arXiv subjects
Publications and source records attributed to Alonso Castillo-Ramirez.
Let $G$ be a group, let $\mathbb{K}$ be a field, and let $V$ be a $\mathbb{K}$-vector space. We prove that there exists a bijective linear cellular automaton $V^G\to V^G$ whose inverse is not a cellular automaton if and only if $G$ is not locally finite and $\dim_{\mathbb{K}}V \geq |\mathbb{K}|^{\aleph_0}$. This answers an open problem proposed by T. Ceccherini-Silberstein and M. Coornaert.
The Norton inequality is one of the fundamental axioms in the theory of Majorana and axial algebras, yet its precise structural consequences have remained only partially understood. In this paper, we show that the Norton inequality alone forces the $0$- and $1$-eigenspace fusion rules for arbitrary idempotents in a commutative real algebra $A$ equipped with a Frobenius form. More precisely, if the Frobenius form is nondegenerate (as in Majorana algebras), we prove that the eigenspaces $A_0(e)$ and $A_1(e)$ of an arbitrary idempotent $e \in A$ are subalgebras and annihilate one another: \[ A_0(e)A_1(e)=\{0\}, \] while in the degenerate case the corresponding inclusions hold modulo the radical of the Frobenius form. This answers a question of T. M. Mudziiri Shumba and S. Shpectorov concerning the closure of the $0$-eigenspace $A_0(e)$.
Let $G$ be a group, let $A$ be a finite alphabet, and let $ϕ: G \to G$ be an endomorphism. We study the topological dynamics of the pullback map $ϕ^* : A^G \to A^G$, given by $ϕ^*(x)=x\circϕ$, a canonical example of a generalized cellular automaton. In the one-dimensional case, where $G=\mathbb Z$ and $ϕ_k(n)=kn$, we prove a sharp dichotomy: $ϕ_k^*$ is equicontinuous precisely for $k\in\{-1,0,1\}$, and cofinitely sensitive otherwise. Although the fixed identity coordinate prevents transitivity on the full shift, the restriction to the natural invariant components is topologically mixing exactly when $k\notin\{-1,0,1\}$. We then extend the analysis to countable groups, showing that $ϕ^*$ is equicontinuous if and only if every element of $G$ is eventually periodic under $ϕ$, while the existence of a non-eventually-periodic element is equivalent to cofinite sensitivity and to the absence of equicontinuous points. Finally, we characterize Bernoulli measure preservation and strong mixing on the punctured configuration space in terms of injectivity and eventual periodicity.
We study the most elementary family of cellular automata defined over an arbitrary group universe $G$ and an alphabet $A$: the lazy cellular automata, which act as the identity on configurations in $A^G$, except when they read a unique active transition $p \in A^S$, in which case they write a fixed symbol $a \in A$. As expected, the dynamical behavior of lazy cellular automata is relatively simple, yet subtle questions arise since they completely depend on the choice of $p$ and $a$. In this paper, we investigate the order of a lazy cellular automaton $τ: A^G \to A^G$, defined as the cardinality of the set $\{ τ^k : k \in \mathbb{N} \}$. In particular, we establish a general upper bound for the order of $τ$ in terms of the fibers of $p$, and we prove that this bound is attained when $p$ is a quasi-constant pattern.
We study two categories of cellular automata. First, for any group $G$, we consider the category $\mathcal{CA}(G)$ whose objects are configuration spaces of the form $A^G$, where $A$ is a set, and whose morphisms are cellular automata of the form $τ: A_1^G \to A_2^G$. We prove that the categorical product of two configuration spaces $A_1^G$ and $A_2^G$ in $\mathcal{CA}(G)$ is the configuration space $(A_1 \times A_2)^G$. Then, we consider the category of generalized cellular automata $\mathcal{GCA}$, whose objects are configuration spaces of the form $A^G$, where $A$ is a set and $G$ is a group, and whose morphisms are $ϕ$-cellular automata of the form $\mathcal{T} : A_1^{G_1} \to A_2^{G_2}$, where $ϕ: G_2 \to G_1$ is a group homomorphism. We prove that a categorical weak product of two configuration spaces $A_1^{G_1}$ and $A_2^{G_2}$ in $\mathcal{GCA}$ is the configuration space $(A_1 \times A_2)^{G_1 \ast G_2}$, where $G_1 \ast G_2$ is the free product of $G_1$ and $G_2$. The previous results allow us to naturally define the product of two cellular automata in $\mathcal{CA}(G)$ and the weak product of two generalized cellular automata in $\mathcal{GCA}$.
For a group $G$ and a finite set $A$, a cellular automaton is a transformation of the configuration space $A^G$ defined via a finite neighborhood and a local map. Although neighborhoods are not unique, every CA admits a unique minimal neighborhood, which consists on all the essential cells in $G$ that affect the behavior of the local map. An active transition of a cellular automaton is a pattern that produces a change on the current state of a cell when the local map is applied. In this paper, we study the links between the minimal neighborhood and the number of active transitions, known as the activity value, of cellular automata. Our main results state that the activity value usually imposes several restrictions on the size of the minimal neighborhood of local maps.
