Rough ideals based on ideal determined varieties

Author

Department of Mathematics, College of science, Persian Gulf University, Bushehr, 7516913817, Iran

Abstract

The paper is devoted to concern a relationship between rough set theory and universal algebra. Notions of lower and upper rough approximations on an algebraic structure induced by an ideal are introduced and some of their properties are studied. Also, notions of rough subalgebras and rough ideals with respect to an ideal of an algebraic structure, which is an extended notion of subalgebras and ideals in an algebraic structure, are introduced and investigated.

Keywords


[1] P. Agliano and A. Ursini, Ideals and other generalizations of congruence classes, J. Aust. Math. Soc. Ser.
A 53 (1992), 103-115.
[2] P. Agliano and A. Ursini, On subtractive varieties II: General properties, Algebra Universalis 36 (1996),
222-259.
[3] R. Biswas and S. Nanda, Rough groups and rough subgroups, Bull. Polish Acad. Sci. Math. 42 (1994),
251-254.
[4] R. Belohlavek and I. chajda, Congruence classes in regular varieties, Acta Math. Univ. Comenianae 68(1)
(1999), 71-75.
[5] Z. Bonikowski, Algebraic structures of rough sets, Rough sets, fuzzy sets and Knowledge discovery. Springer, London, (1994), 242{247.
[6] S. Burris and H. A. Sankappanavar, A Course in Universal Algebra, Springer, Berlin, 1981.
[7] B. Davvaz, Roughness in rings, Inf. Sci. 164 (2004) 147-163.
[8] B. Davvaz, A short note on algebraic T-rough sets, Inf. Sci. 178 (2008) 3247-3252.
[9] B. Davvaz and M. Mahdavipour, Roughness in modules, Inf. Sci. 176 (2006) 3658-3674.
[10] K. Denecke, M. Erne and S. L. Wismath, Galois connections and applications, volume 565. Springer, 2004.
[11] D. Dubois and H. Prade, Rough fuzzy sets and fuzzy rough sets, International Journal of General Systems 17 (1990), 191-209.
[12] F. Feng, C. Li, B. Davvaz and M.I. Ali, Soft sets combined with fuzzy sets and rough sets: a tentative
approach, Soft Computing 14 (2010), 899-911.
[13] K. Fichtner, Varieties of universal algebras with ideals, Mat. Sbornik 75(117) (1968), 445-453. (In Russian.)
[14] K. Fichtner, Eine Bermerkung ber Mannigfaltigkeiten universeller Algebren mit Idealen, Monatsh. d.
Deutsch. Akad. d. Wiss. (Berlin) 12 (1970), 21-25.
[15] B. Ganter and R. Wille, Formal Concept Analysis, Mathematical Foundations, Springer, Berlin, 1999.
[16] G. Gediga, I. Duntsch, Modal-style operators in qualitative data analysis, in: Proceedings of the 2002 IEEE International Conference in Data Mining (2002), 155-162.
[17] H.P. Gumm and A. Ursini, Ideals in universal algebras, Algebra Universalis 19 (1984), 45-55.
[18] F. Garca Pardo, I. P. Cabrera, P. Cordero and M. Ojeda-Aciego, On Galois connections and soft computing,
Lecture Notes in Computer Science, (2013) 224-235.
[19] G. Gratzer, Universal Algebra, Van Nostrand, Princeton, N.J., 1968.
[20] P. J. Higgins, Groups with multiple operators, Proc. London Math. Soc. 3(3) (1956), 366-416.
[21] M. Irfan Ali, B. Davvaz and M. Shabir, Some properties of generalized rough sets, Inf. Sci. 224 (2013),
170-179.
[22] T. Iwinski, Algebraic approach to rough sets, Bull. Pol. Acad. Sci. Math. 35 (1987) 673-683.
