Ranks of finite semigroups of one-dimensional cellular automata

2020

Nat Comput

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For a group G and a finite set A, denote by End(A G ) the monoid of all continuous shift commuting self-maps of A G and by Aut(A G ) its group of units. We study the minimal cardinality of a generating set, known as the rank, of End(A G ) and Aut(A G ). In the first part, when G is a finite group, we give upper and lower bounds for the rank of 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 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.

“…The rank of Aut(A G ) when G is a finite cyclic group has been examined in detail in [5]. Let d(n) be number of divisors of n, including 1 and n itself.…”

Section: Cyclic and Dihedral Groupsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the minimal number of generators of endomorphism monoids of full shifts

2020

Nat Comput

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“…In order to achieve this, we summarise some of the notation and results obtained in [2,3,4]. Definition 2.…”

Section: Regular Finite Cellular Automatamentioning

confidence: 99%

Von Neumann Regular Cellular Automata

Cellular Automata and Discrete Complex Systems

Gadouleau

2017

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For any group G and any set A, a cellular automaton (CA) is a transformation of the configuration space A G defined via a finite memory set and a local function. Let CA(G; A) be the monoid of all CA over A G . In this paper, we investigate a generalisation of the inverse of a CA from the semigroup-theoretic perspective. An element τ ∈ CA(G; A) is von Neumann regular (or simply regular ) if there exists σ ∈ CA(G; A) such that τ • σ • τ = τ and σ • τ • σ = σ, where • is the composition of functions. Such an element σ is called a generalised inverse of τ . The monoid CA(G; A) itself is regular if all its elements are regular. We establish that CA(G; A) is regular if and only if |G| = 1 or |A| = 1, and we characterise all regular elements in CA(G; A) when G and A are both finite. Furthermore, we study regular linear CA when A = V is a vector space over a field F; in particular, we show that every regular linear CA is invertible when G is torsion-free elementary amenable (e.g. when G = Z d , d ∈ N) and V = F, and that every linear CA is regular when V is finite-dimensional and G is locally finite with char(F) ∤ o(g) for all g ∈ G.

“…The theory of cellular automata (CA) has important connections with many areas of mathematics, such as group theory, topology, symbolic dynamics, coding theory, and cryptography. Recently, in [5,6,7], links with semigroup theory have been explored, and, in particular, questions have been considered on the structure of the monoid of all CA and the group of all invertible CA over a given configuration space. The goal of this paper is to bound the minimal number of generators, known in semigroup theory as the rank, of groups of invertible CA.…”

Section: Introductionmentioning

confidence: 99%

“…The question of finding the rank of a monoid is important in semigroup theory; it has been answered for several kinds of transformation monoids and Rees matrix semigroups (e.g., see [1,11]). For the case of monoids of cellular automata over finite groups, the question has been addressed in [6,7]; in particular, the rank of ICA(G; A) when G is a finite cyclic group has been examined in detail in [5].…”

Section: Introductionmentioning

confidence: 99%

Bounding the Minimal Number of Generators of Groups and Monoids of Cellular Automata

Cellular Automata and Discrete Complex Systems

Sánchez-Álvarez

2019

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For a group G and a finite set A, denote by CA(G; A) the monoid of all cellular automata over A G and by ICA(G; A) its group of units. We study the minimal cardinality of a generating set, known as the rank, of 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 ICA(G; A) when G is a finite group, and we apply this to show that, for any infinite abelian group H, the monoid 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 CA(H; A) is finitely generated.