A one-dimensional cellular automaton $τ: A^\mathbb{Z} \to A^\mathbb{Z}$ is a transformation of the full shift defined via a finite neighborhood $S \subset \mathbb{Z}$ and a local function $μ: A^S \to A$. We study the family of cellular automata whose finite neighborhood $S$ is an interval containing $0$, and there exists a pattern $p \in A^S$ satisfying that $μ(z) = z(0)$ if and only if $z \neq p$; this means that these cellular automata have a unique \emph{active transition}. Despite its simplicity, this family presents interesting and subtle problems, as the behavior of the cellular automaton completely depends on the structure of $p$. We show that every cellular automaton $τ$ with a unique active transition $p \in A^S$ is either idempotent or strictly almost equicontinuous, and we completely characterize each one of these situations in terms of $p$. In essence, the idempotence of $τ$ depends on the existence of a certain subpattern of $p$ with a translational symmetry.
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.
For a group $G$ and a finite set $A$, a cellular automaton (CA) is a transformation $τ: A^G \to A^G$ defined via a finite memory set $S \subseteq G$ and a local map $μ: A^S \to A$. Although memory sets are not unique, every CA admits a unique minimal memory set, which consists on all the essential elements of $S$ that affect the behavior of the local map. In this paper, we study the links between the minimal memory set and the generating patterns $\mathcal{P}$ of $μ$; these are the patterns in $A^S$ that are not fixed when the cellular automaton is applied. In particular, we show that when $\vert S \vert \geq 2$ and $\vert \mathcal{P} \vert$ is not a multiple of $\vert A \vert$, then the minimal memory set must be $S$ itself. Moreover, when $\vert \mathcal{P} \vert = \vert A \vert$, $\vert S \vert \geq 3$, and the restriction of $μ$ to these patterns is well-behaved, then the minimal memory set must be $S$ or $S \setminus \{s\}$, for some $s \in S \setminus \{e\}$. These are some of the first general theoretical results on the minimal memory set of a cellular automaton.
Given a finite set $A$ and a group homomorphism $ϕ: H \to G$, a $ϕ$-cellular automaton is a function $\mathcal{T} : A^G \to A^H$ that is continuous with respect to the prodiscrete topologies and $ϕ$-equivariant in the sense that $h \cdot \mathcal{T}(x) = \mathcal{T}( ϕ(h) \cdot x)$, for all $x \in A^G, h \in H$, where $\cdot$ denotes the shift actions of $G$ and $H$ on $A^G$ and $A^H$, respectively. When $G=H$ and $ϕ= \text{id}$, the definition of $\text{id}$-cellular automata coincides with the classical definition of cellular automata. The purpose of this paper is to expand the theory of $ϕ$-cellular automata by focusing on the differences and similarities with their classical counterparts. After discussing some basic results, we introduce the following definition: a $ϕ$-cellular automaton $\mathcal{T} : A^G \to A^H$ has the unique homomorphism property (UHP) if $\mathcal{T}$ is not $ψ$-equivariant for any group homomorphism $ψ: H \to G$, $ψ\neq ϕ$. We show that if the difference set $Δ(ϕ, ψ)$ is infinite, then $\mathcal{T}$ is not $ψ$-equivariant; it follows that when $G$ is torsion-free abelian, every non-constant $\mathcal{T}$ has the UHP. Furthermore, inspired by the theory of classical cellular automata, we study $ϕ$-cellular automata over quotient groups, as well as their restriction and induction to subgroups and supergroups, respectively.
Elementary cellular automata (ECA) are one-dimensional discrete models of computation with a small memory set that have gained significant interest since the pioneer work of Stephen Wolfram, who studied them as time-discrete dynamical systems. Each of the 256 ECA is labeled as rule $X$, where $X$ is an integer between $0$ and $255$. An important property, that is usually overlooked in computational studies, is that the composition of any two one-dimensional cellular automata is again a one-dimensional cellular automaton. In this chapter, we begin a systematic study of the composition of ECA. Intuitively speaking, we shall consider that rule $X$ has low complexity if the compositions $X \circ Y$ and $Y \circ X$ have small minimal memory sets, for many rules $Y$. Hence, we propose a new classification of ECA based on the compositions among them. We also describe all semigroups of ECA (i.e., composition-closed sets of ECA) and analyze their basic structure from the perspective of semigroup theory. In particular, we determine that the largest semigroups of ECA have $9$ elements, and have a subsemigroup of order $8$ that is $\mathcal{R}$-trivial, property which has been recently used to define random walks and Markov chains over semigroups.
For a group $G$ acting on a set $X$, let $\text{End}_G(X)$ be the monoid of all $G$-equivariant transformations, or $G$-endomorphisms, of $X$, and let $\text{Aut}_G(X)$ be its group of units. After discussing few basic results in a general setting, we focus on the case when $G$ and $X$ are both finite in order to determine the smallest cardinality of a set $W \subseteq \text{End}_G(X)$ such that $W \cup \text{Aut}_G(X)$ generates $\text{End}_G(X)$; this is known in semigroup theory as the relative rank of $\text{End}_G(X)$ modulo $\text{Aut}_G(X)$.
