Ranks of finite semigroups of one-dimensional cellular automata

Since first introduced by John von Neumann, the notion of cellular automaton has grown into a key concept in computer science, physics and theoretical biology. In its classical setting, a cellular automaton is a transformation of the set of all configurations of a regular grid such that the image of any particular cell of the grid is determined by a fixed local function that only depends on a fixed finite neighbourhood. In recent years, with the introduction of a generalised definition in terms of transformations of the form τ:AG→AG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau : A^G \rightarrow A^G$$\end{document} (where G is any group and A is any set), the theory of cellular automata has been greatly enriched by its connections with group theory and topology. In this paper, we begin the finite semigroup theoretic study of cellular automata by investigating the rank (i.e. the cardinality of a smallest generating set) of the semigroup CA(Zn;A)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {CA}(\mathbb {Z}_n; A)$$\end{document} consisting of all cellular automata over the cyclic group Zn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_n$$\end{document} and a finite set A. In particular, we determine this rank when n is equal to p, 2k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^k$$\end{document} or 2kp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^kp$$\end{document}, for any odd prime p and k≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k \ge 1$$\end{document}, and we give upper and lower bounds for the general case.


Introduction
Cellular automata (CA) were introduced by John von Neumann as an attempt to design self-reproducing systems that were computationally universal (see [19]).Since then, the theory of CA has grown into an important area of computer science, physics, and theoretical biology (e.g.[4,12,20]).Among the most famous CA are Rule 110 and John Conway's Game of Life, both of which have been widely studied as discrete dynamical systems and are known to be capable of universal computation.
In recent years, many interesting results linking CA and group theory have appeared in the literature (e.g.see [3,4,5]).One of the goals of this paper is to embark in the new task of exploring CA from the point of view of finite semigroup theory.
We shall review the broad definition of CA that appears in [4,Sec. 1.4].Let G be a group and A a set.Denote by A G the set of functions of the form x : G → A. For each g ∈ G, denote by R g : G → G the right multiplication map, i.e. (h)R g := hg for any h ∈ G.We shall emphasise that in this paper we apply maps on the right, while in [4] maps are applied on the left.Definition 1.Let G be a group and A a set.A cellular automaton over G and A is a map τ : A G → A G satisfying the following property: there exists a finite subset S ⊆ G and a local map µ : A S → A such that (g)(x)τ = ((R g • x)| S )µ, for all x ∈ A G , g ∈ G, where (R g • x)| S is the restriction to S of (R g • x) : G → A .
Let CA(G; A) be the set of all cellular automata over G and A; it is straightforward to show that, under composition of maps, CA(G; A) is a semigroup.Most of the literature on CA focus on the case when G = Z d , d ≥ 1, and A is a finite set (see [12]).In this situation, an element τ ∈ CA(Z d ; A) is referred as a d-dimensional cellular automaton.
Although results on semigroups of CA have appeared in the literature before (see [10,18]), the semigroup structure of CA(G; A) remains basically unknown.In particular, the study of the finite semigroups CA(G; A), when G and A are finite, has been generally disregarded, perhaps because some of the classical questions are trivially answered (e.g. the Garden of Eden theorem becomes trivial).However, many new questions, typical of finite semigroup theory, arise in this setting.
One of the fundamental problems in the study of a finite semigroup M is the determination of the cardinality of a smallest generating subset of M ; this is called the rank of M and denoted by Rank(M ): Rank(M ) := min{|H| : H ⊆ M and H = M }.
It is well-known that, if X is any finite set, the rank of the full transformation semigroup Tran(X) (consisting of all functions f : X → X) is 3, while the rank of the symmetric group Sym(X) is 2 (see [7,Ch. 3]).Ranks of various finite semigroups have been determined in the literature before (e.g.see [1,2,8,9,11]).
In order to hopefully bring more attention to the study of finite semigroups of CA, we shall propose the following problem.
Problem 1.For any finite group G and any finite set A, determine Rank(CA(G; A)).
A natural restriction of this problem, and perhaps a more feasible goal, is to determine the ranks of semigroups of CA over finite abelian groups.