[23] W. Krull, Axiomatische Begrundung der allgemeinen Idealtheorie, Sitzungsberichteder Physikalisch Medizinischen Societatder Erlangen 56 (1924), 47-63.
[24] N. Kuroki and P.P. Wang, The lower and upper approximations in a fuzzy group, Inf. Sci. 90 (1996),
203-220.
[25] N. Kuroki, Rough ideals in semigroups, Inf. Sci. 100 (1997), 139-163.
[26] H. Lai, D. Zhang, Concept lattices of fuzzy contexts: Formal concept analysis vs. rough set theory, International Journal of Approximate Reasoning 50 (2009), 695-707.
[27] R. Magari, Su una classe equazionale di algebre, Ann. Mat. Pura Appl. 75(1) (1967), 277-312.
[28] Z. Pawlak, Rough sets, Int. J. Inf. Comput. Sci. 11 (1982), 341-356.
[29] J. Pomykala and J. A. Pomykala, The stone algebra of rough sets, Bull. Pol. Acad. Sci. Math. 36 (1998),
495-508.
[30] S. Rasouli, Heyting Boolean and pseudo-MV lters in residuated lattices, Journal of Multiple Valued Logic
and Soft Computing 31(4) (2018), 287-322.
[31] S. Rasouli and B. Davvaz, Lattices derived from hyperlattices, Communications in Algebra R ⃝ 38(8) (2010), 2720-2737.
[32] S. Rasouli and B. Davvaz, -relations on implicative bounded hyper BCK-algebras, Hacettepe Journal of
Mathematics and Statistics 39(4) (2010), 461-469.
[33] S. Rasouli and B. Davvaz, Roughness in MV-algebras, Inf. Sci. 180(5) (2010), 737-747.
[34] S. Rasouli and B. Davvaz, Homomorphism, Ideals and Binary Relations on Hyper-MV Algebras, Multiplevalued Logic and Soft Computing 17(1) (2011), 47-68.
[35] S. Rasouli, B. Davvaz, An Investigation on Algebraic Structure of Soft Sets and Soft Filters over Residuated Lattices, ISRN Algebra, vol. 2014, Article ID 635783, 8 pages, 2014. doi:10.1155/2014/635783.
[36] S. Rasouli and B. Davvaz, An investigation on Boolean prime lters in BL-algebras, Soft Computing 19(10)
(2015), 2743-2750.
[37] S. Rasouli and B. Davvaz, An investigation on regular relations of universal hyperalgebras, Algebraic Structures and Their Applications 5(1) (2018), 1-21.
[38] S. Rasouli and B. Davvaz, Rough lters based on residuated lattices, Knowledge and Information Systems
58(2) (2018), 399-424.
[39] S. Rasouli and A. Radfar, PMTL lters, Rl lters and PBL lters in residuated lattices, Journal of Multiple
Valued Logic and Soft Computing 29(6) (2017), 551-576.
[40] S. Rasouli, Z. Zarin and A. Hassankhani, Characterization of a new subquasivariety of residuated Lattice,
Journal of applied logics-The IfCoLog journal of logics and their applications 5(1) (2018), 33-63.
[41] A. Ursini, Sulle varieta di algebre con una buona teoria degli ideali, Boll. Unione Mat. Ital. 6(4) (1972),
90-95.
[42] A. Ursini, Osservazioni sulle varieta BIT, Boll. Unione Mat. Ital. 7(4) (1973), 205-211.
[43] A. Ursini, On subtractive varieties I, Algebra Universalis 31 (1994), 204-222.
[44] R. Wille, Restructuring lattice theory: An approach based on hierarchies of concepts, Ordered Sets, I. Rival (Ed.), Reidel, (1982), 445-470.
[45] Q. Xiaoa, Q. Lia and L. Guo, Rough sets induced by ideals in lattices, Inf. Sci. 271 (2014), 82-92.
[46] S. Yamaka, O. Kazancia and B. Davvaz, Generalized lower and upper approximations in a ring, Inf. Sci.
180 (2010) 1759-1768.
[47] Y.Y. Yao, Constructive and algebraic methods of the theory of rough sets, Inf. Sci. 109 (1998), 21-47.