Minimal decomposition theorems and minimal extension principle for picture fuzzy sets

Abstract

Picture fuzzy set theory was originally proposed as a mathematical tool to deal with uncertainty by taking yes, no, neutral memberships of an element of a universal set. It has been studied by a host of researchers theoretically and practically. But still now, the structural properties of picture fuzzy sets are not widely studied. In this article, we propose lower (𝛼, 𝛾, 𝛽)-cut and strong lower (𝛼, 𝛾, 𝛽)-cut of a picture fuzzy set and illustrate some of their properties. Three minimal decomposition theorems for picture fuzzy sets are introduced by lower (𝛼, 𝛾, 𝛽)-cut, strong lower (𝛼, 𝛾, 𝛽)-cut and level set of picture fuzzy sets with illustrations by a numerical example. Some properties of minimal extension principle are also described by using the lower (𝛼, 𝛾, 𝛽)-cut and the strong lower (𝛼, 𝛾, 𝛽)-cut of picture fuzzy sets. Finally, arithmetic operations for picture fuzzy sets are illustrated by using the minimal extension principle.