Algebraic Structure of Atomic Forms over a Fixed Signature

The article develops the algebraic and order-theoretic structure of atomic forms over a fixed finite signature. Atomic forms are defined as finite directed acyclic graphs with ordered input ports and an ordered interface, inducing deterministic endomorphisms . The work introduces interface-preserving isomorphism, establishes a monoid structure under sequential composition, defines a structural embedding relation that yields a partial order, and proves compatibility with the normalization procedure from earlier work. Several local semantics-preserving reduction rules are presented, each strictly decreasing structural size while remaining monotone with respect to embedding. The paper focuses on the endomorphism-level structure and prepares the foundation for subsequent quantitative and categorical developments.