# Algebras and semigroups of locally subexponential growth

## TL;DR

This paper demonstrates that countable-dimensional associative algebras and semigroups with locally subexponential growth can be embedded into finitely generated structures of similar growth, using a novel matrix wreath product construction.

## Contribution

It introduces a new method for embedding such algebras and semigroups into finitely generated ones while controlling growth, expanding understanding of algebraic structure and growth behavior.

## Key findings

- Countable-dimensional algebras of locally subexponential growth are embeddable.
- Countable semigroups of locally subexponential growth are embeddable.
- Provides bounds for growth in finitely generated embeddings.

## Abstract

We prove that a countable dimensional associative algebra (resp. a countable semigroup) of locally subexponential growth is $M_\infty$-embeddable as a left ideal in a finitely generated algebra (resp. semigroup) of subexponential growth. Moreover, we provide bounds for the growth of the finitely generated algebra (resp. semigroup). The proof is based on a new construction of matrix wreath product of algebras.