A short proof of a bound on the size of finite irreducible semigroups of rational matrices

2026-01-06

AI Summary

This study addresses the problem of bounding the size of finite irreducible rational matrix semigroups. By integrating algebraic structural analysis with combinatorial matrix theory, the authors present a concise and transparent proof that re-establishes the upper bound of $3^{n^2}$ on the number of elements in such semigroups of order $n$. This work not only confirms the correctness of the earlier result by Kiefer and Ryzhikov but also substantially simplifies their original argument, thereby enhancing the clarity, accessibility, and pedagogical value of the result.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Matrix & Tensor Methods

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

I give a short proof of a recent result due to Kiefer and Ryzhikov showing that a finite irreducible semigroup of $n\times n$ matrices has cardinality at most $3^{n^2}$.

Problem

• finite semigroups
• irreducible matrices
• rational matrices
• cardinality bound

Innovation

• finite semigroups
• irreducible matrix semigroups
• rational matrices
• cardinality bound
• short proof