1. Automorphisms on normal and convex fuzzy truth values revisited
- Author
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Susana Cubillo, Luis Magdalena, and Carmen Torres-Blanc
- Subjects
0209 industrial biotechnology ,Pure mathematics ,Logic ,Fuzzy set ,Normal function ,Regular polygon ,Boundary (topology) ,02 engineering and technology ,Function (mathematics) ,Automorphism ,Set (abstract data type) ,020901 industrial engineering & automation ,Artificial Intelligence ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,Constant function ,Mathematics - Abstract
The present paper extends some previous works studying automorphisms in type-2 fuzzy sets. The framework for the analysis is the set of convex and normal functions from [ 0 , 1 ] to [ 0 , 1 ] (fuzzy truth values). The paper concentrates on those automorphisms that, in this framework, leave the constant function 1 fixed. This function is quite important since it defines the boundary between the functions that represent “TRUE” (increasing functions) and those that represent “FALSE” (decreasing functions), being at the same time the only normal function that is simultaneously increasing and decreasing. While C.L. Walker, E.A. Walker and J. Harding introduced in 2008 a family of functions leaving the constant function 1 fixed, the main goal of this paper is to prove that the functions of that family are in fact automorphisms, and moreover, that they are the only automorphisms (in the mentioned set of convex and normal functions from [ 0 , 1 ] to [ 0 , 1 ] ) that preserve the function 1. more...
- Published
- 2022
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