Finite Near-Rings, Digital Algebra and Quantum-Inspired Matrix Transformations

Finite near-rings provide a natural algebraic framework for studying nonlinear transformations, finite-state systems, digital computation, and quantum-inspired matrix operations. This paper develops a unified framework connecting finite near-rings with digital algebra over F₂ and quantum-inspired matrix transformations. A genuine finite transformation near-ring of order 8 is constructed from zero-preserving Boolean functions, with pointwise addition and composition multiplication. Finite matrix algebras M₂(F₂) and M₂(F₃) are investigated as finite associative algebras containing linear transformations. The eight-element Heisenberg matrix group over F₂ is analyzed separately, including the operation A ⊕ B = A + B − I, which produces an additive abelian group but does not automatically produce a near-ring with ordinary matrix multiplication. Pauli matrices are introduced as fundamental noncommutative operators and are used as inspiration for finite matrix transformations rather than being identified with finite fields or finite division rings. Connections with digital logic, finite automata, coding theory, graph transformations, and quantum computation are discussed. The paper emphasizes the distinction among finite groups, rings, near-rings, algebras, and division rings.