For any group $G$ and any set $A$, consider the shift action of $G$ on the full shift $A^G$. A configuration $x \in A^G$ has \emph{least period} $H \leq G$ if the stabiliser of $x$ is precisely $H$. Among other things, the number of such configurations is interesting as it provides an upper bound for the size of the corresponding $\text{Aut}(A^G)$-orbit. In this paper we show that if $G$ is finitely generated and $H$ is of finite index, then the number of configurations in $A^G$ with least period $H$ may be computed using the Möbius function of the lattice of subgroups of finite index in $G$. Moreover, when $H$ is a normal subgroup, we classify all situations such that the number of $G$-orbits with least period $H$ is at most $10$.
For a group $G$ and a set $A$, let $\text{End}(A^G)$ be the monoid of all cellular automata over $A^G$, and let $\text{Aut}(A^G)$ be its group of units. By establishing a characterisation of surjunctuve groups in terms of the monoid $\text{End}(A^G)$, we prove that the rank of $\text{End}(A^G)$ (i.e. the smallest cardinality of a generating set) is equal to the rank of $\text{Aut}(A^G)$ plus the relative rank of $\text{Aut}(A^G)$ in $\text{End}(A^G)$, and that the latter is infinite when $G$ has an infinite decreasing chain of normal subgroups of finite index, condition which is satisfied, for example, for any infinite residually finite group. Moreover, when $A=V$ is a vector space over a field $\mathbb{F}$, we study the monoid $\text{End}_{\mathbb{F}}(V^G)$ of all linear cellular automata over $V^G$ and its group of units $\text{Aut}_{\mathbb{F}}(V^G)$. We show that if $G$ is an indicable group and $V$ is finite-dimensional, then $\text{End}_{\mathbb{F}}(V^G)$ is not finitely generated; however, for any finitely generated indicable group $G$, the group $\text{Aut}_{\mathbb{F}}(\mathbb{F}^G)$ is finitely generated if and only if $\mathbb{F}$ is finite.
Let $G$ be a group and $A$ a set equipped with a collection of finitary operations. We study cellular automata $τ: A^G \to A^G$ that preserve the operations of $A^G$ induced componentwise from the operations of $A$. We show that $τ$ is an endomorphism of $A^G$ if and only if its local function is a homomorphism. When $A$ is entropic (i.e. all finitary operations are homomorphisms), we establish that the set $\text{EndCA}(G;A)$, consisting of all such cellular automata, is isomorphic to the direct limit of $\text{Hom}(A^S, A)$, where $S$ runs among all finite subsets of $G$. In particular, when $A$ is an $R$-module, we show that $\text{EndCA}(G;A)$ is isomorphic to the group algebra $\text{End}(A)[G]$. Moreover, when $A$ is a finite Boolean algebra, we establish that the number of endomorphic cellular automata over $A^G$ admitting a memory set $S$ is precisely $(k \vert S \vert)^k$, where $k$ is the number of atoms of $A$.
A code algebra $A_C$ is a nonassociative commutative algebra defined via a binary linear code $C$. In a previous paper, we classified when code algebras are $\mathbb{Z}_2$-graded axial (decomposition) algebras generated by small idempotents. In this paper, for each algebra in our classification, we obtain the Miyamoto group associated to the grading. We also show that the code algebra structure can be recovered from the axial decomposition algebra structure.
For a group $G$ and a finite set $A$, denote by $\text{End}(A^G)$ the monoid of all continuous shift commuting self-maps of $A^G$ and by $\text{Aut}(A^G)$ its group of units. We study the minimal cardinality of a generating set, known as the rank, of $\text{End}(A^G)$ and $\text{Aut}(A^G)$. In the first part, when $G$ is a finite group, we give upper and lower bounds for the rank of $\text{Aut}(A^G)$ in terms of the number of conjugacy classes of subgroups of $G$. In the second part, we apply our bounds to show that if $G$ has an infinite descending chain of normal subgroups of finite index, then $\text{End}(A^G)$ is not finitely generated; such is the case for wide classes of infinite groups, such as infinite residually finite or infinite locally graded groups.
For a group $G$ and a finite set $A$, denote by $\text{CA}(G;A)$ the monoid of all cellular automata over $A^G$ and by $\text{ICA}(G;A)$ its group of units. We study the minimal cardinality of a generating set, known as the rank, of $\text{ICA}(G;A)$. In the first part, when $G$ is a finite group, we give upper bounds for the rank in terms of the number of conjugacy classes of subgroups of $G$. The case when $G$ is a finite cyclic group has been studied before, so here we focus on the cases when $G$ is a finite dihedral group or a finite Dedekind group. In the second part, we find a basic lower bound for the rank of $\text{ICA}(G;A)$ when $G$ is a finite group, and we apply this to show that, for any infinite abelian group $H$, the monoid $\text{CA}(H;A)$ is not finitely generated. The same is true for various kinds of infinite groups, so we ask if there exists an infinite group $H$ such that $\text{CA}(H;A)$ is finitely generated.