Invariant Threshold Symmetry in Bipolar Fuzzy Quasi-Subalgebras of Sheffer–Nelson Algebras

This paper develops a rigorous algebraic framework for quasi-substructures in Sheffer-based Nelson algebras, extending the landscape of fuzzy algebraic theory. By systematically introducing (∈,∈∨q)-bipolar fuzzy quasi-subalgebras and ideals, we analyze their structural properties through generalized belongingness and quasi-coincidence relations. We formalise invariant threshold symmetry as the condition g+(χ)+|g−(χ)|=c for a constant c∈0,2 and every χ∈Ω (Definition ) and prove its structural preservation within (∈,∈∨q)-bipolar fuzzy quasi-subalgebras (Theorem , supported by Theorems , and ). This enables a balanced dual evaluation of positive and negative information. Characterization theorems are established via level subsets, revealing how quasi-substructure properties are governed by bounds at critical membership values. Equivalence results unify classical and bipolar fuzzy perspectives, demonstrating that algebraic constraints preserve structural coherence across crisp and fuzzy environments. Algorithmic verification procedures are provided for practical validation in finite systems, and illustrative examples highlight applications in uncertainty modeling and decision support. Overall, the proposed theory formalizes bipolar fuzzy structures in Sheffer-based Nelson algebras, utilizing invariant threshold symmetry, level-set decomposition, and crisp equivalence to evaluate dual information.