alpha-cut; Fuzzy alpha-cut; Cutability; Partial fuzzy relation; Composition; Dragonfly algebra

Partial fuzzy sets and relations have been recently developed in partial fuzzy set theory to handle undefined values. In this paper, we introduce the definitions of alpha-cut and fuzzy alpha-cut of partial fuzzy relations. Then some basic properties of these cuts of partial fuzzy relations are addressed. Furthermore, we study numerous properties of the compositions of alpha-cut or fuzzy alpha-cut of partial fuzzy relations. Additionally, we focus on the relationship between the compositions of the cuts of partial fuzzy relations on the one hand side, and the cuts of the compositions of the partial fuzzy relations on the other side.