Computing congruences of finite inverse semigroups

2024-06-13

arXiv.org

AI Summary

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.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

In this paper we present a novel algorithm for computing a congruence on an inverse semigroup from a collection of generating pairs. This algorithm uses a myriad of techniques from computational group theory, automata, and the theory of inverse semigroups. An initial implementation of this algorithm outperforms existing implementations by several orders of magnitude.

Problem

• Computes congruences on finite inverse semigroups from generating pairs.
• Uses techniques from groups, automata, and inverse semigroup theories.
• Outperforms existing implementations by several orders of magnitude.

Innovation

• Algorithm computes congruence from generating pairs
• Uses techniques from groups, automata, semigroups theories
• Implementation outperforms existing by orders of magnitude