In this paper we study the finite semigroups CA(Z n ; A), where Z n is the cyclic group of order n ≥ 2 and A is a finite set with at least two elements.By analogy with the classical setting, we may say that the elements of CA(Z n ; A) are one-dimensional cellular automata over Z n and A.
In this paper we shall establish the following theorems.
Theorem 1.Let k ≥ 1 be an integer, p an odd prime, and A a finite set of size q ≥ 2. Then: Theorem 2. Let n ≥ 2 be an integer and A a finite set of size q ≥ 2. Then: where

Preliminary results
For the rest of the paper, let n ≥ 2 an integer and A a finite set of size q ≥ 2. We may assume that A = {0, 1, . . ., q − 1}.When G is a finite group, we may always assume that the finite subset S ⊆ G of Definition 1 is equal to G, so any cellular automaton over G and A is completely determined by the local map µ : A G → A. Therefore, if |G| = n, we have |CA(G; A)| = q q n .It is clear that CA(Z n ; A) is contained in the semigroup of transformations Tran(A n ), where A n is the n-th Cartesian power of A. For any f ∈ Tran(A n ) write f = (f 1 , . . ., f n ), where f i : A n → A is the i-th coordinate function of f .For any semigroup M and σ ∈ M , define the centraliser of σ in M by C M (σ) := {τ ∈ M : τ σ = στ }.
It turns out that CA(Z n ; A) is equal to the centraliser of a certain transformation in Tran(A n ).
For any f ∈ Tran(A n ), define an equivalence relation ∼ on A n as follows: for any x, y ∈ A n , say that x ∼ y if and only if there exist r, s ≥ 1 such that (x)f r = (y)f s .The equivalence classes induced by this relation are called the orbits of f .Lemma 1.Let n ≥ 2 be an integer and A a finite set.Consider the map σ : A n → A n given by (x 1 , . . ., x n )σ = (x n , x 1 , . . ., x n−1 ). Then: Proof.We shall prove each part.
(i) By Definition 1, a map τ : A n → A n is a cellular automaton over G = Z n and A if and only if there exists a map µ : for any 1 ≤ i ≤ n, where the sum in the subindex of x j+i is done modulo n.Hence,

This shows that CA(Z
Therefore, f is a cellular automaton over Z n and A with µ = f n . (ii) This follows directly by the Orbit-Stabiliser Theorem ([6, Theorem 1.4A]).
(iii) Fix τ ∈ CA(Z n ; A), P ∈ O and x ∈ O.By definition of orbit, and since σ is a permutation, for every y ∈ P there is i ∈ Z such that (x)σ i = y.By part (i), (x)τ Lemma 1 (i) is in fact a particular case of a more general result.
Lemma 2. Let G be a finite group and A a finite set.For each g ∈ G, let σ g ∈ CA(G; A) be the cellular automaton with local map µ g : A G → A defined by (x)µ g = (g −1 )x for all x ∈ A G .Then, Proof.The result follows by Curtis-Hedlund Theorem (see [4,Theorem 1.8.1]).
Let ICA(G; A) be the set of invertible cellular automata: It may be shown that ICA(G; We shall use the cyclic notation to denote the permutations in Tran( When D = {b} is a singleton, we write (b → a) instead of ({b} → a).
In the following examples, we identify the elements of A n with their lexicographical order: (a 1 , a 2 , . . ., a n ) ∼ n i=1 a i q i−1 .
If U is a subset of a finite semigroup M , the relative rank of U in M , denoted by Rank(M : U ), is the minimum cardinality of a subset V ⊆ M such that U, V = M .The proof of the main results of this paper are based in the following observation.Lemma 3. Let G be a finite group and A a finite set.Then, In Section 3 we study the rank of ICA(Z n ; A), while in Section 4 we study the relative rank of ICA(Z n ; A) in CA(Z n ; A).
3 The rank of ICA(Z n ; A) Let σ : A n → A n be as defined in Lemma 1.For any d ≥ 1 dividing n, the number of orbits of σ of size d is given by the Moreau's necklace-counting function where µ is the classic Möbius function (see [14]).In particular, if d = p k , where p is a prime number and k ≥ 1, then Remark 1. Observe that α(d, q) = 1 if and only if (d, q) = (2, 2).Hence, the case when n is even and q = 2 is degenerate and shall be analysed separately in the rest of the paper.
