For a formal context, we define an equivalence relation on the set of attributes. Through this equivalence relation, we define the lower and upper approximation operators relative to the family of semiconcepts of the formal context. We study on the two operators with the further properties that are interesting and valuable in the theory of rough set. In addition, we research on the lattice properties of all of semiconcepts in a formal context. Using this lattice, we set up two operators, and find their approximation properties in the theory of rough set. The two ways for giving approximations generalize the idea of Pawlak rough set approximations from one universal set to two non-related universal sets. We provide examples to exam the correct of the two approximation ways in this paper.
Z.Pawlak, Rough sets, International Journal of Computer and Information Sciences11 (1982), 341–356.
2.
Z.Pawlak, Rough Sets: Theoretical Aspects of Reasoning about Data, Kluwer Academic Publishers, Dordrecht, 1991.
3.
V.W.Marek, Characterizing Pawlak’s approximation operators, In:
Peters,J.F.Skowron,A.Marek,V.W.Orlowska,E.SlowinskiR.ZiarkoW.
(Eds.), Transactions on Rough Sets VII, Springer Berlin, Heidelberg, 7, 2007, pp. 140–150.
4.
Y.Y.Yao, On generalizing Pawlak approximation operators,
In:Polkowski,L.Skowron,A.
(Eds.), Rough Sets and Current Trends in Computing, Proceedings ofthe First International Conference, RSCTC’98, LNAI1424,1998, pp. 298–307.
5.
E.A.Marei, Rough set approximations on a semi bitopolog-ical view, International Journal of Scientific and Innovative Mathematical Research3(12) (2015), 59–70.
6.
F.F.Xu, Y.Y.Yao and D.Q.Miao, Rough set approximations in formal concept analysis and knowledge spaces,
In: An,A.et al. (Eds.), ISMIS 2008, LNAI 4994, Springer-Verlag Berlin, Heidelberg, 2008, pp. 319–328.
7.
Y.Y.Yao, Relational interpretations of neighborhood operators and rough set approximation operators, Information Sciences111(1-4) (1998), 239–259.
8.
Y.Y.Yao, The two sides of the theory of rough sets, Knowledge-Based Systems80 (2015), 67–77.
9.
Z.Pawlak and A.Skowron, Rough sets: Some extensions, Information Sciences177(1) (2007), 28–40.
10.
Y.Qian, J.Liang, W.Pedrycz and C.Dang, Positive approximation: An accelerator for attribute reduction in rough set theory, Artificial Intellignece174 (2010), 597–618.
11.
H.M.Abu-Donia, New rough set approximation spaces, Abstract and Applied Analysis2013 (2013), 7. Article ID 189208.
12.
R.Wille, Restructuring lattice theory: An approach based on hierarchies of concepts,
Rival,I.
In: (Eds.), Ordered Sets, Reidel, Dordrecht, 1982, pp. 445–470.
B.Vormbrock, Complete subalgebras of semiconcept algebras and protoconcept algebras,
Ganter,B.Godin,R.
In: (Eds.), ICFCA 2005, LNAI 3403, Springer-Verlag Berlin, Heidelberg, 2005, pp. 329–343.
15.
B.Vormbrock and R.Wille, Semiconcept and protoconcept algebras: The basic theorems,
Ganter,B.
In: et al. (Eds.) Formal Concept Analysis, LNAI 3626, SpringerVerlag Berlin, Heidelberg, 2005, pp. 34–48.
16.
P.Luksch and R.Wille, A mathematical model for conceptual knowledge systems,
Bock,H.H.and
Ihm,P.
In: (Eds.) Classification, Data Analysis, and Knowledge Organisation, Springer-Verlag Berlin, Heidelberg, 1991, pp. 156–162.
J.Klinger, Simple semiconcept graphs: A Boolean logic approach,
In: Delugach,H. and Stumme,G.
(Eds.), ICCS 2001, LNAI 2120, Springer-Verlag Berlin, Heidelberg, 2001, pp. 101–114.
19.
J.Klinger, Semiconcept graphs with variables,
In: U.Priss, D.Corbett and G.Angelova
(Eds.), ICCS2002, LNAI2393, Springer-Verlag Berlin, Heidelberg, 2002, pp. 369–381.
20.
G.Malik, An extension of the theory of information flow to semiconcept and protoconcept graphs,
In: Wolff,K.E.et al. (Eds.), ICCS2004, LNAI3I27, Springer-Verlag Berlin, Heidelberg, 2004, pp. 213–226.
21.
P.Balbiani, Deciding the word problem in pure double Boolean algebras, Journal of Applied Logic19 (2012), 260–273.
22.
S.K.M.Wong, L.S.Wang and Y.Y.Yao, Interval structure: A frame work for representing uncertain information,
Dubois,D.Wellman,M.P.
In: (Eds.), Proceedings of the Eighth Annual Conference on Uncertainty in Artificial Intelligence, 1992, pp. 336–343.
23.
G.L.Liu, Rough set theory based on two universal sets and its applications, Knowledge-Based Systems23 (2010), 110–115.
24.
S.Safari and R.Hooshmandasl, On twelve types of covering-based rough sets, Springer Plus5 (2016), 1003.
25.
H.Mao and R.Kang, Variable precision rough set approximations in concept lattice, Journal of Progressive Research in Mathematics2(1) (2015), 47–56.
26.
W.Ziarko, Variable precision rough set model, Journal of Computer and System Sciences46 (1993), 39–59.
27.
Y.Y.Yao, Rough set approximations: A concept analysis point of view,
In: Ishibuchi,H.
(Eds.), Computational Intelligence—Volume I, Eycyclopedia of Life Support Systems (EOLSS), Abu Dhabi, U.A.E., 2015, pp. 282–296.
