Real vector space of LR-fuzzy intervals with respect to the shape-preserving t-norm-based addition

The construction of the membership function of fuzzy intervals is an important problem in vagueness modeling. The preservation of the shape of fuzzy sets during the addition is a natural requirement in practical computation. The LR-fuzzy intervals introduced by Dubois and Prade, satisfy this requirement if the addition is based on the nilpotent t-norm, generated by L or R shape functions. The shortcoming that not any LR-fuzzy interval has an opposite (inverse related to shape-preserving t-norm-based addition), can be solved, if the set of LR-fuzzy intervals is isomorphically included in an extended set, and this extended set forms a group with respect to shape-preserving t-norm-based addition. In this paper we construct the extended set of these LR-fuzzy intervals. We also show that the extended set is a real vector space with scalar product, and the modal intervals can be considered as the elements of this extended set. Finally, we present the algebraic form of LR-fuzzy intervals and the associated application.