We say that d is a non-trivial divisor of n if d | n and d = 1 (note that, in our definition, d = n is a non-trivial divisor of n).For any integer α ≥ 1, let Sym α and Alt α be the symmetric and alternating groups on [α] = {1, . . ., α}, respectively.
A wreath product of the form [17]).We shall use the additive notation for the elements of (Z d ) α , so the product in where v, w ∈ (Z d ) α , φ, ψ ∈ Sym α , and φ acts on w by permuting the coordinates.We shall identify the elements (v; id) The following result is a refinement of [18,Theorem 9].
Lemma 4. Let n ≥ 2 be an integer and A a finite set of size q ≥ 2. Let d 1 , d 2 , . . ., d ℓ be the non-trivial divisors of n.Then Proof.Let O the set of orbits of σ : A n → A n as defined in Lemma 1. Part (ii) of that lemma shows that CA(Z n ; A) is contained in the semigroup As O contains q singletons and α(d i , q) orbits of size d i ≥ 2, we know by [1, Lemma 2.1] that the group of units of Tran(A n , O) is Clearly, ICA(Z n ; A) ≤ S(A n , O).Let P be an orbit of size d i .Since the restriction of σ to P , denoted by σ| P , is a cycle of length d i , and the centraliser of σ| P in Sym Equality follows as any permutation stabilising the sets of orbits of size d i commutes with σ.
For 1 ≤ i ≤ α, denote by e i the element of (Z d ) α with 1 at the i-th coordinate, and 0 elsewhere.Denote by e 0 the element of (Z d ) α with 0's everywhere.For any α ≥ 2, define permutations z α ∈ Sym α by Note that the order of z α , denoted by o(z α ), is always odd.
In the following Lemma we determine the rank of the generalized symmetric group.
Let M := x, y ≤ Z d ≀ Sym α .Let ρ : Z d ≀ Sym α → Sym α be the natural projection, and note that this is a group homomorphism.Clearly, (M )ρ = Sym α and ker(ρ) = (Z d ) α , so, in order to prove that Now, by Sym α -invariance Suppose that d is even and α is odd.Then, Since Sym α is 2-transitive on the basis of (Z d ) α and y Finally, suppose that d and α are both even.Then, Write α − 1 = 2k + 1, for some k ∈ N. Then We need the following results in order to establish Rank(ICA(Z p , A)) when p is a prime number.
(i) Except for q ∈ {5, 6, 8}, Sym q is generated by an element of order 2 and an element of order 3.
(ii) If p ′ > 3 is a prime number dividing q! and q = 2p ′ − 1, then Sym q is generated by an element of order 2 and an element of order p ′ .
Lemma 7. Let p be a prime number and A a finite set of size q ≥ 2. Then: (i) If q ≥ 3 and p = 2, then Rank(ICA(Z 2 ; A)) = 3.
Proof.If q = p = 2, the result follows by Example 1. Assume (p, q) = (2, 2).By Lemma 4, where α := α(p, q) ≥ 2 is the Moreau's necklace-counting function.We use the basic fact that Rank(G/N ) ≤ Rank(G), for any group G and any normal subgroup N of G. Let U 2 be the Sym αinvariant submodule of (Z p ) α defined in Lemma 6.Then U 2 is a normal subgroup of This implies that there is a normal subgroup N of Z p ≀ Sym α with quotient isomorphic to Define z α and z q as in (2).We shall prove the two cases (i) and (ii).
(i) Suppose that q ≥ 3 and p = 2, so 3 ≤ Rank(W ) by ( 3).We shall show that W = v 1 , v 2 , v 3 where The projections of v 1 , v 2 and v 3 to Sym q generate Sym q , so it is enough to prove that v 1 and zq) .We follow a similar strategy as in the proof of Lemma 5. Note that the projections of v 1 and (v 2 ) o(zq) to Sym α generate Sym α .Now, (Z 2 and so (Z 2 ) α ∩ M = (Z 2 ) α in this case.
