Advertisement

Spectral Lattices of \(\mathbb{\overline R}_{\rm max,+}\)-Formal Contexts

  • Francisco J. Valverde-Albacete
  • Carmen Peláez-Moreno
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4933)

Abstract

In [13] a generalisation of Formal Concept Analysis was introduced with data mining applications in mind, \(\mathcal K\)-Formal Concept Analysis, where incidences take values in certain kinds of semirings, instead of the standard Boolean carrier set. Subsequently, the structural lattice of such generalised contexts was introduced in [15], to provide a limited equivalent to the main theorem of \(\mathcal K\)-Formal Concept Analysis, resting on a crucial parameter, the degree of existence of the object-attribute pairs ϕ. In this paper we introduce the spectral lattice of a concrete instance of \(\mathcal K\)-Formal Concept Analysis, as a further means to clarify the structural and the \(\mathcal K\)-Concept Lattices and the choice of ϕ. Specifically, we develop techniques to obtain the join- and meet-irreducibles of a \(\mathbb{\overline R}_{\rm max,+}\)-Concept Lattice independently of ϕ and try to clarify its relation to the corresponding structural lattice.

Keywords

Closure Operator Natural Order Concept Lattice Formal Context Confusion Matrice 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Akian, M., Bapat, R., Gaubert, S.: Handbook of Linear Algebra. In: Max-Plus Algebra, CRC Press, Boca Raton (2006)Google Scholar
  2. 2.
    Baccelli, F., et al.: Synchronization and Linearity. Wiley, Chichester (1992)zbMATHGoogle Scholar
  3. 3.
    Cohen, G., Gaubert, S., Quadrat, J.–P.: Duality and separation theorems in idempotent semimodules. Linear Algebra and Its Applications 379, 395–422 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Erné, M.: Adjunctions and Galois connections: Origins, History and Development. In: Mathematics and Its Applications, vol. 565, pp. 1–138. Kluwer Academic, Dordrecht (2004)Google Scholar
  5. 5.
    Gaubert, S.: Théorie des systèmes linéaires dans les dioïdes. Thèse, École des Mines de Paris (July 1992)Google Scholar
  6. 6.
    Golan, J.S.: Power Algebras over Semirings. With Applications in Mathematics and Computer Science. In: Mathematics and its applications, vol. 488, Kluwer Academic, Dordrecht (1999)Google Scholar
  7. 7.
    Golan, J.S.: Semirings and Their Applications. Kluwer Academic Publishers, Dordrecht (1999)zbMATHGoogle Scholar
  8. 8.
    Moreau, J.J.: Inf-convolution, sous-additivité, convexité des fonctions numériques. J. Math. pures et appl. 49, 109–154 (1970)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Sergeev, S.: Max-plus definite matrix closures and their eigenspaces. Linear Algebra and its Applications 421(2-3), 182–201 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Singer, I.: Abstract Convex Analysis. Monographs and Advanced Texts. Wiley-Interscience (1997)Google Scholar
  11. 11.
    Singer, I. (*,s)-dualities. Journal of Mathematical Sciences 115(4), 2506–2541 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Singer, I.: Some relations between linear mappings and conjugations in idempotent analysis. Journal of Mathematical Sciences 115(5), 2610–2630 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Valverde-Albacete, F.J., Peláez-Moreno, C.: Towards a generalisation of Formal Concept Analysis for data mining purposes. In: Missaoui, R., Schmidt, J. (eds.) Formal Concept Analysis. LNCS (LNAI), vol. 3874, pp. 161–176. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  14. 14.
    Valverde-Albacete, F.J., Peláez-Moreno, C.: Further Galois connections between semimodules over idempotent semirings. In: Diatta, J., Eklund, P. (eds.) Proceedings of the 4th Conference on Concept Lattices and Applications (CLA 2007), October 2007, Montpellier, pp. 199–212 (2007)Google Scholar
  15. 15.
    Valverde-Albacete, F.J., Peláez-Moreno, C.: Galois connections between semimodules and applications in data mining. In: Kuznetsov, S.O., Schmidt, S. (eds.) ICFCA 2007. LNCS (LNAI), vol. 4390, pp. 181–196. Springer, Heidelberg (2007)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Francisco J. Valverde-Albacete
    • 1
  • Carmen Peláez-Moreno
    • 1
  1. 1.Dpto. de Teoría de la Señal y de las ComunicacionesUniversidad Carlos III de MadridLeganésSpain

Personalised recommendations