Functionally recursive rings of matrices-Two examples

TL;DR

This paper introduces two rings generated by functionally recursive matrices, analyzing their structure, growth, and properties, including isomorphisms and Gelfand-Kirillov dimension, contributing to the understanding of recursive matrix rings.

Contribution

The paper defines finite-state and functionally recursive matrices, introduces two specific rings generated by these matrices, and analyzes their algebraic properties and growth behavior.

Findings

First ring is isomorphic to the 2-generated free ring.

Second ring has a nil multiplicative semigroup of monomials of degree 5.

Second ring's Gelfand-Kirillov dimension is 1 + log(2)/log(a).

Abstract

We define the notions of finite-state and functionally recursive matrices and their growth. We also introduce two rings generated by functionally recursive matrices. The first is isomorphic to the 2-generated free ring. The second is a 2-generated monomial ring such that the multiplicative semigroup of monomials in the generators is nil of degree 5 and the ring has Gelfand Kirillov dimension 1 + log(2)/log(a) where a=1/2(1+sqrt(5)).