# Quadratic automaton algebras and intermediate growth

## TL;DR

This paper provides examples of quadratic automaton algebras with complex relation structures and introduces a quadratic algebra of intermediate growth, addressing open questions in algebraic theory.

## Contribution

It presents the first example of a quadratic automaton algebra with no finite Gr"obner basis and a new example of a quadratic algebra exhibiting intermediate growth.

## Key findings

- Automaton algebra with no finite Gr"obner basis
- Quadratic algebra of intermediate growth
- Answers to Ufnarovski's question

## Abstract

We present an example of a quadratic algebra given by three generators and three relations, which is automaton (the set of normal words forms a regular language) and such that its ideal of relations does not possess a finite Gr\"obner basis with respect to any choice of generators and any choice of a well-ordering of monomials compatible with multiplication. This answers a question of Ufnarovski. Another result is a simple example (4 generators and 7 relations) of a quadratic algebra of intermediate growth.