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The Existence of Generalized Inverses of Fuzzy Matrices

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Interactions Between Computational Intelligence and Mathematics Part 2

Part of the book series: Studies in Computational Intelligence ((SCI,volume 794))

Abstract

In this paper we show that every fuzzy matrix with entries in a complete residuated lattice possess the generalized inverses of certain types, and in particular, it possess the greatest generalized inverses of these types. We also provide an iterative method for computing these greatest generalized inverses, which terminates in a finite number of steps, for example, for all fuzzy matrices with entries in a Heyting algebra. For other types of generalized inverses we determine criteria for the existence, given in terms of solvability of particular systems of linear matrix equations. When these criteria are met, we prove that there is the greatest generalized inverse of the given type and provide a direct method for its computing.

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References

  1. Baksalary, O.M., Trenkler, G.: Core inverse of matrices. Linear Multilinear Algebra 58, 681–697 (2010)

    Article  MathSciNet  Google Scholar 

  2. Bandler, W., Kohout, L.J.: Fuzzy relational products as a tool for analysis and synthesis of the behaviour of complex natural and artificial systems. In: Wang, S.K., Chang, P.P. (eds.) Fuzzy Sets: Theory and Application to Policy Analysis and Information Systems, pp. 341–367. Plenum Press, New York (1980)

    Chapter  Google Scholar 

  3. Bělohlávek, R.: Similarity relations and BK-relational products. Inf. Sci. 126, 287–295 (2000)

    Article  MathSciNet  Google Scholar 

  4. Ben-Israel, A., Greville, T.N.E.: Generalized Inverses: Theory and Applications, 2nd edn. Springer, New York (2003)

    MATH  Google Scholar 

  5. Bogdanović, S., Ćirić, M., Popović, Ž.: Semilattice Decompositions of Semigroups. University of Niš, Faculty of Economics (2011)

    Google Scholar 

  6. Cen, J.-M.: Fuzzy matrix partial orderings and generalized inverses. Fuzzy Sets Syst. 105, 453–458 (1999)

    Article  MathSciNet  Google Scholar 

  7. Chen, J.L., Patricio, P., Zhang, Y.L., Zhu, H.H.: Characterizations and representations of core and dual core inverses. Canad. Math. Bull. 60, 269–282 (2017)

    Article  MathSciNet  Google Scholar 

  8. Ćirić, M., Stanimirović, P., Ignjatović, J.: Outer and inner inverses in semigroups belonging to the prescribed Green’s equivalence classes, to appear

    Google Scholar 

  9. Crvenković, S.: On *-regular semigroups. In: Proceedings of the Third Algebraic Conference, Beograd, pp. 51–57 (1982)

    Google Scholar 

  10. Cui-Kui, Z.: On matrix equations in a class of complete and completely distributive lattices. Fuzzy Sets Syst. 22, 303–320 (1987)

    Article  MathSciNet  Google Scholar 

  11. Cui-Kui, Z.: Inverses of L-fuzzy matrices. Fuzzy Sets Syst. 34, 103–116 (1990)

    Article  MathSciNet  Google Scholar 

  12. Dolinka, I.: A characterization of groups in the class of *-regular semigroups. Novi Sad J. Math. 29, 215–219 (1999)

    MathSciNet  MATH  Google Scholar 

  13. Drazin, M.P.: A class of outer generalized inverses. Linear Algebra Appl. 436, 1909–1923 (2012)

    Article  MathSciNet  Google Scholar 

  14. Han, S.-C., Li, H.-X., Wang, J.-Y.: Resolution of matrix equations over arbitrary Brouwerian lattices. Fuzzy Sets Syst. 159, 40–46 (2008)

    Article  MathSciNet  Google Scholar 

  15. Hartwig, R.: Block generalized inverses. Arch. Ration. Mech. Anal. 61, 197–251 (1976)

    Article  MathSciNet  Google Scholar 

  16. Hashimoto, H.: Subinverses of fuzzy matrices. Fuzzy Sets Syst. 12, 155–168 (1984)

    Article  MathSciNet  Google Scholar 

  17. Howie, J.M.: Fundamentals of Semigroup Theory. Clarendon Press, Oxford (1995)

    MATH  Google Scholar 

  18. Ignjatović, J., Ćirić, M.: Moore-Penrose equations in involutive residuated semigroups and involutive quantales. Filomat 31(2), 183–196 (2017)

