J2 Radical in Automata Nearrings

TL;DR

This paper investigates the structure of automata over group alphabets viewed as nearrings, focusing on the Jacobson 2-radical to understand their complexity and identify conditions for explicit radical computation and semisimple images.

Contribution

It provides bounds on the Jacobson 2-radical in automata nearrings and identifies cases where this radical can be explicitly computed, enhancing understanding of their algebraic structure.

Findings

Bounds on the Jacobson 2-radical established

Explicit radical computation possible in certain groups

Semisimple images determined for specific cases

Abstract

Looking at the automata defined over a group alphabet as a nearring, we see that they are a highly complicated structure. As with ring theory, one method to deal with complexity is to look at semisimplicity modulo radical structures. We find some bounds on the Jacobson 2-radical and show that in certain groups, this radical can be explicitly found and the semisimple image determined.