Behavioral Ontology Part 2: An Algebra of Dynamic Behavioral Ontology

A companion paper, Behavioral Ontology: A General Theory of Declarative Automation (Argibay, 2026), defines a formal system for specifying the behavioral ontology of automated systems through five orthogonal components and establishes that the resulting specification language is context-free. This paper develops the algebraic structure that those specifications admit. It defines two equivalence relations on Dynamic Behavioral Ontology (DBO) specifications — structural equivalence (isomorphism) and behavioral equivalence (identical outcomes under identical conditions) — and shows that the latter is the natural equivalence for intent-level comparison. It defines a partial order (the subset relation), two binary operations (difference and union), and a normalization procedure for composed specifications. It proposes that these structures form the foundation of a formal algebra of DBO, providing tools for comparing systems across implementations, certifying that modifications preserve intent, and establishing when two apparently different systems are, at the level of intent, the same. This is a first draft; the formal proofs and completeness results remain to be developed.