Von Neumann regularity, split epicness and elementary cellular automata

TL;DR

This paper characterizes Von Neumann regularity in cellular automata on mixing subshifts of finite type, linking it to split epicness and providing a decision procedure for elementary cellular automata.

Contribution

It establishes a precise criterion for Von Neumann regularity in cellular automata and demonstrates its decidability for all elementary CA.

Findings

Von Neumann regularity corresponds to split epicness onto the image.

Decidability of Von Neumann regularity for elementary CA.

Characterization applies to cellular automata on mixing subshifts of finite type.

Abstract

We show that a cellular automaton on a mixing subshift of finite type is a Von Neumann regular element in the semigroup of cellular automata if and only if it is split epic onto its image in the category of sofic shifts and block maps. It follows from [S.-T\"orm\"a, 2015] that Von Neumann regularity is decidable condition, and we decide it for all elementary CA.