%0 Journal Article
%T -Approximation: A New Approach to Algebraic Approximation
%A M. R. Hooshmandasl
%A A. Shakiba
%A A. K. Goharshady
%A A. Karimi
%J Journal of Discrete Mathematics
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/909684
%X We intend to study a new class of algebraic approximations, called -approximations, and their properties. We have shown that -approximations can be used for applied problems which cannot be modeled by inclusion based approximations. Also, in this work, we studied a subclass of -approximations, called -approximations, and showed that this subclass preserves most of the properties of inclusion based approximations but is not necessarily inclusionbased. The paper concludes by studying some basic operations on -approximations and counting the number of -min functions. 1. Introduction Uncertainty is often present in real-life applications. Uncertainty in noncrisp sets is characterized by nonempty boundary regions, in which nothing can be said about their elements with certainty. In classical set theory, a subset of a universe induces a partition over that universe. This partition can be interpreted as a knowledge on elements of ; that is, elements in are indiscernible to each other and also the same thing holds for items in . This knowledge may be improved to another partition, for example, , whose items in each partition are indiscernible to each other. In consequence, for a subset of , the problem of whether belongs to or not, with respect to knowledge , may become undecidable; that is, some elements indiscernible to with respect to knowledge may be in , whereas some other indiscernible elements to with knowledge may not belong to . To cope with such uncertainty, some tools were invented such as the Dempster-Shafer theory of evidence [1], theory of fuzzy sets [2¨C5], and theory of rough sets [6¨C8]. Rough set theory and fuzzy set theory are two independent approaches for uncertainty. There is a connection between rough set theory and Dempster-Shafer theory. Strictly speaking, lower and upper approximations of rough set theory correspond to the inner and outer reductions from Dempster-Shafer theory [9]. Rough set theory and its generalizations are all based on the inclusion relation [7, 8, 10¨C15], which is a limitation in approximations. In this work, we introduce a new concept named -approximation set. This concept is independent of inclusion relation and contains rough sets and their generalizations as special cases. We provide some examples of approximations using this new concept, which cannot be obtained by rough set theory. This paper is organized as follows. The notion of -approximation sets is proposed in Section 2, followed by considering some operations on them. The definition of conditioned rough sets is proposed in Section 3 and the number of such
%U http://www.hindawi.com/journals/jdm/2014/909684/