# Fuzzy mathematics

Fuzzy mathematics forms a branch of mathematics related to fuzzy set theory and fuzzy logic. It started in 1965 after the publication of Lotfi Asker Zadeh's seminal work Fuzzy sets. A fuzzy subset A of a set X is a function A:X→L, where L is the interval [0,1]. This function is also called a membership function. A membership function is a generalization of a characteristic function or an indicator function of a subset defined for L = {0,1}. More generally, one can use a complete lattice L in a definition of a fuzzy subset A .

The evolution of the fuzzification of mathematical concepts can be broken down into three stages:

1. straightforward fuzzification during the sixties and seventies,
2. the explosion of the possible choices in the generalization process during the eighties,
3. the standardization, axiomatization and L-fuzzification in the nineties.

Usually, a fuzzification of mathematical concepts is based on a generalization of these concepts from characteristic functions to membership functions. Let A and B be two fuzzy subsets of X. Intersection A ∩ B and union A ∪ B are defined as follows: (A ∩ B)(x) = min(A(x),B(x)), (A  B)(x) = max(A(x),B(x)) for all xX. Instead of min and max one can use t-norm and t-conorm, respectively , for example, min(a,b) can be replaced by multiplication ab. A straightforward fuzzification is usually based on min and max operations because in this case more properties of traditional mathematics can be extended to the fuzzy case.

A very important generalization principle used in fuzzification of algebraic operations is a closure property. Let * be a binary operation on X. The closure property for a fuzzy subset A of X is that for all x,yX, A(x*y) ≥ min(A(x),A(y)). Let (G,*) be a group and A a fuzzy subset of G. Then A is a fuzzy subgroup of G if for all x,y in G, A(x*y−1) ≥ min(A(x),A(y−1)).

A similar generalization principle is used, for example, for fuzzification of the transitivity property. Let R be a fuzzy relation in X, i.e. R is a fuzzy subset of X×X. Then R is transitive if for all x,y,z in X, R(x,z) ≥ min(R(x,y),R(y,z)).

## Some fields of mathematics using fuzzy set theory

Fuzzy subgroupoids and fuzzy subgroups were introduced in 1971 by A. Rosenfeld . Hundreds of papers on related topics have been published. Recent results and references can be found in  and.

Main results in fuzzy fields and fuzzy Galois theory are published in a 1998 paper.

Fuzzy topology was introduced by C.L. Chang in 1968 and further was studied in many papers.

Main concepts of fuzzy geometry were introduced by Tim Poston in 1971, A. Rosenfeld in 1974, by J.J. Buckley and E. Eslami in 1997 and by D. Ghosh and D. Chakraborty in 2012-14  

Basic types of fuzzy relations were introduced by Zadeh in 1971.

The properties of fuzzy graphs have been studied by A. Kaufman, A. Rosenfel, and by R.T. Yeh and S.Y. Bang. Recent results can be found in a 2000 article.

Possibility theory, nonadditive measures, fuzzy measure theory and fuzzy integrals are studied in the cited articles and treatises.

Main results and references on formal fuzzy logic can be found in these citations.