1. bookVolume 41 (2021): Issue 2 (November 2021)
Journal Details
License
Format
Journal
eISSN
2084-0373
First Published
16 Apr 2017
Publication timeframe
2 times per year
Languages
English
access type Open Access

Generalized Rough Sets Applied to BCK/BCI-Algebras

Published Online: 06 Sep 2021
Volume & Issue: Volume 41 (2021) - Issue 2 (November 2021)
Page range: 343 - 360
Received: 24 Mar 2019
Accepted: 23 Sep 2020
Journal Details
License
Format
Journal
eISSN
2084-0373
First Published
16 Apr 2017
Publication timeframe
2 times per year
Languages
English
Abstract

The concept of a (strong) set-valued BCK/BCI-morphism in BCK/BCI-algebras is considered, and several properties are investigated. Conditions for a set-valued mapping to be a set-valued BCK/BCI-morphism are given. Using the concept of generalized approximation space, generalized rough subalgebra (ideal) in BCK/BCI-algebras are introduced, and investigate their properties. Using the concept of generalized approximation space and ideal of BCK/bCI-algebra, another type of generalized lower and upper approximations based on the ideal is considered, and then several properties are investigated.

Keywords

MSC 2010

[1] B. Davvaz, Roughness in rings, Inform. Sci. 164 (2004) 147–163. https://doi.org/10.1016/j.ins.2003.10.00110.1016/j.ins.2003.10.001 Search in Google Scholar

[2] B. Davvaz, A short note on algebraic T -rough sets, Inform. Sci. 178 (2008) 3247–3252. https://doi.org/10.1016/j.ins.2008.03.01410.1016/j.ins.2008.03.014 Search in Google Scholar

[3] B. Davvaz and M. Mahdavipour, Rough approximations in a general approximation space and their fundamental properties, Int. J. Gen. Syst. 37 (2008), 373–386. https://doi.org/10.1080/0308107070125099410.1080/03081070701250994 Search in Google Scholar

[4] W.A. Dudek, Y.B. Jun and H.S. Kim, Rough set theory applied to BCI-algebras, Quasigroups and Related Systems 9 (2002) 45–54. Search in Google Scholar

[5] Y. Huang, BCI-algebra (Science Press, Beijing, China, 2006). Search in Google Scholar

[6] K. Iséki, On BCI-algebras, Math. Seminar Notes 8 (1980) 125–130. Search in Google Scholar

[7] K. Iséki and S. Tanaka, An introduction to the theory of BCK-algebras, Math. Japon. 23 (1978) 1–26. Search in Google Scholar

[8] M. Kondo, On the structure of generalized rough sets, Inform. Sci. 176 (2006) 589–600. https://doi.org/10.1016/j.ins.2005.01.00110.1016/j.ins.2005.01.001 Search in Google Scholar

[9] M. Kryszkiewicz, Rough set approach to incomplete information systems, Inform. Sci. 112 (1998) 39–49. https://doi.org/10.1016/S0020-0255(98)10019-110.1016/S0020-0255(98)10019-1 Search in Google Scholar

[10] N. Kuroki, Rough ideals in semigroups, Inform. Sci. 100 (1997) 139–163. https://doi.org/10.1016/S0020-0255(96)00274-510.1016/S0020-0255(96)00274-5 Search in Google Scholar

[11] V. Leoreanu-Fotea and B. Davvaz, Roughness in n-ary hypergroups, Inform. Sci. 178 (2008) 4114–4124. https://doi.org/10.1016/j.ins.2008.06.01910.1016/j.ins.2008.06.019 Search in Google Scholar

[12] J. Liang, J. Wang and Y. Qian, A new measure of uncertainty based on knowledge granulation for rough sets, Inform. Sci. 179 (2009) 45–470. https://doi.org/10.1016/j.ins.2008.10.01010.1016/j.ins.2008.10.010 Search in Google Scholar

[13] J. Meng and Y.B. Jun, BCK-Algebras, Kyungmoon Sa Co. (Seoul, Korea, 1994). Search in Google Scholar

