Fuzzy addition in the L-fuzzy real line

Abstract

Abstract Under the hypothesis L is a chain, we construct a binary operation ⊕ on the L-fuzzy real line R (L) which reduces to the usual addition on R if ⊕ is restricted to the embedded image of R in R (L), which yields a partially ordered, abelian cancellation semigroup with identity, and which is jointly fuzzy continuous on R (L). We show ⊕ is unique, i.e. it is the only extension of addition to R (L) which is consistent. We study the relationship between ⊕ and other fuzzy continuous extensions of the usual addition. We also show that fuzzy translation is a weak fuzzy homeomorphism and, under certain conditions, a fuzzy homeomorphism.