Some general comments on fuzzy sets of type-2

Type-2 fuzzy sets were introduced by Zadeh in 1975 in a series of papers. The basic idea is to generalize the notion of ordinary fuzzy sets (type-1), a¤ording wider applicability. This paper gives a general overview of some of the mathematical aspects of type-2 fuzzy set theory, emphasizing precisely how type-2 generalizes type-1 and interval-valued fuzzy set theory. We will summarize the theory, at least what we know of it, and make some general comments on the mathematical setting and signi…cance of various theorems. One purpose of these comments is to show that type-2 fuzzy sets are, in a strict mathematical sense, an appropriate generalization of type-1 and of interval-valued fuzzy sets. The fundamental object for a “fuzzy set theory”is an algebra of truth values. For ordinary fuzzy sets, or fuzzy sets of type-1, that algebra, which we denote by I, is the unit interval with the usual max, min, and negation, and constants 0 and 1. To understand the algebraic properties of the algebra of fuzzy subsets of a set S, it su¢ ces to understand those of I. This is a special case of a general phenomenon. A fuzzy set is a mapping f : S ! [0; 1], and operations of these mappings come pointwise from operations on [0; 1]. There are many ways to generalize this notion. For example, instead of taking mappings f : S ! [0; 1] one could take other relations in S [0; 1]. But in the spirit of fuzzy sets, one wants a mapping that associates with an element s 2 S something that is a measure of a degree of "belonging". So the natural thing to do is to generalize the algebra I of truth values. What does this mean? A reasonable meaning is to use an algebra A of truth values that contains I as a subalgebra, or more precisely that contains a subalgebra isomorphic to I. Then, mappings of a set into A when restricted to that subalgebra would be ordinary fuzzy sets. But I should be contained in A in a special way. There should be a "natural" copy of it in A. A common generalization of the algebra I is the algebra I consisting of pairs (a; b) with 0 a b 1, and with appropriate coordinate operations. Mappings f : S ! I are interval-valued fuzzy sets, and have come to play a big role in applications. Type-2 fuzzy sets are mappings into fuzzy subsets of the unit interval. The usual operations put on Map([0; 1]; [0; 1]) to make it into an appropriate algebra are certain convolutions of the operations of I, and are the ones proposed by Zadeh, resulting in an algebra