CELLULAR AUTOMATA OVER SEMI-DIRECT PRODUCT GROUPS: REDUCTION AND INVERTIBILITY RESULTS

2006 ◽  
Vol 16 (06) ◽  
pp. 1071-1085
Author(s):  
SILVIO CAPOBIANCO
Keyword(s):  
Cellular Automata ◽  
Direct Product ◽  
Configuration Spaces ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
Current State ◽  
Global Properties ◽  
Finite Factor ◽  
Finiteness Properties

Cellular automata are transformations of configuration spaces over finitely generated groups, such that the next state in a point only depends on the current state of a finite neighborhood of the point itself. Many questions arise about retrieving global properties from such local descriptions, and finding algorithms to perform these tasks. We consider the case when the group is a semi-direct product of two finitely generated groups, and show that a finite factor (whatever it is) can be thought of as part of the alphabet instead of the group, preserving both the dynamics and some "finiteness" properties. We also show that, under reasonable hypotheses, this reduction is computable: this leads to some reduction theorems related to the invertibility problem.

scholarly journals Bounds on the Action Degree of Groups

MATEMATIKA ◽  
2019 ◽  
Vol 35 (2) ◽  
pp. 271-282
Author(s):  
S. Alrehaili ◽  
Charef Beddani
Keyword(s):  
Finite Group ◽  
Direct Product ◽  
Finite Groups ◽  
Random Element ◽  
General Relation ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
A Finite Group

The commutativity degree is the probability that a pair of elements chosen randomly from a group commute. The concept of  commutativity degree has been widely discussed by several authors in many directions.  One of the important generalizations of commutativity degree is the probability that a random element from a finite group G fixes a random element from a non-empty set S that we call the action degree of groups. In this research, the concept of action degree is further studied where some inequalities and bounds on the action degree of finite groups are determined.  Moreover, a general relation between the action degree of a finite group G and a subgroup H is provided. Next, the action degree for the direct product of two finite groups is determined. Previously, the action degree was only defined for finite groups, the action degree for finitely generated groups will be defined in this research and some bounds on them are going to be determined.

Collectives of Automata in Finitely Generated Groups

2020 ◽  
Vol 108 (5-6) ◽  
pp. 671-678
Author(s):  
D. V. Gusev ◽  
I. A. Ivanov-Pogodaev ◽  
A. Ya. Kanel-Belov
Keyword(s):  
Finitely Generated ◽  
Finitely Generated Groups

scholarly journals Some properties of the growth and of the algebraic entropy of group endomorphisms

2017 ◽  
Vol 20 (4) ◽  
Author(s):  
Anna Giordano Bruno ◽  
Pablo Spiga
Keyword(s):  
Finite Groups ◽  
Addition Theorem ◽  
Finitely Generated ◽  
Growth Type ◽  
Locally Finite ◽  
Locally Finite Groups ◽  
Algebraic Entropy ◽  
Finitely Generated Groups ◽  
Identity Automorphism ◽  
Inner Automorphisms

AbstractWe study the growth of group endomorphisms, a generalization of the classical notion of growth of finitely generated groups, which is strictly related to algebraic entropy. We prove that the inner automorphisms of a group have the same growth type and the same algebraic entropy as the identity automorphism. Moreover, we show that endomorphisms of locally finite groups cannot have intermediate growth. We also find an example showing that the Addition Theorem for algebraic entropy does not hold for endomorphisms of arbitrary groups.

The word problem for semigroups satisfying x3 = x

1978 ◽  
Vol 84 (1) ◽  
pp. 11-19 ◽  
Author(s):  
J. A. Gerhard
Keyword(s):  
Word Problem ◽  
Upper Bound ◽  
Finitely Generated ◽  
Finitely Generated Groups

In the paper (4) of Green and Rees it was established that the finiteness of finitely generated semigroups satisfying xr = x is equivalent to the finiteness of finitely generated groups satisfying xr−1 = 1 (Burnside's Problem). A group satisfying x2 = 1 is abelian and if it is generated by n elements, it has at most 2n elements. The free finitely generated semigroups satisfying x3 = x are thus established to be finite, and in fact the connexion with the corresponding problem for groups can be used to give an upper bound on the size of these semigroups. This is a long way from an algorithm for a solution of the word problem however, and providing such an algorithm is the purpose of the present paper. The case x = x3 is of interest since the corresponding result for x = x2 was done by Green and Rees (4) and independently by McLean(6).

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