Fuzzy order relative to fuzzy B-algebras

Abstract

Let (X; ∗, 0) be a B-algebra and µ a fuzzy B-algebra defined on X.  For each x ∈ X, the fuzzy order of x, denoted by \(FO_{µ}^{B}(x)\), is the least positive integer n such that µ(xⁿ) = µ(0).  Also, the order of an element x ∈ X, denoted by |x|_B, is the least positive integer n such that xⁿ = 0.  In this paper, some properties of \(FO_{µ}^{B}(x)\) are established.  Also, the relationship between \(FO_{µ}^{B}(x)\) and the order |x|_B of x is also explored.