This chapter examines the algebraic foundations of first-order non-alethic logical systems Nn*, which integrate both paraconsistent and paracomplete features by rejecting the Principle of Non-contradiction and Principle of Excluded Middle. These hybrid systems, derived from the synthesis of Cn* and Pn* calculi, permit the formal treatment of simultaneously inconsistent and paracomplete information. The chapter presents the syntactic structure and axiomatic basis of Nn*, showing that it preserves essential schemes of classical predicate logic where appropriate. A key contribution is the development of N1*-monadic algebras, algebraic structures that serve as models for quantified non-alethic logic. These algebras generalize monadic and Curry algebras by including dual quantifier operators and maintaining compatibility with the logical behavior of strong negation. Representation theorems are established, demonstrating the correspondence between N1*-monadic algebras and Boolean-valued functional algebras. This algebraic approach lays the groundwork for formalizing non-alethic reasoning in systems subject to uncertainty, vagueness, and contradiction.