Sheffer Stroke Hoop Algebras: From Axiomatization to Isomorphism Theorems

In this paper, we introduce and formalize the theory of Sheffer stroke hoop algebras, providing a minimal axiomatization for bounded hoop algebras utilizing a single binary operation. We systematically establish the fundamental algebraic properties of this novel structure, beginning with the logical independence of its core axioms. We equip the algebra with an induced partial order, proving it constitutes a well-defined ∧-semilattice, and demonstrate a bidirectional structural translation: bounded hoop algebras satisfying the Double Negation Property (DNP) can be equivalently expressed as Sheffer stroke hoop algebras, and vice versa. Furthermore, we investigate the internal algebraic architecture by introducing sub-algebras, filters, positive implicative filters, and ideals, rigorously establishing the inherent structural duality between filters and ideals. By proving that proper filters naturally induce full algebraic congruences, we successfully construct quotient Sheffer stroke hoop algebras. We characterize prime filters by demonstrating that a quotient algebra forms a chain if and only if its generating filter is prime. Finally, we complete the theoretical framework by establishing the Correspondence Theorem and the First Isomorphism Theorem, seamlessly embedding classical universal algebraic principles into the Sheffer stroke setting.