Existence, uniqueness, calculus and properties of triangular approximations of fuzzy numbers under a general condition

• We consider a general family of parameters associated to a fuzzy number.
• We study the existence and uniqueness of triangular fuzzy numbers which preserve a fixed parameter in the family.
• We compute the nearest triangular fuzzy number of a fuzzy number which preserves the fixed parameter.
• Properties of the obtained approximation operator are studied.

We determine the set PtPt of real parameters associated with a fuzzy number which belong to a general family including all the important parameters, with the property that for any given fuzzy number there exists a triangular fuzzy number with the same value of the parameter. We propose a method to compute the nearest triangular fuzzy number which preserves p∈Ptp∈Pt and we study the properties of identity, scale and translation invariance, additivity and continuity of the obtained approximation operator. We give examples related with recent results in this topic and an application to the nearest triangular approximation of a fuzzy number preserving the value.