arXiv subjects

A. Castillo-Ramirez

Publications and source records attributed to A. Castillo-Ramirez.

3 recordsLinked to original sources

arXiv2026

A categorical framework for cellular automata

This paper proposes a generalized framework for cellular automata using the language of category theory, extending the classical definition beyond set-theoretic constraints. For an arbitrary category $\mathscr{C}$ with products, we define $\mathscr{C}$-cellular automata as morphisms $\tau : A^G \to B^G$ in $\mathscr{C}$, where the alphabets $A$ and $B$ are objects in $\mathscr{C}$ and the universe is a group $G$. We show that $\mathscr{C}$-cellular automata form a subcategory of $\mathscr{C}$ closed under finite products, and that they satisfy a categorical version of the Curtis-Hedlund-Lyndon theorem. For two arbitrary group universes $G$ and $H$, we extend our theory to define generalized $\mathscr{C}$-cellular automata as morphisms $\tau : A^G \to B^H$ constructed via a group homomorphism $\phi : H \to G$. Finally, we prove that generalized $\mathscr{C}$-cellular automata form a subcategory of $\mathscr{C}$ with a finite weak product involving the free product of the underlying group universes. This framework unifies existing concepts and provides purely categorical proofs of foundational results in the theory of cellular automata.

cs.FL↗

arXiv2022

A generalization of cellular automata over groups

Let $G$ be a group and let $A$ be a finite set with at least two elements. A cellular automaton (CA) over $A^G$ is a function $\tau : A^G \to A^G$ defined via a finite memory set $S \subseteq G$ and a local function $\mu :A^S \to A$. The goal of this paper is to introduce the definition of a generalized cellular automaton (GCA) $\tau : A^G \to A^H$, where $H$ is another arbitrary group, via a group homomorphism $\phi : H \to G$. Our definition preserves the essence of CA, as we prove analogous versions of three key results in the theory of CA: a generalized Curtis-Hedlund Theorem for GCA, a Theorem of Composition for GCA, and a Theorem of Invertibility for GCA. When $G=H$, we prove that the group of invertible GCA over $A^G$ is isomorphic to a semidirect product of $\text{Aut}(G)^{op}$ and the group of invertible CA. Finally, we apply our results to study automorphisms of the monoid $\text{CA}(G;A)$ consisting of all CA over $A^G$. In particular, we show that every $\phi \in \text{Aut}(G)$ defines an automorphism of $\text{CA}(G;A)$ via conjugation by the invertible GCA defined by $\phi$, and that, when $G$ is abelian, $\text{Aut}(G)$ is embedded in the outer automorphism group of $\text{CA}(G;A)$.

math.GR↗

arXiv2016

Lengths of words in transformation semigroups generated by digraphs

Given a simple digraph $D$ on $n$ vertices (with $n\ge2$), there is a natural construction of a semigroup $\langle D\rangle$ associated with $D$. For any edge $(a,b)$ of $D$, let $a\to b$ be the idempotent of defect $1$ mapping $a$ to $b$ and fixing all vertices other than $a$; then define $\langle D\rangle$ to be the semigroup $\langle a\to b:(a,b)\in E(D)\rangle$. For $\alpha \in \langle D \rangle$, let $\ell(D,\alpha)$ be the minimal length of a word in $E(D)$ expressing $\alpha$. When $D=K_n$ is the complete undirected graph, Howie and Iwahori, independently, obtained a formula to calculate $\ell(K_n,\alpha)$, for any $\alpha \in \langle K_n \rangle = \text{Sing}_n$; however, no analogous nontrivial results are known when $D \neq K_n$. In this paper, we characterise all simple digraphs $D$ such that either $\ell(D,\alpha)$ is equal to Howie-Iwahori's formula for all $\alpha \in \langle D \rangle$, or $\ell(D,\alpha) = n - \text{fix}(\alpha)$ for all $\alpha \in \langle D \rangle$, or $\ell(D,\alpha) = n - \text{rk}(\alpha)$ for all $\alpha \in \langle D \rangle$. When $D$ is an acyclic digraph and $\alpha \in \langle D \rangle$, we find a tight upper bound for $\ell(D,\alpha)$. Finally, we study the case when $D$ is a strong tournament (which corresponds to a smallest generating set of idempotents of defect $1$ of $\text{Sing}_n$), and we propose some conjectures.

math.GR↗