[14] Z. Meng and Z. Shi, A fast approach to attribute reduction in incomplete decision systems with tolerance relation-based rough sets, Inform. Sci. 179 (2009) 2774–2793. https://doi.org/10.1016/j.ins.2009.04.00210.1016/j.ins.2009.04.002 Search in Google Scholar

[15] D.Q. Miao, Y. Zhao, Y.Y. Yao, H.X. Li and F.F. Xu, Relative reducts in consistent and inconsistent decision tables of the Pawlak rough set model, Inform. Sci. 179 (2009) 4140–4150. https://doi.org/10.1016/j.ins.2009.08.02010.1016/j.ins.2009.08.020 Search in Google Scholar

[16] J.N. Mordeson, Rough set theory applied to (fuzzy) ideal theory, Fuzzy Sets and Systems 121 (2001) 315–324. https://doi.org/10.1016/S0165-0114(00)00023-310.1016/S0165-0114(00)00023-3 Search in Google Scholar

[17] Z. Pawlak, Rough sets, Int. J. Inform. Comp. Sci. 11 (1982) 341–356. https://doi.org/10.1007/BF0100195610.1007/BF01001956 Search in Google Scholar

[18] Z. Pawlak and A. Skowron, Rudiments of rough sets, Inform. Sci. 177 (2007) 3–27. https://doi.org/10.1016/j.ins.2006.06.00310.1016/j.ins.2006.06.003 Search in Google Scholar

[19] Z. Pawlak and A. Skowron, Rough sets: some extensions, Inform. Sci. 177 (2007) 28–40. https://doi.org/10.1016/j.ins.2006.06.00610.1016/j.ins.2006.06.006 Search in Google Scholar

[20] Z. Pawlak and A. Skowron, Rough sets and boolean reasoning, Inform. Sci. 177 (2007) 41–73. https://doi.org/10.1016/j.ins.2006.06.00710.1016/j.ins.2006.06.007 Search in Google Scholar

[21] D. Pei, On definable concepts of rough set models, Inform. Sci. 177 (2007) 4230–4239. https://doi.org/10.1016/j.ins.2007.01.02010.1016/j.ins.2007.01.020 Search in Google Scholar

[22] Y. Qian, J. Liang, D. Li, H. Zhang and C. Dang, Measures for evaluating the decision performance of a decision table in rough set theory, Inform. Sci. 178 (2008) 181–202. https://doi.org/10.1016/j.ins.2007.08.01010.1016/j.ins.2007.08.010 Search in Google Scholar

[23] A. Skowron and J. Stepaniuk, Generalized approximation spaces, in: T.Y. Lin, A.M. Wildberger (Eds.), Soft Computing, The Society for Computer Simulation (San Diego, 1995) 18–21. Search in Google Scholar

[24] J. Stepaniuk, Approximation spaces in extensions of rough set theory, in: L. Polkowski, A. Skowron (Eds.), RSCTC98, LNAI 1424 (1998) 290–297. https://doi.org/10.1007/3-540-69115-4_4010.1007/3-540-69115-4_40 Search in Google Scholar

[25] Q. Wang and J. Zhan, Rough semigroups and rough fuzzy semigroups based on fuzzy ideals, Open Math. 14 (2016) 1114–1121. https://doi.org/10.1515/math-2016-010210.1515/math-2016-0102 Search in Google Scholar

[26] W.Z. Wu and W.X. Zhang, Constructive and axiomatic approaches of fuzzy approximation operators, Inform. Sci. 159 (2004) 233–254. https://doi.org/10.1016/j.ins.2003.08.00510.1016/j.ins.2003.08.005 Search in Google Scholar

[27] W.Z. Wu, J.S. Mi and W.X. Zhang, Generalized fuzzy rough sets, Inform. Sci. 151 (2003) 263–282. https://doi.org/10.1016/S0020-0255(02)00379-110.1016/S0020-0255(02)00379-1 Search in Google Scholar