    Article  MathSciNet  Google Scholar 

  19. Ignjatović, J., Ćirić, M., Bogdanović, S.: On the greatest solutions to weakly linear systems of fuzzy relation inequalities and equations. Fuzzy Sets Syst. 161, 3081–3113 (2010)

    Article  MathSciNet  Google Scholar 

  20. Kim, K.H., Roush, F.W.: Generalized fuzzy matrices. Fuzzy Sets Syst. 4, 293–315 (1980)

    Article  MathSciNet  Google Scholar 

  21. Pradhan, R., Pal, M.: Some results on generalized inverse of intuitionistic fuzzy matrices. Fuzzy Inf. Eng. 6, 133–145 (2014)

    Article  MathSciNet  Google Scholar 

  22. Prasada Rao, P.S.S.N.V., Bhaskara Rao, K.P.S.: On generalized inverses of Boolean matrices. Linear Algebra Appl. 11, 135–153 (1975)

    Article  MathSciNet  Google Scholar 

  23. Rakić, D.S., Dinčić, N.C., Djordjević, D.S.: Group Moore-Penrose, core and dual core inverse in rings with involution. Linear Algebra Appl. 463, 115–133 (2014)

    Article  MathSciNet  Google Scholar 

  24. Rao, C.R.: On generalized inverses of Boolean valued matrices, presented at the Conference on Combinatorial Mathematics, Delhi (1972)

    Google Scholar 

  25. Roman, S.: Lattices and Ordered Sets. Springer, New York (2008)

    MATH  Google Scholar 

  26. Schein, B.: Regular elements of the semigroup of all binary relations. Semigroup Forum 13, 95–102 (1976)

    Article  MathSciNet  Google Scholar 

  27. Wang, G., Wei, Y., Qiao, S.: Generalized Inverses: Theory and Computations. Science Press, Beijing (2003)

    Google Scholar 

  28. Wang, Z.-D.: \(T\)-type regular \(L\)-relations on a complete Brouwerian lattice. Fuzzy Sets Syst. 145, 313–322 (2004)

    Google Scholar 

  29. Xu, S.Z., Chen, J.L., Zhang, X.X.: New characterizations for core inverses in rings with involution. Front. Math. China. 12(1), 231–246 (2017)

    Article  MathSciNet  Google Scholar 

  30. Zhu, H., Chen, J., Patricio, P.: Further results on the inverse along an element in semigroups and rings. Linear Multilinear Algebra 64, 393–403 (2016)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors are very grateful to the reviewers whose valuable comments have had a significant impact on improving the quality of the paper.

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Authors and Affiliations

  1. Faculty of Sciences and Mathematics, University of Niš, Višegradska 33, 18000, Niš, Serbia

    Miroslav Ćirić & Jelena Ignjatović

Authors
  1. Miroslav Ćirić
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  2. Jelena Ignjatović
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Corresponding author

Correspondence to Miroslav Ćirić .

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Editors and Affiliations

  1. Department of Information Technology, Széchenyi István University, Győr, Hungary

    László T. Kóczy

  2. Department of Mathematics, Faculty of Science, University of Cádiz, Cádiz, Spain

    Jesús Medina-Moreno

  3. Department of Mathematics, Faculty of Economic and Business Sciences, University of Cádiz, Cádiz, Spain

    Eloísa Ramírez-Poussa

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Ćirić, M., Ignjatović, J. (2019). The Existence of Generalized Inverses of Fuzzy Matrices. In: Kóczy, L., Medina-Moreno, J., Ramírez-Poussa, E. (eds) Interactions Between Computational Intelligence and Mathematics Part 2. Studies in Computational Intelligence, vol 794. Springer, Cham. https://doi.org/10.1007/978-3-030-01632-6_2

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