The notion of intersectional soft subalgebras of a BE-algebra is introduced, and related properties are investigated. Characterization of an intersectional soft subalgebra is discussed. The problem of classifying intersectional soft subalgebras by their inclusive subalgebras will be solved. 1. Introduction In 1966, Imai and Iséki [1] and Iséki [2] introduced two classes of abstract algebras: BCK-algebras and BCI-algebras. It is known that the class of BCK-algebras is a proper subclass of the class of BCI-algebras. Ma et al. studied -tyle (interval-valued) fuzzy ideals in BCI-algebras and soft -algebras (see [3–5]). As a generalization of a BCK-algebra, H. S. Kim and Y. H. Kim [6] introduced the notion of a BE-algebra and investigated several properties. In [7], Ahn and So introduced the notion of ideals in BE-algebras. They gave several descriptions of ideals in BE-algebras. Song et al. [8] considered the fuzzification of ideals in BE-algebras. They introduced the notion of fuzzy ideals in -algebras and investigated related properties. They gave characterizations of a fuzzy ideal in BE-algebras. Various problems in system identification involve characteristics which are essentially nonprobabilistic in nature [9]. In response to this situation, Zadeh [10] introduced fuzzy set theory as an alternative to probability theory. Uncertainty is an attribute of information. In order to suggest a more general framework, the approach to uncertainty is outlined by Zadeh [11]. To solve complicated problem in economics, engineering, and environment, we cannot successfully use classical methods because of various uncertainties typical for those problems. There are three theories: theory of probability, theory of fuzzy sets, and the interval mathematics which we can consider as mathematical tools for dealing with uncertainties. But all these theories have their own difficulties. Uncertainties cannot be handled using traditional mathematical tools but may be dealt with using a wide range of existing theories such as probability theory, theory of (intuitionistic) fuzzy sets, theory of vague sets, theory of interval mathematics, and theory of rough sets. However, all of these theories have their own difficulties which are pointed out in [12]. Maji et al. [13] and Molodtsov [12] suggested that one reason for these difficulties may be due to the inadequacy of the parametrization tool of the theory. To overcome these difficulties, Molodtsov [12] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties
References
[1]
Y. Imai and K. Iséki, “On axiom systems of propositional calculi. XIV,” Proceedings of the Japan Academy, vol. 42, pp. 19–22, 1966.
[2]
K. Iséki, “An algebra related with a propositional calculus,” Proceedings of the Japan Academy, vol. 42, pp. 26–29, 1966.
[3]
X. Ma, J. Zhan, and Y. B. Jun, “New types of fuzzy ideals of BCI-algebras,” Neural Computing and Applications, vol. 21, pp. S19–S27, 2012.
[4]
X. Ma, J. Zhan, and Y. B. Jun, “Soft -algebras based on fuzzy sets,” Journal of Multiple-Valued Logic and Soft Computing, vol. 19, pp. 547–563, 2012.
[5]
X. Ma, J. Zhan, B. Davvaz, and Y. B. Jun, “Some kinds of ( )-interval-valued fuzzy ideals of BCI-algebras,” Information Sciences, vol. 178, no. 19, pp. 3738–3754, 2008.
[6]
H. S. Kim and Y. H. Kim, “On BE-algebras,” Scientiae Mathematicae Japonicae, vol. 66, no. 1, pp. 113–116, 2007.
[7]
S. S. Ahn and K. S. So, “On ideals and upper sets in -algebras,” Scientiae Mathematicae Japonicae, vol. 68, no. 2, pp. 279–285, 2008.
[8]
S. Z. Song, Y. B. Jun, and K. J. Lee, “Fuzzy ideals in BE-algebras,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 33, no. 1, pp. 147–153, 2010.
[9]
L. A. Zadeh, “From circuit theory to system theory,” Proceedings of the Institute of Radio Engineers, vol. 50, pp. 856–865, 1962.
[10]
L. A. Zadeh, “Fuzzy sets,” Information and Computation, vol. 8, pp. 338–353, 1965.
[11]
L. A. Zadeh, “Toward a generalized theory of uncertainty (GTU)—an outline,” Information Sciences, vol. 172, no. 1-2, pp. 1–40, 2005.
