semigroup theory

This paper systematically investigates the algebraic structure and computational properties of finite semigroupoids—i.e., categories without identities—in automata theory. Addressing the key issue that associativity and type consistency are logically independent, it introduces a rigorous framework distinguishing strict versus lax homomorphisms for arrow-typed semigroupoids and establishes the first systematic enumeration method. Using relational and declarative programming techniques, the work enables abstract generation of partial composition tables, automated homomorphism checking, and construction of minimal transformation representations. Contributions include: (1) the first open-source tool supporting enumeration, homomorphism analysis, and representation reduction for semigroupoids; (2) verification of existence and essential distinctions among multiple type structures; and (3) strengthened algebraic characterization of typed computation, providing a novel foundation for categorical semantics and formal verification.

This study addresses the computational inefficiency in enumerating congruences on finite inverse semigroups. Methodologically, it introduces a novel algorithm integrating computational group theory, automata theory, and the algebraic structure of inverse semigroups: it constructs a state-transition automaton from generating pairs, leverages group-action orbit decompositions and the normal series property of inverse semigroups to efficiently enumerate and verify congruences, and—uniquely—systematically unifies these three theoretical frameworks to enable algebraically guided pruning and parallelization. Experiments demonstrate that the algorithm achieves speedups of 2–4 orders of magnitude over existing methods and successfully computes congruences for inverse semigroups of order exceeding 1,000, substantially extending the feasible problem scale. The primary contribution is the establishment of a cross-theoretical framework for congruence computation on inverse semigroups, providing a new paradigm for automated algebraic analysis.

This study addresses ambiguity in both the initial distribution and transition mechanism of continuous-time Markov processes. To jointly model these two sources of uncertainty, we introduce the novel concept of *imprecise Markov semigroups*. We establish geometric and topological ergodicity criteria—applicable to Euclidean spaces, Riemannian manifolds, and general measurable spaces—and rigorously prove sufficient conditions for exponential decay of uncertainty over time. Our methodology integrates convex analysis, operator semigroup theory, differential geometry, and measure theory. This work constitutes the first extension of classical ergodicity theory to settings with imprecise probabilistic specifications. The resulting framework provides a verifiable theoretical foundation and a unified analytical toolset for robust machine learning and uncertainty-aware visual modeling.

This paper uncovers the algebraic essence underlying convex analysis, Gaussian probability, and quadratic structure. To this end, we introduce Graphical Quadratic Algebra (GQA)—a novel algebraic framework based on chordal graphs—that uniformly models quadratic relations, Gaussian stochastic processes, and nondeterministic Gaussian processes via rotation-invariant quadratic generators. We provide the first sound and complete axiomatic characterization of three fundamental models: least-squares estimation, Gaussian randomness, and nondeterminism—revealing their shared conditional algebraic structure. Our method integrates string diagram theory, categorical semantics, and formal semantics of probabilistic programming. Theoretical contributions include soundness and completeness proofs for all three models within GQA. Applications demonstrate efficacy in linear regression, probabilistic programming, and noisy circuit modeling.

This work addresses the computation of finite-index congruences on finitely presented semigroups and monoids. We propose two novel algorithms: one for infinite semigroups/monoids, generalizing Sims’ low-index subgroup algorithm; and another for finite cases, based on principal congruences and their unions. Our approach introduces, for the first time, the integration of relative Green’s relations and a monoid analogue of the Schreier lemma into the generation of principal congruences, significantly reducing computational redundancy. We develop the first open-source, cross-platform congruence computation toolkit—unified across GAP, C++ (libsemigroups), and Python—and integrate it into both the GAP Semigroups package and the main libsemigroups library. Experimental evaluation demonstrates that our method efficiently enumerates all finite-index left, right, and two-sided congruences on diverse classical semigroups, outperforming existing implementations in both speed and scalability.

The semantic relationship between nonnegative real-weighted automata and probabilistic automata remains poorly understood, particularly due to the lack of a systematic characterization of their stochastic structure. This work addresses this gap by introducing a semantics-preserving weight rescaling technique that normalizes finitely weighted automata into locally stochastic probabilistic automata, thereby disentangling their exponential growth component from their stochastic behavior. Leveraging Perron–Frobenius theory and spectral radius analysis, we establish an equivalence between the two models under normalization. We further propose a characterization via stochastic regular expressions based on geometrically weighted Kleene stars and develop a general behavioral decomposition framework that effectively constructs a locally stochastic probabilistic automaton from any given weighted automaton with finite total mass.

This work unifies the modeling of inference rules and proof structures of formal systems within a categorical framework. It introduces a met-variable context representation based on Cartesian PROPs, encodes assumptions and conclusions using spans, and constructs a symmetric monoidal category of proofs with met-variable substitution as the sole primitive operation. This approach is the first to uniformly embed both inference rules and proof structures into the semantics of symmetric monoidal categories, thereby supporting compositional and reusable handling of hypotheses. The authors implement an open-source verification algorithm and surface syntax, successfully encoding formulas, axioms, and representative derivations of first-order logic, and release a functional proof checker.

This study addresses the problem of embedding word structures into low-dimensional complex matrix semigroups, aiming to overcome their inherent structural limitations. By integrating combinatorial word theory with representation-theoretic techniques from complex matrix groups such as SL(2, ℂ), the work proposes a novel approach that successfully constructs word representations for Euclidean Bianchi groups. This method transcends the constraints of conventional low-dimensional matrix embeddings and establishes the first symbolic word embedding framework tailored to 2×2 complex matrix semigroups. The resulting framework provides a rigorous theoretical foundation for the systematic analysis of fundamental decision problems—including membership and equivalence—within this algebraic setting.

This work addresses the formal modeling and efficient analysis of quantitative properties in financial systems driven by uncertain economic variables. It introduces Weighted Finite Financial Automata (WFFA) and Weighted Financial Regular Expressions, establishing a scenario-based, compositional language-theoretic framework. For the first time in financial contexts, the study formulates a Kleene–Schützenberger correspondence between weighted automata and regular expressions and designs a declarative specification language that supports effective translation between them. The proposed approach enables computable analysis of extreme payoffs for financial instruments and trading strategies, identifies expressive yet computationally tractable subclasses, and facilitates efficient formal verification across multi-market scenarios.