Algebraic structure of square matrices over residuated lattices

Year 2019, Volume: 68 Issue: 2, 2216 - 2237, 01.08.2019

Arsham Borumand Saeid , E Eslami S Balaei

https://doi.org/10.31801/cfsuasmas.423277

Abstract

In this paper, we introduce the algebra Mnn(L) of square matrices over residuated lattice L. The operations are induced by the corresponding operations of L. It is shown that the defined algebra behaves like a residuated lattice, but there are some slight differences. The properties of this algebra with respect to special residuated lattices are
investigated. The notions of filter and ideal together with their roles are specified.

Keywords

Residuated lattices, matrices over residuated lattices, operations on matrices, filter, ideal

References

• Adak, A. K., Bhowmik, M. and Pal, M., Some properties of generalized intuitionistic fuzzy nilpotent matrices over distributive lattices, Fuzzy Inf. Eng, 4 (2012), 371-387.
• Atanassov, K., intuitionistic fuzzy sets, in V.Sgurev, ed.,Vii ITKRS Session, Sofia, Tune 1983.
• Belohlavek, R., Concept lattices and order in fuzzy logic. emphAnn. Pure Appl. Logic, 128(3) (2004), 277-298.
• Ciungu, L. C, Classes of residuated lattices, Annals of University of Craiova, Math. Comp, Sci. Ser., 33 (2006), 189-207.
• Galatos, N., Jipsen, P., Kowalski, T. and Ono, H., Residuated lattices: An Algebraic Gelimpse at Substructural Logics, Volume 151 of studies in Logic and the Foundations of Mathematics, Elsevier, Amesterdam, 2007.
• Ganter, B. and Wille, R., Formal Concepts Analysis, Mathematical Foundations. Springer, Berlin, 1990.
• Hosseinyazdi, M., The optimization problem over a distributive lattice, Journal of Global Optimization, 41 (2008), 283-298.
• Kim K. H. and Roush, F. W., Generalized fuzzy matrices, Fuzzy sets and systems, 4 (1980), 243-315.
• Pal, M., Intuitionistic fuzzy determinat, V.U.J. physical sciences, 7 (2001), 87-93.
• Pal, M., Khan, S. K. and Shyamal, A. K., Intuitionistic fuzzy matrices. Notes on Intuitionistic fuzzy sets, 8(2) (2002), 51- 62.
• Piciu, D., Algebras of Fuzzy Logic, Universitaria din Craiova, 2007.
• Sriram S. and Murugadas, P., On semiring of intuitionistic fuzzy matrices, Applied Mathematical. Sciences, 4(23) ( 2010), 1099-1105.
• Thomason, M. G., Convergence of powers of a fuzzy matrix, J. Math. Anal. Appl, 57 (1977), 476-486.
• Wilding, D., Linear algebra over semiring, university of Manchester, 2014.
• Wille. R., Restructuring lattice theory: An approach based on concepts. In I. Rival, editor, Ordered sets, volume 83 of NATO Advanced Study Institute Series, pages 445- 470, Springer, Dordrecht, 1982.
• Xiao, Y., Xue, T., Xue, Z. and Cheng, H., A new kind of intuitionistic fuzzy implication, Journal of Information and Computational Sciences, 8(13) (2011), 2839-2849.