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Adjunctions on the lattices of partitions and of partial partitions

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Abstract

The complete lattice Π(E) of partitions of a space E has been extended into Π*(E), the one of partial partitions of E (where the space covering axiom is removed). We recall the main properties of Π*(E), and exhibit two adjunctions (residuations) between Π(E) and Π*(E). Given two spaces E 1 and E 2 (distinct or equal), we analyse adjunctions between Π*(E 1) and Π*(E 2), in particular those where the lower adjoint applies a set operator to each block of the partial partition; we also show how to build such adjunctions from adjunctions between \({\mathcal{P}(E_2)}\) and \({\mathcal{P}(E_2)}\) (the complete lattices of subsets of E 1 and E 2). They are then extended to adjunctions between Π(E 1) and Π(E 2). We obtain as particular case the adjunction on Π(E) that was defined by Serra (for the upper adjoint) and Ronse (for the lower adjoint). We also study dilations from Π*(E 1) to an arbitrary complete lattice L; a particular case is given, for \({L \subseteq [0,+\infty]}\) , by ultrametrics; then the adjoint erosion provides the corresponding hierarchy. We briefly discuss possible applications in image processing and in data clustering.

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References

  1. Bauer A., Birkedal L., Scott D.: Equilogical spaces. Theor. Comput. Sci. 315, 35–59 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Benzécri, J.P.: L’Analyse de Données, I: la Taxinomie. Dunod (1973)

  3. Birkhoff, G.: Lattice Theory, American Mathematical Society Colloquium Publications, vol. 25, 8th printing, 3rd edn. American Mathematical Society, Providence (1995)

  4. Bloch I., Heijmans H., Ronse C.: Mathematical morphology. In: Aiello, M., Pratt-Hartmann, I., van Benthem, J. (eds) Handbook of Spatial Logics, Chap. 14., pp. 857–944. Springer, Berlin (2007)

    Chapter  Google Scholar 

  5. Blyth T.: Lattices and Ordered Algebraic Structures. Springer, London (2005)

    MATH  Google Scholar 

  6. Bruns G.: A lemma on directed sets and chains. Arch. Math. 18, 561–563 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  7. Davey B., Priestley H.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press, Cambridge, UK (2002)

    MATH  Google Scholar 

  8. Denecke, K., Erné, M., Wismath, S. (eds): Galois Connections and Applications. Kluwer, Dordrecht (2004)

    MATH  Google Scholar 

  9. Dras̆kovic̆ová H.: The lattice of partitions in a set. Acta Fac. Rerum Natur. Univ. Comenian. Math. Publ. 24, 37–65 (1970)

    Google Scholar 

  10. Dras̆kovic̆ová H.: Congruence relations on the lattice of partitions in a set. Matematický C̆asopis (Slovenska Akademia Vied) 21, 141–153 (1971)

    Google Scholar 

  11. Dubreil P., Dubreil-Jacotin M.: Théorie algébriques des relations d’équivalence. J. Math. Pures Appl. (Ser. IX) 18, 63–95 (1939)

    MATH  Google Scholar 

  12. Duquenne V., Monjardet B.: Relations binaires entre partitions. Math. Sci. Hum. 80, 5–37 (1982)

    MathSciNet  MATH  Google Scholar 

  13. Gierz G., Hofmann K., Keimel K., Lawson J., Mislove M., Scott D.: Continuous Lattices and Domains, Encyclopedia of Mathematics and its Applications, vol 93. Cambridge University Press, Cambridge (2003)

    Google Scholar 

  14. Grätzer G.: General Lattice Theory, 2nd edn. Birkhäuser, Basel (2003)

    MATH  Google Scholar 

  15. Hadwiger H.: Minkowskische Addition und Subtraktion beliebiger Punktmengen und die Theoreme von Erhard Schmidt. Math. Z. 53, 210–218 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  16. Heijmans H.: Morphological Image Operators. Advances in Electronics and Electron Physics Series. Academic Press, Boston (1994)

    Google Scholar 

  17. Heijmans H., Ronse C.: The algebraic basis of mathematical morphology – part I: Dilations and erosions. Comput. Vis. Graph. Image Process. 50(3), 245–295 (1990)

    Article  MATH  Google Scholar 

  18. Johnson S.: Hierarchical clustering schemes. Psychometrika 32(3), 241–254 (1967)

    Article  Google Scholar 

  19. Jordens O., Sturm T.: Closure systems of equivalences with a local property. Math. Jpn. 36(2), 245–250 (1991)

    MathSciNet  MATH  Google Scholar 

  20. Jordens O., Sturm T.: Equivalences with closed equivalence classes. Math. Jpn. 36(2), 291–304 (1991)

    MathSciNet  MATH  Google Scholar 

  21. Krasner M.: Nombres semi-réels et espaces ultramétriques. Compt. Rendus hebdomadaires des séances l’Acad. Sci. 219, 433–435 (1944)

