Large Semigroups of Cellular Automata

TL;DR

This paper investigates the properties of semigroups of cellular automata transformations, focusing on their 'largeness' properties ID and MC, and how these depend on the number of symbols and algebraic structure.

Contribution

It characterizes when certain semigroups of cellular automata are ID and MC based on the number of symbols and algebraic properties, linking one-sided and two-sided shift spaces.

Findings

Multiplication semigroup is ID and MC iff s is not a prime power.

Linear semigroup is always MC; ID iff s is prime.

When symbols form a finite field, the linear semigroup is both ID and MC.

Abstract

In this article we consider semigroups of transformations of cellular automata which act on a fixed shift space. In particular, we are interested in two properties of these semigroups which relate to "largeness". The first property is ID and the other property is maximal commutativity (MC). A semigroup has the ID property if the only infinite invariant closed set (with respect to the semigroup action) is the entire space. We shall consider two examples of semigroups: one is spanned by cellular automata transformations that represent multiplications by integers on the one-dimensional torus and the other one consists of all the cellular automata transformations which are linear (when the symbols set is the ring of integers mod n). It will be shown that the two properties of these semigroups depend on the number of symbols s. The multiplication semigroup is ID and MC if and only if s is not a power of prime. The linear semigroup over the mentioned ring is always MC but is ID if and only if s is prime. When the symbol set is endowed with a finite field structure (when possible) the linear semigroup is both ID and MC. In addition, we associate with each semigroup which acts on a one sided shift space a semigroup acting on a two sided shift space, and vice versa, in such a way that preserves the ID and the MC properties.