (2001)

  • Volume: 42, Issue: 4, page 729-740
  • ISSN: 0010-2628

Abstract

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The structure of solution-sets for the equation F ( x ) = G ( y ) is discussed, where F , G are given residuated functions mapping between partially-ordered sets. An algorithm is proposed which produces a solution in the event of finite termination: this solution is maximal relative to initial trial values of x , y . Properties are defined which are sufficient for finite termination. The particular case of max-based linear algebra is discussed, with application to the synchronisation problem for discrete-event systems; here, if data are rational, finite termination is assured. Numerical examples are given. For more general residuated real functions, lower semicontinuity is sufficient for convergence to a solution, if one exists.

How to cite

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Cuninghame-Green, Ray A., and Zimmermann, Karel. "Equation with residuated functions." Commentationes Mathematicae Universitatis Carolinae 42.4 (2001): 729-740. <http://eudml.org/doc/248794>.

@article{Cuninghame2001,
abstract = {The structure of solution-sets for the equation $F(x)=G(y)$ is discussed, where $F,G$ are given residuated functions mapping between partially-ordered sets. An algorithm is proposed which produces a solution in the event of finite termination: this solution is maximal relative to initial trial values of $x,y$. Properties are defined which are sufficient for finite termination. The particular case of max-based linear algebra is discussed, with application to the synchronisation problem for discrete-event systems; here, if data are rational, finite termination is assured. Numerical examples are given. For more general residuated real functions, lower semicontinuity is sufficient for convergence to a solution, if one exists.},
author = {Cuninghame-Green, Ray A., Zimmermann, Karel},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {systems of nonlinear equations; residuation theory; max-algebras; structure of solution sets; residuated functions; synchronization; discrete-event systems},
language = {eng},
number = {4},
pages = {729-740},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Equation with residuated functions},
url = {http://eudml.org/doc/248794},
volume = {42},
year = {2001},
}

TY - JOUR
AU - Cuninghame-Green, Ray A.
AU - Zimmermann, Karel
TI - Equation with residuated functions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2001
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 42
IS - 4
SP - 729
EP - 740
AB - The structure of solution-sets for the equation $F(x)=G(y)$ is discussed, where $F,G$ are given residuated functions mapping between partially-ordered sets. An algorithm is proposed which produces a solution in the event of finite termination: this solution is maximal relative to initial trial values of $x,y$. Properties are defined which are sufficient for finite termination. The particular case of max-based linear algebra is discussed, with application to the synchronisation problem for discrete-event systems; here, if data are rational, finite termination is assured. Numerical examples are given. For more general residuated real functions, lower semicontinuity is sufficient for convergence to a solution, if one exists.
LA - eng
KW - systems of nonlinear equations; residuation theory; max-algebras; structure of solution sets; residuated functions; synchronization; discrete-event systems
UR - http://eudml.org/doc/248794
ER -

References

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  1. Baccelli F.L., Cohen G., Olsder G.J., Quadrat J.-P., Synchronization and Linearity, An Algebra for Discrete Event Systems, Wiley, Chichester, 1992. Zbl0824.93003MR1204266
  2. Blyth T.S., Janowitz M., Residuation Theory, Pergamon, Oxford, 1972. Zbl0301.06001MR0396359
  3. Cuninghame-Green R.A., Butkovic P., The Equation A ø t i m e s x = B ø t i m e s y over ( { - ∞ } ∪ ℝ , max , + ) , Theoretical Computer Science, Special Issue on Algebra, to appear. MR1957609
  4. Cuninghame-Green R.A., Cechlarova K., Residuation in fuzzy algebra and some applications, Fuzzy Sets and Systems 71 227-239 (1995). (1995) Zbl0845.04007MR1329610
  5. Cuninghame-Green R.A., Minimax Algebra, Lecture Notes in Economics and Mathematical Systems No. 166, Springer-Verlag, Berlin, 1979. Zbl0739.90073MR0580321
  6. Walkup E.A., Borriello G., A General Linear Max-Plus Solution Technique, in Idempotency (ed. J. Gunawardena), Cambridge, 1998. Zbl0898.68035
  7. Zimmermann U., Linear and Combinatorial Optimization in Ordered Algebraic Structures, North Holland, Amsterdam, 1981. Zbl0466.90045MR0609751

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