The skew-growth function on the monoid of square matrices

TL;DR

This paper develops a divisibility theory for the monoid of square matrices over a principal ideal domain, introducing a skew growth function that decomposes into Euler products, revealing new algebraic structures.

Contribution

It introduces an elementary divisibility framework for matrix monoids and defines a skew growth function with Euler product decomposition, extending growth theory to matrices.

Findings

Any finite subset has a least left common multiple up to a right unit.

The skew growth function decomposes into Euler products.

The theory applies to matrices over residue finite principal ideal domains.

Abstract

We develop an elementary theory of divisibility on the monoid $M(n,R)^\times$ consisting of all square matrices of size $n\ge 1$ of non-zero determinants with coefficients in a principal ideal domain $R$. In particular, we show that any finite subset of the monoid has the least left common multiple up to a right unit factor. When $R$ is residue finite, we consider a signed generating series, called the skew growth function, of least common multiples of finite right equivalence classes of irreducible elements. As an elementary application of the divisibility theory, we show that the skew-growth function decomposes into Euler products.