    MathSciNet  MATH  Google Scholar 

  22. Matheron G.: Random Sets and Integral Geometry. Wiley, New York (1975)

    MATH  Google Scholar 

  23. Meyer F.: Morphological segmentation revisited. In: Bilodeau, M., Meyer, F., Schmitt, M. (eds) Space, Structure and Randomness—Contributions in Honor of Georges Matheron in the Fields of Geostatistics, Random Sets, and Mathematical Morphology, Lecture Notes in Statistics, vol. 183, pp. 315–347. Springer, Berlin (2005)

    Google Scholar 

  24. Meyer F., Najman L.: Segmentation, arbre de poids minimum et hiérarchies. In: Najman, L., Talbot, H. (eds) Morphologie Mathématique 1—approches déterministes, chap 2, pp. 201–233. Hermès Lavoisier, Paris (2008)

    Google Scholar 

  25. Minkowski H.: Volumen und Oberfläche. Math. Ann. 57, 447–495 (1903)

    Article  MathSciNet  MATH  Google Scholar 

  26. Mitchell J.: Foundations for Programming Languages. MIT Press, Cambridge (1996)

    Google Scholar 

  27. Najman, L.: Ultrametric watersheds: a bijection theorem for hierarchical edge-segmentation. CoRR abs/1002.1887 (2010)

  28. Ore O.: Theory of equivalence relations. Duke Math. J. 9, 573–627 (1942)

    Article  MathSciNet  MATH  Google Scholar 

  29. Ronse C.: Partial partitions, partial connections and connective segmentation. J. Math. Imaging Vis. 32(2), 97–125 (2008)

    Article  MathSciNet  Google Scholar 

  30. Ronse C.: Reconstructing masks from markers in non-distributive lattices. Appl. Algebra Eng. Commun. Comput. 19(1), 51–85 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  31. Ronse C., Serra J.: Fondements algébriques de la morphologie. In: Najman, L., Talbot, H. (eds) Morphologie Mathématique 1—approches déterministes, Chap 2, pp. 49–96. Hermès Lavoisier, Paris (2008)

    Google Scholar 

  32. Sabelfeld A., Sands D.: A per model of secure information flow in sequential programs. Higher Order Symbol. Comput. 14(1), 59–91 (2001)

    Article  MATH  Google Scholar 

  33. Serra J.: Image Analysis and Mathematical Morphology. Academic Press, London (1982)

    MATH  Google Scholar 

  34. Serra, J. (eds): Image Analysis and Mathematical Morphology, II: Theoretical Advances. Academic Press, London (1988)

    Google Scholar 

  35. Serra J.: Mathematical morphology for Boolean lattices. In: Serra, J. (eds) Image Analysis and Mathematical Morphology, II: Theoretical Advances, Chap 2, pp. 37–58. Academic Press, London (1988)

    Google Scholar 

  36. Serra J.: Morphological operators for the segmentation of colour images. In: Bilodeau, M., Meyer, F., Schmitt, M. (eds) Space, Structure and Randomness—Contributions in Honor of Georges Matheron in the Fields of Geostatistics, Random Sets, and Mathematical Morphology, Lecture Notes in Statistics, vol. 183, pp. 223–255. Springer, Berlin (2005)

    Google Scholar 

  37. Serra J.: A lattice approach to image segmentation. J. Math. Imaging Vis. 24(1), 83–130 (2006)

    Article  MathSciNet  Google Scholar 

  38. Soille P.: On morphological operators based on rank filters. Pattern Recogn. 35(2), 527–535 (2002)

    Article  MATH  Google Scholar 

  39. Soille P.: Morphological Image Analysis: Principles and Applications, 2nd edn. Springer, Berlin (2003)

    MATH  Google Scholar 

  40. Soille, P.: On genuine connectivity relations based on logical predicates. In: Proceedings of 14th International Conference on Image Analysis and Processing, Modena, Italy, pp. 487–492. IEEE Computer Society Press (2007)

  41. Soille P.: Constrained connectivity for hierarchical image partitioning and simplification. IEEE Trans. Pattern Anal. Mach. Intell. 30(7), 1132–1145 (2008)

    Article  Google Scholar 

  42. Soille P., Grazzini J.: Advances in constrained connectivity. In: Coeurjolly, D., Sivignon, I., Tougne, L., Dupond, F. (eds) Discrete Geometry for Computer Imagery, LNCS, vol. 4992, pp. 423–433. Springer, Berlin (2008)

    Chapter  Google Scholar 

  43. Sturm T.: Verbände von Kernen isotoner Abbildungen. Czekoslovak Math. J. 22(97), 126–144 (1972)

    MathSciNet  Google Scholar 

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  1. LSIIT UMR 7005 CNRS-UdS, Parc d’Innovation, Boulevard Sébastien Brant, BP 10413, 67412, Illkirch Cedex, France

    Christian Ronse

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  1. Christian Ronse
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Correspondence to Christian Ronse.

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Ronse, C. Adjunctions on the lattices of partitions and of partial partitions. AAECC 21, 343–396 (2010). https://doi.org/10.1007/s00200-010-0129-x

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  • DOI: https://doi.org/10.1007/s00200-010-0129-x

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