arXiv subjects

Edgar Alcalá-Arroyo

Publications and source records attributed to Edgar Alcalá-Arroyo.

4 recordsLinked to original sources

arXiv2026

Generation by lazy and invertible cellular automata over the Klein four-group

Let $G=C_2\times C_2$ and $A=\{0,1\}$. We characterize the submonoid of $\text{CA}(G;A)$ generated by lazy and invertible cellular automata. A cellular automaton belongs to this submonoid if and only if, whenever it maps the two free $G$-orbits onto the same free orbit, it maps both constant configurations and some two-point orbit to the same constant. The proof gives a factorization using elementary orbit collapses and three two-cell lazy maps. The criterion yields the cardinality $59\,712$ and identifies the maps missed by a weaker constant-separation invariant. Adjoining one idempotent generates the full monoid, and the generated submonoid is contained in exactly one maximal proper submonoid, of cardinality $65\,024$. The submonoid generated by lazy maps alone has $4\,007$ elements, and for both generating sets the maximum shortest-word length is six. The criterion and the cardinalities $59\,712$ and $4\,007$ are formally verified in Lean.

math.GR↗

arXiv2026

Idempotence criteria for lazy cellular automata

A lazy cellular automaton $τ:A^G\to A^G$ is determined by a finite neighborhood $S\subseteq G$ containing the group identity $e$, a pattern $p\in A^S$, and a writing symbol $a\in A\setminus\{p(e)\}$: its local rule is equal to the projection to $e$ except when it reads the pattern $p$, in which case it writes $a$. For a nonempty subset $T$ of the fiber $S_a:= p^{-1}(a)$, we introduce compatibility conditions on $p$ along the right translates $St$, $t\in T$. We show that, over every group, the existence of a compatible set is necessary for $τ$ to be non-idempotent, and that it is also sufficient whenever $st = ts$ for all $t \in S_a$, $s \in S_{p(e)}$. In particular, compatibility characterizes non-idempotence over abelian groups, which reduces deciding idempotence to a search over subsets of $S_a$. Finally, for every dihedral group $D_n$ with $n\geq 4$, we construct an idempotent lazy cellular automaton that admits a compatible set, showing that the commutation hypothesis cannot simply be dropped.

math.GR↗

arXiv2026

When do lazy and invertible cellular automata generate all cellular automata?

Let $G$ be a group and let $A$ be a set with at least two elements. A lazy cellular automaton over $A^G$ acts as the identity except at the cells where a fixed pattern occurs, where it replaces the central symbol by a fixed different symbol. Let $M(G;A)$ be the submonoid of $\mathrm{CA}(G;A)$ generated by the lazy and the invertible cellular automata. We study when $M(G;A) = \mathrm{CA}(G;A)$. Since no lazy cellular automaton is surjective, every surjective element of $M(G;A)$ is invertible; hence $M(G;A) \neq \mathrm{CA}(G;A)$ whenever $A$ is infinite or $G$ has an element of infinite order. Moreover, if $|A| = 2$ and $G$ has a quotient isomorphic to the Klein four-group $C_2 \times C_2$, then $M(G;A) \neq \mathrm{CA}(G;A)$. For locally finite $G$, and in particular for finite $G$, these are the only obstructions: we prove that $M(G;A) = \mathrm{CA}(G;A)$ if and only if $3 \leq |A| < \infty$, or $|A| = 2$ and $G$ has no quotient isomorphic to $C_2 \times C_2$.

math.GR↗

arXiv2026

On the order of lazy cellular automata

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.

cs.FL↗