Fuzzy GE-norms and fuzzy normed GE-algebras

Abstract

In this paper, we introduce the notion of a fuzzy GE-norm on a GE-algebra (X, ∗, 1X) as a scale-dependent fuzzy analogue of the GE-norm introduced in [2]. We established fundamental structural properties of fuzzy GE-norms, derived several identities and generalizations of classical norm inequalities, and developed examples demonstrating the variety of fuzzy behaviors that arise in GE-algebraic settings. Particular attention was given to the interaction between fuzzy GE-norms and GE-morphisms. We showed that a fuzzy GE-norm is not automatically preserved under mappings induced by a GE-morphism. Positive preservation results were obtained when the morphism is injective or surjective and satisfies natural compatibility requirements. A central result of the paper is a complete characterization of fuzzy normability: a GE-algebra admits a fuzzy GE-norm if and only if its induced order is transitive. This theorem identifies transitivity as the precise structural requirement for norm-like behavior in GE-algebras. Additional results include uniqueness of fuzzy limits in commutative GE-algebras, continuity of left and right translations, fuzzy convergence criteria, and Lipschitz-type estimates for the rational fuzzy GE-norm.