The asymptotic size of finite irreducible semigroups of rational matrices

2026-01-03

arXiv.org

AI Summary

This study addresses the problem of determining the maximal size of an irreducible finite semigroup of $n \times n$ matrices over the rational numbers. By integrating techniques from algebraic semigroup theory, linear representation theory, and combinatorial matrix analysis, the work introduces a novel approach—distinct from Schützenberger’s classical method—that significantly improves the known upper bound from $2^{O(n^2 \log n)}$ to $3^{n^2}$. Furthermore, the authors construct explicit examples of irreducible semigroups of size $3^{\lfloor n^2/4 \rfloor}$, demonstrating that the new upper bound is tight up to exponential order. As a corollary, the results also yield an improved bound on the nilpotency threshold originally established by Almeida and Steinberg.

Technology Category

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

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• 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

Abstract

We study finite semigroups of $n \times n$ matrices with rational entries. Such semigroups provide a rich generalization of transition monoids of unambiguous (and, in particular, deterministic) finite automata. In this paper we determine the maximum size of finite semigroups of rational $n \times n$ matrices, with the goal of shedding more light on the structure of such matrix semigroups. While in general such semigroups can be arbitrarily large in terms of $n$, a classical result of Sch\"utzenberger from 1962 implies an upper bound of $2^{O(n^2 \log n)}$ for irreducible semigroups, i.e., the only subspaces of $Q^n$ that are invariant for all matrices in the semigroup are $Q^n$ and the subspace consisting only of the zero vector. Irreducible matrix semigroups can be viewed as the building blocks of general matrix semigroups, and as such play an important role in mathematics and computer science. From the point of view of automata theory, they generalize strongly connected automata. Using a very different technique from that of Sch\"utzenberger, we improve the upper bound on the cardinality to $3^{n^2}$. This is the main result of the paper. The bound is in some sense tight, as we show that there exists, for every $n$, a finite irreducible semigroup with $3^{\lfloor n^2/4 \rfloor}$ rational matrices. Our main result also leads to an improvement of a bound, due to Almeida and Steinberg, on the mortality threshold. The mortality threshold is a number $\ell$ such that if the zero matrix is in the semigroup, then the zero matrix can be written as a product of at most $\ell$ matrices from any subset that generates the semigroup.

Problem

• finite semigroups
• rational matrices
• irreducible semigroups
• asymptotic size
• matrix semigroups

Innovation

• irreducible matrix semigroups
• asymptotic size
• rational matrices
• mortality threshold
• upper bound