• Jaime Gil-Aluja
Part of the Applied Optimization book series (APOP, volume 32)


When attempting to find the mechanism of the brain of a living being that is capable of bringing to light the most elemental symptom of intelligence, one must surely pay a great deal of attention to the concept of relation. If a domestic animal is accustomed to eat from a certain container, when it hears the particular noise it makes then it will run towards it: it associates and relates, noise-food. From our very earliest childhood we are taught to improve the devices that lead to relation, by developing in the very smallest, the most diverse kinds of connections: between things and colours, effort and prizes, between certain objects and others,... Teachers are very much aware of the importance of relation in the mental progress of a child. And the truth of the matter is that whatever activity is carried on, it is difficult to imagine any reasoning whatsoever in which relations are not present as a very basic support.


Fuzzy Relation Transitive Relation Fuzzy Graph Fuzzy Sphere Boolean Matrix 
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.


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  1. 1.
    See, for example, in this respect Kaufmann, A and Gil Aluja, J: Grafos neuronales para la economla y gestion de empresas. Ed. Pirdmide. Madrid 1995, page 18.Google Scholar
  2. 2.
    We have specifically place zeros when µ = 0 with the object of making obvious the custom of doing without the arcs in this case, when resorting to the arrow form representation.Google Scholar
  3. 3.
    In these types of square matrices we are going to use, for easy and visual effectiveness, the type of arrow form representation we show in this case.Google Scholar
  4. 4.
    We are not referring here to the case in which the intensity coincides in the relation a, with ai with that of the relation al with a,.Google Scholar
  5. 5.
    It would be advisable not to confuse “class of similarity” with “class of equivalence” or “strongly connected sub-graph”. In fact, for matrix [B], the sub-set formed byy fat, a4, as) is a class of equivalence but not a class of similarity, while in [B] it is one thing and the other.Google Scholar
  6. 6.
    Here we disregard the normal rules used for converting a fuzzy relation into a boolean relation, since we are only attempting to illustrate, by an example, the concept of pre-order.Google Scholar
  7. 7.
    We repeat the necessity to distinguish between similarity and equivalency class. The formation of maximum strongly connected sub-graphs or equivalency classes will be dealt with later on.Google Scholar
  8. 8.
    See in this respect Kaufmann, A, and Gil Aluja, J.: Nuevas técnicas para la direccidn estratégica. Publicaciones Universidad de Barcelona, Barcelona, 1991, page 117–147.Google Scholar
  9. 9.
    The development of the relations of incidence has permitted the formulation of the notion of forgotten effect. Anyone interested in this subject can consult: Kaufmann, A. and Gil Aluja, J.: Modelos para la investigaciOn de efectos olvidados. Ed Milladoiro. Santiago de Compostela, 1988.Google Scholar
  10. 10.
    Kaufmann, A. and Gil Aluja, J.: Modelos para la investigaciOn de efectos olvidados. Ed. Milladoiro. Santiago de Compostela, 1988.Google Scholar
  11. 11.
    Kaufmann, A. and Gil Aluja, J.: Técnicas operativas de gestion para el tratamiento de la incertidumbre. Ed. Hispano Europea. Barcelona 1987, pages 147–148.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1999

Authors and Affiliations

  • Jaime Gil-Aluja
    • 1
    • 2
  1. 1.Barcelona UniversitySpain
  2. 2.Paris-Dauphine UniversityFrance

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