Background Classical fuzzy sets have been generalized to better model uncertainty, leading to developments such as intuitionistic, Pythagorean, and Fermatean fuzzy sets. Fermatean fuzzy sets (ffss), characterized by the cube sum constraint on membership and non-membership degrees, provide a more flexible framework for handling complex uncertainty. BN-algebras are important algebraic structures with applications in logic and information theory. Methods This study integrates Fermatean fuzzy sets with BN-algebras by introducing the notions of Fermatean fuzzy subalgebras and Fermatean fuzzy ideals. Level cuts of Fermatean fuzzy sets are defined and analyzed, and algebraic techniques are employed to investigate their structural properties. Results It is shown that Fermatean fuzzy level subalgebras and Fermatean fuzzy level ideals correspond to classical subalgebras and ideals of BN-algebras. Several characterizations and closure properties are established, supported by illustrative examples and rigorous proofs. Conclusions The results demonstrate that Fermatean fuzzy sets significantly enrich the theory of BN-algebras by accommodating higher degrees of uncertainty. This framework provides a solid theoretical foundation for further studies and potential applications in algebraic logic and uncertainty-based systems.
