Rough soft lattice implication algebras and corresponding decision making methods
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ORIGINAL ARTICLE
Rough soft lattice implication algebras and corresponding decision making methods Hua Zhang1 • Jianming Zhan2
Received: 14 September 2015 / Accepted: 22 January 2016 Ó Springer-Verlag Berlin Heidelberg 2016
Abstract Soft set theory and rough set theory are two mathematical tools for dealing with uncertainties. By combining rough sets and soft sets, Feng put forth rough soft set theory. In this paper, we apply rough soft set theory to lattice implication algebras. Rough soft (implicative, associative) filters with respect to a filter over lattice implication algebras are investigated. Finally, we put forward two kinds of decision making methods for rough soft lattice implication algebras. In particular, two applied examples are also given. Maybe it would be served as a foundation of more complicated soft set models in decision making methods. Keywords Lattice implication algebra (implicative, associative) filter Rough soft (implicative associative) filter Decision making
1 Introduction It is well known that non-classical logic has become an important tool for information sciences and artificial intelligence to copy with fuzzy information and uncertain information. Many-valued logic, a great extension of classical logic [3], has been a crucial direction in non& Jianming Zhan [email protected] Hua Zhang [email protected] 1
College of information Engineering, Hubei University for Nationalities, Enshi 445000, Hubei, China
2
Department of Mathematics, Hubei University for Nationalities, Enshi 445000, Hubei, China
classical logic. In order to study logical system whose propositional value is given in a lattice, Xu [38] put forth the concept of lattice implication algebras from the semantic viewpoint. Later on, Xu and Qin [40] discussed the properties of implicative filters in a lattice implicative algebra. Further, Jun [15–19] investigated some kinds of (fuzzy) filters in lattice implication algebras and some important results were obtained. After that, this logical algebras have been extensively investigated by several researchers, (cf., [23–25, 31, 41, 45]). For more details, the reader is refereed to the book [42]. Rough set theory introduced by Pawlak [30], an important mathematical approach to deal with inexact and uncertain information, has received attention on the research clues in both theory as well as applications. We know that the Pawlak approximation operators are defined by an equivalence relation, and these equivalences in Pawlak rough sets are too restrictive for many applied areas. To overcome this some more general models have been proposed (cf., e.g., [33, 43, 44, 50]). At present, this theory has been widely applied in pattern recognition, machine learning, information systems, intelligent systems, three-way decision, and others directions of knowledge, (cf., [20, 35, 36, 48, 49]). Soft set theory introduced by Molodtsov [29], a new tool for dealing with uncertainties from the viewpoint of parametrization, has been attracted by amount of researchers recently. I
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