Large Semigroups of Cellular Automata

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.

💡 Research Summary

The paper investigates semigroups of cellular automaton (CA) transformations acting on a fixed shift space, focusing on two notions of “largeness”: the ID (Infinite‑Dense) property and the MC (Maximal‑Commutative) property. A semigroup has the ID property if the only infinite closed set invariant under its action is the whole space; it has the MC property if it is commutative and cannot be properly extended to a larger commutative semigroup. These concepts provide a dynamical and algebraic measure of how richly the semigroup mixes the configuration space.

Two families of semigroups are examined. The first consists of CA that implement multiplication by integers on the one‑dimensional torus, realized by taking the alphabet Σ of size s as the cyclic group ℤₛ and defining, for each integer m, a global map μₘ that multiplies every cell value by m modulo s. This “multiplication semigroup” is inherently commutative because integer multiplication is abelian. The authors prove that the ID property holds precisely when s is not a prime power. If s = pᵏ (p prime, k ≥ 1), there exist non‑trivial infinite closed invariant subsets—e.g., configurations whose p‑adic expansions follow a fixed pattern—so the semigroup fails to be ID. When s has at least two distinct prime factors, any non‑trivial closed set eventually spreads to the whole space under the action of the μₘ, guaranteeing ID. Consequently, the multiplication semigroup is ID ∧ MC exactly for composite s with at least two distinct prime divisors.

The second family consists of linear CA over the ring ℤₙ (n = s). Here each cell updates as a linear combination of a finite neighbourhood, which can be represented by a matrix A with entries in ℤₙ; the global transformation is the linear map induced by A. The set of all such linear CA forms a semigroup that is always MC: any two linear maps can be composed in either order, and the resulting map still belongs to the same semigroup, so no larger commutative semigroup can contain it without losing closure. The ID property, however, depends on the arithmetic of n. If n is a prime, ℤₙ is a field, and every linear CA is a permutation of the configuration space; there are no proper infinite closed invariant subsets, so the semigroup is ID. If n is composite, one can construct invariant subshifts (e.g., configurations whose symbols are all multiples of a non‑trivial divisor of n) that remain closed and infinite, violating ID. Hence the linear semigroup is always MC and is ID iff s is prime.

When the alphabet admits a finite field structure—i.e., s = pᵏ and the symbols are identified with the field GF(pᵏ)—the linear semigroup enjoys both properties regardless of k. The field structure guarantees that every non‑zero linear map is invertible, eliminating any non‑trivial invariant closed sets, while commutativity remains maximal as before.

A further contribution is the construction of a correspondence Φ between semigroups acting on one‑sided shifts Σ^ℕ and those acting on two‑sided shifts Σ^ℤ. Φ extends a one‑sided CA to a two‑sided CA by applying the same local rule to all coordinates, and conversely restricts a two‑sided CA to the non‑negative coordinates. The authors prove that Φ preserves both ID and MC: a semigroup has ID (resp. MC) on Σ^ℕ if and only if its image under Φ has the same property on Σ^ℤ. This shows that the largeness results are intrinsic to the algebraic nature of the semigroups rather than to the directionality of the shift space.

Methodologically, the paper blends dynamical systems techniques (analysis of invariant closed sets, use of shift‑invariant topologies), algebraic tools (semigroup theory, commutative ring and field theory), and number‑theoretic arguments (prime factorisation, Chinese remainder theorem) to obtain necessary and sufficient conditions for ID and MC. The proofs involve explicit construction of invariant subshifts when the conditions fail, and rigorous arguments that any infinite closed invariant set must be the whole space when the conditions hold.

In summary, the authors establish a clear dichotomy: the multiplication semigroup is ID ∧ MC exactly when the alphabet size s is not a prime power; the linear semigroup is always MC and is ID precisely when s is prime, becoming both ID and MC whenever the alphabet carries a finite field structure. The preservation of these properties under the one‑sided ↔ two‑sided correspondence further underscores their fundamental nature. These results deepen the understanding of how algebraic properties of the symbol set influence the dynamical richness of cellular automaton semigroups, with potential implications for symbolic dynamics, cryptographic constructions, and the theory of computation on infinite configurations.