Decompositions and representations of  L -fuzzy natural numbers

Abstract

In this paper, decompositions of L-fuzzy natural numbers and cut sets of L-fuzzy sets on finite sets are presented. Then, based on some unified nature of L-fuzzy natural numbers, the notions of ${\mathbb{N}}_{\beta }^L$-nested mappings and ${\mathbb{N}}_{\alpha }^L$-nested mappings are introduced and some representations of them are obtained. It is showed that if a ${\mathbb{N}}_{\beta }^L$-nested mapping (resp. ${\mathbb{N}}_{\alpha }^L$-nested mapping) is generated by an L-fuzzy natural number, then its representation is exactly equal to itself. As for applications, the addition and the exponentiation of two ${\mathbb{N}}_{\beta }^L$-nested mappings (resp. two ${\mathbb{N}}_{\alpha }^L$-nested mappings) are introduced. It is also showed that, given two ${\mathbb{N}}_{\beta }^L$-nested mappings (resp. two ${\mathbb{N}}_{\alpha }^L$-nested mappings), cut sets that correspond to the representations of their addition (or, their exponentiation) are equal to cut sets of the addition (or, the exponentiations) of the given mappings.