(untitled) (European Journal of Pure and Applied Mathematics, 2026)
The interplay between fuzzy structures and algebraic systems has been an active area of research due to its theoretical significance and potential applications in fuzzy logic, approximate reasoning, and non-classical algebraic frameworks. In this paper, we introduce and investigate the notions of fuzzy translations and fuzzy multiplications of fuzzy subalgebras and fuzzy ideals within Hilbert algebras. We establish fundamental properties, provide necessary and sufficient conditions for these operations to preserve algebraic structures, and examine their interrelations through the concept of fuzzy S-extensions. Several illustrative examples and counterexamples are presented to demonstrate that the converses of certain results do not hold in general. Our results not only extend previous studies on fuzzification in Hilbert algebras but also offer new structural insights that may be applicable to related algebraic systems and fuzzy logical models.