Relatively Free Semigroups of Intermediate Growth

Shneerson, L. M.

doi:10.1006/jabr.2000.8503

Cited by 18 publications

(28 citation statements)

References 19 publications

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“…Finally, all the rules (N1-10) derive from the rules s 2 = 1 (N1) and r n → r ′ n for n ≥ 2 (N4, [8][9][10], and from graphical equalities (N2-3,5-7). Each of the rules r n → r ′ n is strictly length-shortening, so N is precisely the set of irreducible words with respect to the rules, and consists of minimal representative words.…”

Section: 2mentioning

confidence: 99%

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A Mealy Machine With Polynomial Growth of Irrational Degree

Bartholdi

Reznykov

2008

Int. J. Algebra Comput.

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Abstract. We consider a very simple Mealy machine (two non-trivial states over a two-symbol alphabet), and derive some properties of the semigroup it generates. It is an infinite, finitely generated semigroup, and we show that the growth function of its balls behaves asymptotically like n α , for α = 1 + log 2/log; that the semigroup satisfies the identity g 6 = g 4 ; and that its lattice of two-sided ideals is a chain.

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Section: 2mentioning

confidence: 99%

“…Belov and Ivanov [3,4] construct finitely presented semigroups with non-integer GK dimension. Shneerson [10] constructs relatively free (i.e., free relative to an identity) semigroups of intermediate growth (asymptotically m log m ). His proof involves Fibonacci numbers, as does the present construction.…”

Section: Introductionmentioning

confidence: 99%

A Mealy Machine With Polynomial Growth of Irrational Degree

Bartholdi

Reznykov

2008

Int. J. Algebra Comput.

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“…This paper is a continuation of the author's paper [12]. Here we discuss the connections between the growth of semigroups satisfying a given system of identities and unavoidable words arising from some natural problems of the theory of formal languages [6,8] and playing an important role in modern algebra.…”

Section: Introductionmentioning

confidence: 97%

“…We note that the proof of this conjecture uses a method for constructing f.g. semigroups of subexponential growth given in [12] and some ideas from M. Sapir's papers [9,10] (also see [6]), where for the first time the connections between symbolic dynamical systems and Burnside-type problems were explored. These connections are based on the notion of a uniformly recurrent word.…”

Section: Introductionmentioning

confidence: 99%

Growth, unavoidable words, and M. Sapir's conjecture for semigroup varieties

Shneerson

2004

Journal of Algebra

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Let ᑬ be a nonperiodic semigroup variety satisfying the nontrivial identity Z n = W , where Z n is the nth element of the so-called Zimin's sequence Z 1 = u 1 , Z n+1 = Z n u n+1 Z n (n = 1, 2, . . .) of unavoidable words. It is shown that every finitely generated (f.g.) semigroup S ∈ ᑬ has subexponential growth and every uniformly recurrent infinite word which is geodesic and lexicographically reduced relative to S is periodic. Furthermore, the length of the period is bounded above by a constant which depends only on the number n and on the number of generators of S. The first of these results gives a positive answer to the problem posed by M. Sapir. Some applications including examples of f.g. relatively free semigroups having intermediate growth and maximal (equals 1) superdimension are considered.

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“…Various Mealy automata of intermediate growth were found in later years (see, for example, [6,7]). But the properties of the growth of groups, semigroups and Mealy automata are different in kind (see, for example, [8,9]). …”

Section: Introductionmentioning

confidence: 99%

On the 3-state Mealy automata over an m-symbol alphabet of growth order [nlogn/2logm]

Reznykov¹,

Sushchansky

2006

Journal of Algebra

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On the 3-state Mealy automata over an m-symbol alphabet of growth order [n log n/2 log m ] AbstractWe consider sequence {J m , m 2} of 3-state Mealy automata over an m-symbol alphabet such that the growth of J m is intermediate of order [n log n/2 log m ]. For each automaton J m we describe the transformation monoid S J m , defined by it, provide generating series for the growth functions, and consider some properties of S J m and J m .

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Relatively Free Semigroups of Intermediate Growth

Cited by 18 publications

References 19 publications

A Mealy Machine With Polynomial Growth of Irrational Degree

A Mealy Machine With Polynomial Growth of Irrational Degree

Growth, unavoidable words, and M. Sapir's conjecture for semigroup varieties

On the 3-state Mealy automata over an m-symbol alphabet of growth order [nlogn/2logm]

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