The Intersection Algebra of Safety, Cosafety, Liveness, and Coliveness over Linear Temporal Logic

We study the algebra generated by intersections of the four fundamental temporal property classes — safety, cosafety, liveness, and coliveness — over Linear Temporal Logic (LTL) on infinite traces. Building on the Cantor-topological characterisation of these classes and on the Safety/Liveness decomposition theorem, we develop a unified algebraic and topological view of their pairwise and higher-order intersections, identifying which combinations collapse to trivial classes and which generate proper sublattices of LTL-definable omega-languages. The resulting intersection algebra clarifies the structural boundary between monitorability and unmonitorability and provides a uniform foundation for reasoning about runtime verification, governance properties, and combined safety/progress guarantees. We discuss applications to the design of LTL governance frameworks and to the classification of agentic obligations expressed as combinations of safety and liveness constraints.