Idempotent cellular automata and their natural order

TL;DR

This paper investigates idempotent cellular automata on a group $G$ formed by reading a fixed pattern $p:S\to A$ and replacing it with a symbol $a$, thereby acting as the identity except at pattern matches. It characterizes when such pattern-defined CA are idempotent, distinguishing constant, symmetrical, and quasi-constant patterns, and shows that the minimal memory set is $S$ and that the image coincides with a subshift defined by the pattern when idempotence holds. It then develops a framework for the natural partial order on idempotent CA, giving an exact criterion in terms of their images and kernels, and proves rich order-theoretic phenomena: if $G$ is infinite there exist infinitely many pairwise incomparable idempotents, and if $G$ contains an element of infinite order there is an infinite ascending chain of idempotents. Collectively, results bridge pattern-defined CA with semigroup order theory and subshift dynamics across arbitrary groups $G$ and finite alphabets $A$ with $|A|\,\geq2$.

Abstract

Motivated by the search for idempotent cellular automata (CA), we study CA that act almost as the identity unless they read a fixed pattern $p$. We show that constant and symmetrical patterns always produce idempotent CA, and we characterize the quasi-constant patterns that produce idempotent CA. Our results are valid for CA over an arbitrary group $G$. Moreover, we study the semigroup theoretic natural partial order defined on idempotent CA. If $G$ is infinite, we prove that there is an infinite independent set of idempotent CA, and if $G$ has an element of infinite order, we prove that there is an infinite increasing chain of idempotent CA.

Theorems & Definitions (50)

• Theorem 1
• Theorem 2
• Theorem 3
• Definition 1
• Example 1
• Remark 1
• Example 2
• Definition 2: Def. 1.4.1 in CSC10
• Example 3
• Definition 3: Sec. 1.5 in CSC10