Recall that for any integer n ≥ 2, we denote by d(n) the number of divisors of n (including 1 and n itself) and by d + (n) the number of even divisors of n (so d + (n) = 0 if and only if n is odd).Theorem 4. Let n ≥ 2 be an integer and A a finite set of size q ≥ 2.
In the proof of the following result we shall use the notion of kernel of a transformation τ : A n → A n as the partition of A n induced by the equivalence relation {(x, y) ∈ A n × A n : (x)τ = (y)τ }.Lemma 9. Let n ≥ 2 be an integer and A a finite set of size q ≥ 2. Then: Proof.Let O be the set of orbits of σ : A n → A n , as defined in Lemma 1.Let d 1 , . . ., d ℓ be all the divisors of n ordered as follows For 1 ≤ i ≤ ℓ, let α i := α(d i , q) and denote by O i the subset of O of orbits of size d i .Let Suppose that q = 2 or n is odd, so α i ≥ 2 for all i.For any pair of divisors d j and d i such that d j | d i , fix ω j ∈ B j and ω i ∈ B i in distinct orbits.Denote the orbits that contains ω i by [ω i ].Define idempotents τ i,j ∈ CA(Z n ; A) in the following way: Note that τ i,j collapses [ω i ] to [ω j ] and fixes everything else.We claim that H := ICA(Z n ; A), τ i,j : Let ξ ∈ CA(Z n ; A).For 1 ≤ i ≤ ℓ, and define Clearly ξ i ∈ CA(Z n ; A).By Lemma 1, we have (B i )ξ ⊆ j≤i B i , so ξ = ξ 1 ξ 2 . . .ξ ℓ .
We shall prove that ξ i ∈ H for all 1 ≤ i ≤ ℓ.Decompose ξ i as ξ i = ξ ′ i ξ ′′ i , where (B i )ξ ′ i ⊆ j<i B j and (B i )ξ ′′ i ⊆ B i .
1. We show that ξ ′ i ∈ H.If B i = ∪ α i s=1 P s is the decomposition of B i into orbits, we may write ξ ′ i = ξ ′ i | P 1 . . .ξ ′ i | Pα i , where ξ ′ i | Ps acts as ξ ′ i on P s and fixes everything else.In this case, Q s := (P s )ξ ′ i | Ps is an orbit contained in B j for some j < i.By Lemma 4, there is φ s ∈ Sym α i ×Sym α j ≤ ICA(Z n ; A) such that φ s acts as the double transposition ([ω i ], P s )([ω j ], Q s ), and 2. We show that ξ ′′ i ∈ H.In this case, ξ ′′ i ∈ Tran(B i ).In fact, as ξ ′′ i preserves the partition of B i into orbits, ξ ′′ i ∈ σ| B i ≀Tran α i .As α i ≥ 2, the semigroup Tran α i is generated by Sym α i ≤ ICA(Z n ; A) together with the idempotent τ i,i .Hence, ξ ′′ i ∈ H.
This establishes that the relative rank of ICA(Z n ; A) in CA(Z n ; A) is at most E(n).
For the converse, suppose that ICA(Z n ; A), U = CA(Z n ; A), where |U | < E(n).Hence, we may assume that, for some d j | d i , U ∩ ICA(Z n ; A), τ i,j = ∅. (5) By Lemma 1, there is no τ ∈ CA(Z n ; A) such that (X)τ ⊆ Y for X ∈ O a , Y ∈ O b with d b ∤ d a .This, together with (5), implies that U has no element with kernel of the form {{x, y}, {z} : x ∈ P, y ∈ Q, z ∈ A n \ (P ∪ Q)} for any P ∈ O i , Q ∈ O j .Thus, there is no element in ICA(Z n ; A), U with kernel of such form, which is a contradiction (because τ i,j ∈ CA(Z n ; A) has indeed this kernel).The case when q = 2 and n is even follows similarly, except that now, as there is a unique orbit of size 2 in O, there is no idempotent τ 2,2 .

Acknowledgment
This work was supported by the EPSRC grant EP/K033956/1.
Proof.As ICA(G; A) is the group of units of CA(G; A), this follows by[2, Lemma 3.1].