[12]
D. Molodtsov, “Soft set theory—first results,” Computers & Mathematics with Applications, vol. 37, no. 4-5, pp. 19–31, 1999.
[13]
P. K. Maji, A. R. Roy, and R. Biswas, “An application of soft sets in a decision making problem,” Computers & Mathematics with Applications, vol. 44, no. 8-9, pp. 1077–1083, 2002.
[14]
P. K. Maji, R. Biswas, and A. R. Roy, “Soft set theory,” Computers & Mathematics with Applications, vol. 45, no. 4-5, pp. 555–562, 2003.
[15]
D. Chen, E. C. C. Tsang, D. S. Yeung, and X. Wang, “The parameterization reduction of soft sets and its applications,” Computers & Mathematics with Applications, vol. 49, no. 5-6, pp. 757–763, 2005.
[16]
N. ？a？man, F. ？itak, and S. Enginoglu, “FP-soft set theory and its applications,” Annals of Fuzzy Mathematics and Informatics, vol. 2, no. 2, pp. 219–226, 2011.
[17]
F. Feng, “Soft rough sets applied to multicriteria group decision making,” Annals of Fuzzy Mathematics and Informatics, vol. 2, no. 1, pp. 69–80, 2011.
[18]
H. Akta？ and N. ？a？man, “Soft sets and soft groups,” Information Sciences, vol. 177, no. 13, pp. 2726–2735, 2007.
[19]
U. Acar, F. Koyuncu, and B. Tanay, “Soft sets and soft rings,” Computers & Mathematics with Applications, vol. 59, no. 11, pp. 3458–3463, 2010.
[20]
A. O. Atagün and A. Sezgin, “Soft substructures of rings, fields and modules,” Computers & Mathematics with Applications, vol. 61, no. 3, pp. 592–601, 2011.
[21]
F. Feng, Y. B. Jun, and X. Zhao, “Soft semirings,” Computers & Mathematics with Applications, vol. 56, no. 10, pp. 2621–2628, 2008.
[22]
Y. B. Jun, “Soft BCK/BCI-algebras,” Computers & Mathematics with Applications, vol. 56, no. 5, pp. 1408–1413, 2008.
[23]
Y. B. Jun, H. S. Kim, and J. Neggers, “Pseudo -algebras,” Information Sciences, vol. 179, no. 11, pp. 1751–1759, 2009.
[24]
Y. B. Jun, K. J. Lee, and C. H. Park, “Soft set theory applied to ideals in -algebras,” Computers & Mathematics with Applications, vol. 57, no. 3, pp. 367–378, 2009.
[25]
Y. B. Jun, K. J. Lee, and J. Zhan, “Soft -ideals of soft BCI-algebras,” Computers & Mathematics with Applications, vol. 58, no. 10, pp. 2060–2068, 2009.
[26]
Y. B. Jun, K. J. Lee, and A. Khan, “Soft ordered semigroups,” Mathematical Logic Quarterly, vol. 56, no. 1, pp. 42–50, 2010.
[27]
Y. B. Jun and C. H. Park, “Applications of soft sets in ideal theory of BCK/BCI-algebras,” Information Sciences, vol. 178, no. 11, pp. 2466–2475, 2008.
[28]
C. H. Park, Y. B. Jun, and M. A. ？ztürk, “Soft WS-algebras,” Communications of the Korean Mathematical Society, vol. 23, no. 3, pp. 313–324, 2008.
[29]
J. Zhan and Y. B. Jun, “Soft -algebras based on fuzzy sets,” Computers & Mathematics with Applications, vol. 59, no. 6, pp. 2037–2046, 2010.
[30]
N. ？a？man and S. Engino？lu, “Soft set theory and uni-int decision making,” European Journal of Operational Research, vol. 207, no. 2, pp. 848–855, 2010.
[31]
Y. B. Jun, M. S. Kang, and K. J. Lee, “Intersectional soft sets and applications to BCK/BCI-algebras,” Communications of the Korean Mathematical Society, vol. 28, pp. 11–12, 2013.
[32]
Y. B. Jun, K. J. Lee, and E. H. Roh, “Intersectional soft BCK/BCI-ideals,” Annals of Fuzzy Mathematics and Informatics, vol. 4, no. 1, pp. 1–7, 2012.
