Semigroup Codes for Diagnostic Sequences

This paper develops a semigroup-theoretic framework for the algebraic modelling of diagnostic processes. Diagnostic sequences are represented as words over finite alphabets of test actions, and their structural properties are examined via kernel congruences and Krohn–Rhodes-type decompositions. Beyond establishing these foundational results, we show that the canonical reduction of diagnostic words leads to algebraically minimal pathways that eliminate redundant clinical transitions, thereby improving computational and operational efficiency. The decomposition of diagnostic semigroups into reversible and irreversible components further yields a principled hierarchical modelling strategy that mirrors the layered structure of clinical decision-making. Stability properties derived from the semigroup action provide an algebraic criterion for robustness under repeated evaluations, offering insights into diagnostic reliability. We also demonstrate how the associated kernel congruence aligns with Myhill–Nerode equivalence, enabling a direct interface with automata-theoretic and symbolic computation methods. Taken together, the theory provides not only a rigorous algebraic foundation for diagnostic sequence optimization but also practical implications for workflow design, decision support systems, and the formal analysis of diagnostic protocols.