Double-Framed Soft Sets with Applications in BCK/BCI-Algebras

In order to lay a foundation for providing a soft algebraic tool in considering many problems that contain uncertainties, the notion of double-framed soft sets is introduced, and applications in BCK/BCI-algebras are discussed. The notions of double-framed soft algebras in BCK/BCI-algebras are introduced, and related properties are investigated. Characterizations of double-framed soft algebras are considered. Product and int-uni structure of double-framed soft algebras are discussed, and several examples are provided. 1. Introduction The real world is inherently uncertain, imprecise, and vague. Various problems in system identification involve characteristics which are essentially nonprobabilistic in nature [1]. In response to this situation, Zadeh [2] 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 [3]. To solve a 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 [4]. Maji et al. [5] and Molodtsov [4] 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 [4] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. Molodtsov pointed out several directions for the applications of soft sets. Worldwide, there has been a rapid growth in interest in soft set theory and its applications in recent years. Evidence of this can be found in the increasing number of high-quality articles on soft sets and related topics that have been published in a variety of international journals, symposia, workshops, and international conferences in recent years. Maji et al. [5] described the