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Abstract

The Mareay’s rough set has been regarded as an approximation processing model in an approximation space induced by an arbitrary binary relation on the single universe. This is an effective mathematical tool for dealing with uncertain knowledge and vagueness in data for the single universe. Based on this induced notion, we firstly establish and verify two new classes, called a successor class and a core of a successor class induced by an arbitrary binary relation between two universes. Then we propose a generalization of the Mareay’s rough set in an approximation space based on cores of successor classes induced by an arbitrary binary relation between two universes, with a corresponding example. Some interesting algebraic properties of the new approximation processing model are investigated. We develop the use of the novel rough set in a semigroup under a preorder and compatible relation. We establish the notions of rough semigroups, rough ideals and rough completely prime ideals. Then we provide sufficient conditions of rough semigroups, rough ideals and rough completely prime ideals. Under homomorphism problems in semigroups, the relationship between the rough semigroup (resp. rough ideal and rough completely prime ideal) and the homomorphic image of the rough semigroup (resp. rough ideal and rough completely prime ideal) is demonstrated.

Keywords

Rough sets approximation spaces semigroups rough semigroups rough ideals rough completely prime ideals binary relations preorder relations compatible relations.

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References

Pawlak , Rough sets, International Journal of Information and Computer Security 11 (1982), 341–356.

Li ,

Mei and

Yuejin , Incomplete decision contexts: Approximate concept construction, rule acquisition and knowledge reduction, International Journal of Approximate Reasoning 54(1) (2013), 149–165.

Y.Y.

Yao , Interval sets and three-way concept analysis in incomplete contexts, International Journal of Machine Learning and Cybernetics 8(1) (2017), 3–20.

Giampiero ,

Davide ,

Tommaso and

Federico , Rough set theory and digraphs, Journal of Intelligent and Fuzzy Systems 159(4) (2017), 291–325.

Maini ,

Kumar and

Misra , Rough set based feature selection using swarm intelligence with distributed sampled initialisation, Proceedings of the 6th IEEE International Conference on Computer Applications In Electrical Engineering-Recent Advances, 2017.

Yu ,

Yang ,

Lin ,

Meng and

Wu , Application of rough set theory for NVNA phase reference uncertainty analysis in hybrid information system, Computers and Electrical Engineering 69 (2018), 893–906.

Zouache and

F.B.

Abdelaziz , A cooperative swarm intelligence algorithm based on quantum-inspired and rough sets for feature selection, Computers and Industrial Engineering 115 (2018), 26–36.

Biswas and

Nanda , Rough groups and rough subgroups, Bulletin of the Polish Academy of Sciences Mathematics 42 (1994), 251–254.

Kuroki and

J.N.

Mordeson , Structure of rough sets and rough groups, Journal of Fuzzy Mathematics 5(1) (1997), 183–191.

10.

Waqas ,

Waqas and

Shin Min , A comparison between lower and upper approximations in groups with respect to group homomorphisms, Journal of Intelligent and Fuzzy Systems 35(1) (2018), 693–703.

11.

Kuroki , Rough ideals in semigroup, Information Sciences 100 (1997), 139–163.

12.

Q.M.

Xiao and

Z.L.

Zhang , Rough prime ideals and rough fuzzy prime ideals in semigroups, Information Sciences 176 (2006), 725–733.

13.

Yaqoob ,

Aslam and

Chinram , Rough prime bi-ideals and rough fuzzy prime bi-ideals in semigroups, Annals of Fuzzy Mathematics and Informatics 3 (2012), 203–211.

14.

Wang and

Zhan , Rough semigroups and rough fuzzy semigroups based on fuzzy ideals, Open Mathematics 14 (2016), 1114–1121.

15.

Y.B.

Jun , Roughness of ideals in BCK-algebras, Scientiae Mathematicae Japonicae 57(1) (2003), 165–169.

16.

Davvaz , Roughness in rings, Information Sciences 164(1) (2004), 147–163.

17.

Davvaz , Roughness based on fuzzy ideals, Information Sciences 176 (2006), 2417–2437.

18.

Davvaz and

Mahdavipour , Roughness in modules, Information Sciences 176 (2006), 3658–3674.

19.

M.I.

Ali ,

Shabir and

Tanveer , Roughness in hemirings, Neural Computing and Applications 21 (2012), 171–180.

20.

Yang and

Xu , Roughness in quantales, Information Sciences 220 (2013), 568–579.

21.

Rehman ,

Park ,

S.I.

Ali Shah and

Ali , On generalized roughness in LA-semigroups, Mathematics 6(7) (2018), 1–8.

22.

Songtao ,

Xiaohong ,

Chunxin and

Choonkil , Multigranulation rough filters and rough fuzzy filters in Pseudo-BCI algebras, Journal of Intelligent and Fuzzy Systems 34(6) (2018), 4377–4386.

23.

C.L.

Nehaniv and

Ito , Algebraic Engineering, Proceedings of the International Workshop on Formal Languages and Computer Systems, 1999.

24.

Y.Y.

Yao and

T.Y.

Lin , Generalization of rough sets using modal logic, Intelligent Automation and Soft Computing International Journal 2 (1996), 103–120.

25.

Y.Y.

Yao , Constructive and algebraic methods of the theory of rough sets, Information Sciences 109 (1998), 21–47.

26.

Y.Y.

Yao , Relational interpretations of neighborhood operators and rough set approximation operators, Information Sciences 111 (1998), 239–259.

27.

Slowinski and

Vanderpooten , A generalized definition of rough approximations based on similarity, IEEE Transactions on Knowledge and Data Engineering 12(2) (2000), 331–336.

28.

Greco ,

Matarazzo and

Slowinski , Rough approximation by dominance relations, International Journal of Intelligent Systems 17 (2002), 153–171.

29.

Mareay , Generalized rough sets based on neighborhood systems and topological spaces, Journal of the Egyptian Mathematical Society 24 (2016), 603–608.

30.

J.M.

Howie , An Introduction to semigroup theory, Academic Press, 1976.

31.

J.N.

Mordeson ,

D.S.

Malik and

Kuroki , Fuzzy semigroups, Springer-Verlag, Berlin, Heidelberg, New York, 2010.

32.

Zach , Sets, logic, and computation. University of Calgary, 2017.

Generalizations of rough sets induced by binary relations approach in semigroups

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