Quantifying quantum-like structure in rough set lattices: Numerical indices for complex and intelligent systems

Certain rough set lattices induced by binary relations form almost disjoint unions of Boolean algebras, a subclass of orthomodular lattices that captures quantum-like structure. In empirical data analysis, however, rough set lattices rarely realize this ideal form exactly. To quantify approximate similarity to such structures, we introduce numerical indices for rough set lattices. Simulation experiments based on controlled perturbations of binary relations show that the proposed indices distinguish structured perturbations from density-matched random baselines and remain informative under substantial perturbation. We further apply the framework to rough set lattices derived from machine-learning confusion matrices. In the examples studied, ordinary classification outcomes do not exhibit the characteristic signatures of almost disjoint unions of Boolean algebras, whereas a superposition-based setting does. The proposed approach provides a quantitative methodology for detecting latent lattice-theoretic organization in complex and intelligent systems, with potential relevance to cognitive and other self-organizing information systems.