Kybernetika 55 no. 2, 252-272, 2019

A binary operation-based representation of a lattice

Mourad Yettou, Abdelaziz Amroune and Lemnaouar ZedamDOI: 10.14736/kyb-2019-2-0252

Abstract:

In this paper, we study and characterize some properties of a given binary operation on a lattice. More specifically, we show necessary and sufficient conditions under which a binary operation on a lattice coincides with its meet (resp. its join) operation. Importantly, we construct two new posets based on a given binary operation on a lattice and investigate some cases that these two posets have a lattice structure. Moreover, we provide some representations of a given lattice based on these new constructed lattices.

Keywords:

binary operation, lattice, neutral element, lattice representation

Classification:

06B05, 06B15

References:

  1. M. Ashraf, S. Ali and C. Haetinger: On derivations in rings and their applications. Aligarh Bull. Math. 25 (2006), 79-107.   CrossRef
  2. B. Bede: Mathematics of Fuzzy Sets and Fuzzy Logic. Springer, Berlin 2013.   CrossRef
  3. G. Beliakov, A. Pradera and T. Calvo: Aggregation Functions: A Guide for Practitioners. Springer, Heidelberg 2007.   CrossRef
  4. G. Birkhoff: Lattice Theory. Third edition. Amer. Math. Soc., Providence 1967.   CrossRef
  5. T. S. Blyth: Set theory and abstract algebra. Longman, London, New York 1975.   CrossRef
  6. G. D. Cooman and E. E. Kerre: Order norms on bounded partially ordered sets. J. Fuzzy Math. 2 (1994), 281-310.   CrossRef
  7. B. A. Davey and H. A. Priestley: Introduction to Lattices and Order. Second edition. Cambridge University Press, 2002.   DOI:10.1017/cbo9780511809088
  8. D. S. Dummit and R. M. Foote: Abstract Algebra. Third edition. Hoboken, Wiley 2004.   CrossRef
  9. L. Ferrari: On derivations of lattices. Pure Math. Appl. 12 (2001), 365-382.   CrossRef
  10. G. Grätzer and F. Wehrung: Lattice Theory: Special Topics and Applications. Volume 1. Springer International Publishing Switzerland, 2014.   DOI:10.1007/978-3-319-06413-0
  11. G. Grätzer and F. Wehrung: Lattice theory: Special Topics and Applications. Volume 2. Springer International Publishing Switzerland, 2016.   DOI:10.1007/978-3-319-44236-5
  12. R. Halaš and J. Pócs: On the clone of aggregation functions on bounded lattices. Inform. Sci. 329 (2016), 381-389.   DOI:10.1016/j.ins.2015.09.038
  13. T. Jwaid, B. De Baets, J. Kalická and R. Mesiar: Conic aggregation functions. Fuzzy Sets Systems 167 (2011), 3-20.   DOI:10.1016/j.fss.2010.07.004
  14. F. Karaçal and M. N. Kesicio\v glu: A t-partial order obtained from t-norms. Kybernetika 47 (2011), 300-314.   CrossRef
  15. F. Karaçal and R. Mesiar: Aggregation functions on bounded lattices. Int. J. General Systems 46 (2017), 37-51.   DOI:10.1080/03081079.2017.1291634
  16. B. Kolman, R. C. Busby and S. C. Ross: Discrete Mathematical Structures. Fourth edition. Prentice-Hall, Inc., 2003.   CrossRef
  17. M. Komorníková and R. Mesiar: Aggregation functions on bounded partially ordered sets and theirs classification. Fuzzy Sets Systems 175 (2011), 48-56.   DOI:10.1016/j.fss.2011.01.015
  18. R. Lidl and G. Pilz: Applied Abstract Algebra. Second edition. Springer-Verlag, New York, Berlin, Heidelberg 1998.   DOI:10.1007/978-1-4757-2941-2
  19. S. Lipschutz: Discrete Mathematics. Third edition. McGraw-Hill, 2007.   DOI:10.1201/b13782
  20. R. Martínez, J. Massó, A. Neme and J. Oviedo: On the lattice structure of the set of stable matchings for a many to one model. Optimization 50 (2001), 439-457.   DOI:10.1080/02331930108844574
  21. J. Medina: Characterizing when an ordinal sum of t-norms is a t-norm on bounded lattices. Fuzzy Sets and Systems 202 (2012), 75-88.   DOI:10.1016/j.fss.2012.03.002
  22. R. Mesiar and M. Komorníková: Aggregation functions on bounded posets. 35 Years of Fuzzy Set Theory, Springer, Berlin, Heidelberg 261 (2010), pp. 3-17.   DOI:10.1007/978-3-642-16629-7\_1
  23. D. Ponasse and J. C. Carrega: Algèbre et tobologie boléennes. Masson, Paris 1979.   CrossRef
  24. E. P. Risma: Binary operations and lattice structure for a model of matching with contracts. Math. Soc. Sci. 73 (2015), 6-12.   DOI:10.1016/j.mathsocsci.2014.11.001
  25. S. Roman: Lattices and Ordered Sets. Springer Science and Business Media, New York 2008.   DOI:10.1007/978-0-387-78901-9
  26. A. Rosenfeld: An Introduction to Algebraic Structures. Holden-Day, San Francisco 1968.   CrossRef
  27. B. S. Schröder: Ordered Sets. Birkhauser, Boston 2003.   DOI:10.1007/978-1-4612-0053-6
  28. G. Szász: Translationen der verbände. Acta Fac. Rer. Nat. Univ. Comenianae 5 (1961), 449-453.   DOI:10.1007/bf01238704
  29. G. Szász: Derivations of lattices. Acta Sci. Math. 37 (1975), 149-154.   DOI:10.1016/s0020-1693(00)93924-0
  30. X. L. Xin, T. Y. Li and J. H. Lu: On derivations of lattices. Inform. Sci. 178 (2008), 307-316.   DOI:10.1016/j.ins.2007.08.018