Algebraic characteristics of extended fuzzy numbers

Extended fuzzy numbers, previously called fuzzy intervals, are discussed by using the resolution identity and the extension principle. The regularity and the spread are defined for describing the algebraic properties of extended fuzzy numbers. Arithmetic operations on α-level set intervals are suggested instead of general set operations in order to reduce the amount of computation. A sufficient and necessary condition for solving A + X = C is derived. The exact solution for A + X = C is obtained. Finally, A − A = θ (a fuzzification of the crisp 0), which is a natural extension from the nonfuzzy field, is proved.