The asymptotic size of finite irreducible semigroups of rational matrices

TL;DR

This paper determines the maximum size of finite irreducible semigroups of rational matrices, improving existing bounds from exponential to a tighter exponential bound, and explores their structure and implications in automata theory.

Contribution

It provides a new upper bound of 3^{n^2} for the size of irreducible rational matrix semigroups, improving upon the classical bound and establishing near-tightness.

Findings

Upper bound of 3^{n^2} for irreducible semigroup size

Existence of semigroups with size close to the upper bound

Improved bound on the mortality threshold in automata theory

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.