[28] [54] W.Z. Wu, Y. Leung and J.-S. Mi, On characterizations of (ℐ, 𝒯 )-fuzzy rough approximation operators, Fuzzy Sets and Systems 154 (2005) 76–102. https://doi.org/10.1016/j.fss.2005.02.01110.1016/j.fss.2005.02.011 Search in Google Scholar

[29] W.-Z. Wu and W.-X. Zhang, Neighborhood operator systems and approximations, Inform. Sci. 144 (2002) 201–217. https://doi.org/10.1016/S0020-0255(02)00180-910.1016/S0020-0255(02)00180-9 Search in Google Scholar

[30] U. Wybraniec-Skardowska, On a generalization of approximation space, Bull. Polish Acad. Sci. Math. 37 (1989) 51–65. Search in Google Scholar

[31] O.G. Xi, Fuzzy BCK-algebra, Math. Japon. 36 (1991) 935–942. Search in Google Scholar

[32] Q.-M. Xiao and Z.-L. Zhang, Rough prime ideals and rough fuzzy prime ideals in semigroups, Inform. Sci. 176 (2006) 725–733. https://doi.org/10.1016/j.ins.2004.12.01010.1016/j.ins.2004.12.010 Search in Google Scholar

[33] S. Yamak, O. Kazanci and B. Davvaz, Generalized lower and upper approximations in a ring, Inform. Sci. 180 (2010) 1759–1768. https://doi.org/10.1016/j.ins.2009.12.02610.1016/j.ins.2009.12.026 Search in Google Scholar

[34] Y. Yao, Three-way decisions with probabilistic rough sets, Inform. Sci. 180 (2010) 341–353. https://doi.org/10.1016/j.ins.2009.09.02110.1016/j.ins.2009.09.021 Search in Google Scholar

[35] Y.Y. Yao and T.Y. Lin, Generalization of rough sets using modal logic, Intel. Automat. Soft Comp., Int. J. 2 (1996) 103–120. https://doi.org/10.1080/10798587.1996.1075066010.1080/10798587.1996.10750660 Search in Google Scholar

[36] Y. Yao, On generalizing Pawlak approximation operators, LNAI 1424 (1998) 298–307. https://doi.org/10.1007/3-540-69115-4_4110.1007/3-540-69115-4_41 Search in Google Scholar

[37] Y.Y. Yao, Constructive and algebraic methods of the theory of rough sets, Inform. Sci. 109 (1998) 2–47. https://doi.org/10.1016/S0020-0255(98)00012-710.1016/S0020-0255(98)00012-7 Search in Google Scholar

[38] L.A. Zadeh, Toward a generalized theory of uncertainty (GTU) – an outline, Inform. Sci. 172 (2005) 1–40. https://doi.org/10.1109/GRC.2005.154722710.1109/GRC.2005.1547227 Search in Google Scholar

[39] W. Zhu, Relationship between generalized rough sets based on binary relation and covering, Inform. Sci. 179 (2009) 210–225. https://doi.org/10.1016/j.ins.2008.09.01510.1016/j.ins.2008.09.015 Search in Google Scholar

[40] W. Zhu, Generalized rough sets based on relations, Inform. Sci. 177 (22) (2007) 4997–5011. https://doi.org/10.1016/j.ins.2007.05.03710.1016/j.ins.2007.05.037 Search in Google Scholar

[41] W. Zhu and Fei-Yue Wang, On three types of covering rough sets, IEEE Transactions On Knowledge and Data Engineering 19 (8) (2007) 1131–1144. https://doi.org/10.1109/TKDE.2007.104410.1109/TKDE.2007.1044 Search in Google Scholar

[42] W. Zhu, Topological approaches to covering rough sets, Inform. Sci. 177 (2007) 1499–1508. https://doi.org/10.1016/j.ins.2006.06.00910.1016/j.ins.2006.06.009 Search in Google Scholar

[43] W. Zhu, Relationship among basic concepts in covering-based rough sets, Inform. Sci. 179 (2009) 2478–2486. https://doi.org/10.1016/j.ins.2009.02.01310.1016/j.ins.2009.02.013 Search in Google Scholar

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