Characterizations of Length and Mean-Fuzzy Ideals in Hilbert Algebras over Hyperstructures

This paper introduces and studies the notions of the length and the mean of a hyperstructure in Hilbert algebras.By employing these notions, we define the concepts of length fuzzy ideals and mean fuzzy ideals of Hilbert algebras and investigate their fundamental properties.Several characterization theorems are established, describing the necessary and sufficient conditions for a hyperstructure to form a length (resp., mean) k-fuzzy ideal for k ∈ {1, 2, 3, 4}.Relationships between length (resp., mean) fuzzy ideals and hyperfuzzy ideals are examined in detail.Furthermore, we provide results connecting these classes of fuzzy ideals with their upper-, lower-, and equal-level subsets, and demonstrate that particular combinations of constant and fuzzy structures naturally lead to length-or mean-type fuzzy ideals.Illustrative examples and counterexamples are presented to validate the main results.