On Some Algebraic Structures of Type 2 Fuzzy Multisets

Introduction

Georg Cantor (1845-1918, German mathematician and logician) created and formulated a mathematical discipline called Theory of sets which is fundamental for whole Mathematics. It grew out of his investigations of some concrete problems regarding certain types of infinite sets of real numbers. In the notion of a set a sharp and unambiguous distinction exists between the members and nonmembers (Stoll, 1963). But the modern world is full of uncertainty, ambiguity, vagueness, imprecision and inexactness. To get rid of these situations, many tools were introduced. A wide range of existing theories such as Probability Theory, Fuzzy set Theory, Rough set Theory, Vague set Theory, Soft set Theory, Interval Mathematics etc are often used as mathematical tools to overcome the uncertainties and vagueness of real life problems (Klir &Yaun, 1995; Molodtsov, 1999; Pawlak, 1982; Zadeh, 1965).

Fuzzy set theory proposed by L. Zadeh is the most successful one to understand and manipulate imperfect knowledge for solving complicated problems of modern world. In this approach sets are defined by partial membership, in contrast to crisp membership used in classical definition of a set. (Dey & Pal, 2015) presents a method to construct fuzzy complex sets from ordinary fuzzy sets by introduced the ordered sequences of membership functions. As an extension of ordinary fuzzy set Zadeh introduced the concept of type 2 fuzzy set in which membership grades are ordinary fuzzy sets itself. Type 2 fuzzy sets are very useful in circumstances where it is difficult to determine an exact membership function for a fuzzy set (Mizumoto & Tanaka,1976). So they are useful for incorporating linguistic uncertainties, eg., the words that are used in linguistic knowledge can mean different things to different people (Zadeh, 1975). By using type 2 fuzzy sets one can improve certain kinds of inference better than ordinary fuzzy sets with increasing imprecision, uncertainty and vagueness in information, type 2 fuzzy sets are gaining more popularity. The basic concepts of type 2 fuzzy set theory and its extensions, as well as some practical applications, can be found in (Mendel 2014, Mendel, John & Liu, 2001; Miyamoto, 2001; Mizumoto & Tanaka, 1981; Shinoj, Baby & John, 2015; Qin &Liu, 2015). Fuzzy surfaces are constructed from incomplete datasets or from data that contain uncertainty which has not statistical nature. The concept of level sets for the fuzzy surfaces are extended to type 2 fuzzy environment (Farahani, Rahmany & Basir, 2015).

To handle the practical problems in which there are enormous repetitions one can use multisets, a mathematical structure introduced by Cerf et al, in which repetitions of elements are allowed. In the 20th century, mathematicians formalized multisets and began to study them as a precise mathematical structure. The development of multiset theory can be seen in (Blizard, 1989; Bedregal, Beliakor, Bustince, Calvo, Mesiar & Paternain, 2012; Girish & John, 2012; Syropoulos, 2016). Compared to other traditional structures the hybrid structure of type 2 fuzzy multiset is an effective method which provide additional degrees of freedom to present the vagueness and uncertainty of the real world and sustain a realistic evaluation (Paul & John, 2015).