scholarly journals Relatively Free Semigroups of Intermediate Growth

2001 ◽  
Vol 235 (2) ◽  
pp. 484-546 ◽  
Author(s):  
L.M Shneerson
Keyword(s):  
Intermediate Growth ◽  
Free Semigroups

TYPES OF GROWTH AND IDENTITIES OF SEMIGROUPS

2005 ◽  
Vol 15 (05n06) ◽  
pp. 1189-1204
Author(s):  
L. M. SHNEERSON
Keyword(s):  
Finitely Generated ◽  
Intermediate Growth ◽  
Free Semigroups

In this paper we discuss the problem: how large can the intermediate growth of nilpotent and relatively free semigroups be. We construct a sequence {Sn} of finitely-generated semigroups such that the growth of the semigroup Sn+1 (n = 1,2,…) is intermediate and larger than or equal to the growth of exp (m/φn(m)), where φn(m) is the nth iteration of ln m. All semigroups Sn are nilpotent in the sense of Malcev. We also find the sequence of relatively free semigroups with the same types of growth.

scholarly journals Integrable and Chaotic Systems Associated with Fractal Groups

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 237
Author(s):  
Rostislav Grigorchuk ◽  
Supun Samarakoon
Keyword(s):  
Operator Algebras ◽  
Probabilistic Approach ◽  
Schur Complement ◽  
Automata Theory ◽  
Random Schrödinger Operators ◽  
Rational Maps ◽  
Holomorphic Dynamics ◽  
Von Neumann ◽  
Intermediate Growth ◽  
Quasi Crystals

Fractal groups (also called self-similar groups) is the class of groups discovered by the first author in the 1980s with the purpose of solving some famous problems in mathematics, including the question of raising to von Neumann about non-elementary amenability (in the association with studies around the Banach-Tarski Paradox) and John Milnor’s question on the existence of groups of intermediate growth between polynomial and exponential. Fractal groups arise in various fields of mathematics, including the theory of random walks, holomorphic dynamics, automata theory, operator algebras, etc. They have relations to the theory of chaos, quasi-crystals, fractals, and random Schrödinger operators. One important development is the relation of fractal groups to multi-dimensional dynamics, the theory of joint spectrum of pencil of operators, and the spectral theory of Laplace operator on graphs. This paper gives a quick access to these topics, provides calculation and analysis of multi-dimensional rational maps arising via the Schur complement in some important examples, including the first group of intermediate growth and its overgroup, contains a discussion of the dichotomy “integrable-chaotic” in the considered model, and suggests a possible probabilistic approach to studying the discussed problems.

Completely simple semigroups: free products, free semigroups and varieties

1981 ◽  
Vol 88 (3-4) ◽  
pp. 293-313 ◽  
Author(s):  
P. R. Jones
Keyword(s):  
Matrix Representation ◽  
Maximal Subgroups ◽  
Lattice Of Varieties ◽  
Free Products ◽  
Completely Simple Semigroups ◽  
Countable Rank ◽  
Free Semigroups ◽  
Abelian Subgroups ◽  
And Inversion ◽  
Completely Simple

SynopsisThe class CS of completely simple semigroups forms a variety under the operations of multiplication and inversion (x−1 being the inverse of x in its ℋ-class). We determine a Rees matrix representation of the CS-free product of an arbitrary family of completely simple semigroups and deduce a description of the free completely simple semigroups, whose existence was proved by McAlister in 1968 and whose structure was first given by Clifford in 1979. From this a description of the lattice of varieties of completely simple semigroups is given in terms of certain subgroups of a free group of countable rank. Whilst not providing a “list” of identities on completely simple semigroups it does enable us to deduce, for instance, the description of all varieties of completely simple semigroups with abelian subgroups given by Rasin in 1979. It also enables us to describe the maximal subgroups of the “free” idempotent-generated completely simple semigroups T(α, β) denned by Eberhart et al. in 1973 and to show in general the maximal subgroups of the “V-free” semigroups of this type (which we define) need not be free in any variety of groups.

scholarly journals Partitioning of Retained Energy in Broilers and Birds with Intermediate Growth Rate

2007 ◽  
Vol 86 (10) ◽  
pp. 2162-2171 ◽  
Author(s):  
G. Lopez ◽  
K. de Lange ◽  
S. Leeson
Keyword(s):  
Growth Rate ◽  
Intermediate Growth

Group-Free Semigroups

2001 ◽  
pp. 227-258
Author(s):  
P. A. Grillet
Keyword(s):  
Free Semigroups

scholarly journals An example of an indexed language of intermediate growth

1999 ◽  
Vol 215 (1-2) ◽  
pp. 325-327 ◽  
Author(s):  
R.I. Grigorchuk ◽  
A. Machí
Keyword(s):  
Intermediate Growth

Left large subsets of free semigroups and groups that are not right large

2014 ◽  
Vol 90 (2) ◽  
pp. 374-385 ◽  
Author(s):  
Neil Hindman ◽  
Lakeshia Legette Jones ◽  
Monique Agnes Peters
Keyword(s):  
Free Semigroups

scholarly journals RESERCH ON THE INTERMEDIATE GROWTH AND SUBSTRATE OF SEA CUCUMBER IN A FISHING PORT ANCHORAGE

2018 ◽  
Vol 74 (2) ◽  
pp. I_342-I_347
Author(s):  
Masami OHASHI ◽  
Rumiko KAJIHARA ◽  
Toshiaki ITO ◽  
Yuji ANAGUCHI ◽  
Masaki KATAYAMA ◽  
...  
Keyword(s):  
Sea Cucumber ◽  
Intermediate Growth

Numerical upper bounds on growth of automaton groups

2021 ◽  
pp. 1-33
Author(s):  
Jérémie Brieussel ◽  
Thibault Godin ◽  
Bijan Mohammadi
Keyword(s):  
Group Theory ◽  
Upper Bounds ◽  
Geometric Invariant ◽  
Finitely Generated ◽  
Grigorchuk Group ◽  
Finitely Generated Group ◽  
Intermediate Growth ◽  
Mealy Automata ◽  
Optimal Bounds ◽  
Algorithmic Procedure

The growth of a finitely generated group is an important geometric invariant which has been studied for decades. It can be either polynomial, for a well-understood class of groups, or exponential, for most groups studied by geometers, or intermediate, that is between polynomial and exponential. Despite recent spectacular progresses, the class of groups with intermediate growth remains largely mysterious. Many examples of such groups are constructed using Mealy automata. The aim of this paper is to give an algorithmic procedure to study the growth of such automaton groups, and more precisely to provide numerical upper bounds on their exponents. Our functions retrieve known optimal bounds on the famous first Grigorchuk group. They also improve known upper bounds on other automaton groups and permitted us to discover several new examples of automaton groups of intermediate growth. All the algorithms described are implemented in GAP, a language dedicated to computational group theory.

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