A description of the finite right-congruences of finitely generated free semigroups

A. Ádám

doi:10.1007/bf02029171

FREE SEMIGROUP PRESENTATIONS

International Journal of Mathematical Physics ◽

10.18063/ijmp.v1i1.671 ◽

2018 ◽

Vol 1 (1) ◽

Author(s):

Nwawuru Francis

Keyword(s):

Direct Product ◽

Free Semigroup ◽

Finitely Generated ◽

Free Semigroups ◽

Finitely Presented

Let and be two free semigroups. We define external direct product of two free semigroups as an ordered pair of words such that and .We investigate the presentations of external direct product of free semigroups, state and prove under some conditions that the external direct product of two finitely generated free semigroups is finitely generated, also the external direct product of two finitely presented free semigroups is finitely presented.

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TYPES OF GROWTH AND IDENTITIES OF SEMIGROUPS

International Journal of Algebra and Computation ◽

10.1142/s021819670500275x ◽

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.

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On finitary properties for fiber products of free semigroups and free monoids

Semigroup Forum ◽

10.1007/s00233-020-10127-0 ◽

2020 ◽

Vol 101 (2) ◽

pp. 326-357

Author(s):

Ashley Clayton

Keyword(s):

Sufficient Conditions ◽

Free Monoid ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Finitely Generated ◽

Free Monoids ◽

Fiber Product ◽

Finite Presentability ◽

Free Semigroups ◽

Necessary And Sufficient

Abstract We consider necessary and sufficient conditions for finite generation and finite presentability for fiber products of free semigroups and free monoids. We give a necessary and sufficient condition on finite fiber quotients for a fiber product of two free monoids to be finitely generated, and show that all such fiber products are also finitely presented. By way of contrast, we show that fiber products of free semigroups over finite fiber quotients are never finitely generated. We then consider fiber products of free semigroups over infinite semigroups, and show that for such a fiber product to be finitely generated, the quotient must be infinite but finitely generated, idempotent-free, and $$\mathcal {J}$$ J -trivial. Finally, we construct automata accepting the indecomposable elements of the fiber product of two free monoids/semigroups over free monoid/semigroup fibers, and give a necessary and sufficient condition for such a product to be finitely generated.

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INDICABLE GROUPS AND ENDOMORPHIC PRESENTATIONS

Glasgow Mathematical Journal ◽

10.1017/s0017089511000632 ◽

2011 ◽

Vol 54 (2) ◽

pp. 335-344

Author(s):

MUSTAFA GÖKHAN BENLI

Keyword(s):

Normal Subgroup ◽

Semidirect Product ◽

Finitely Generated ◽

Finitely Presented Groups ◽

Finitely Presented Group ◽

Free Semigroups ◽

Finitely Presented ◽

Indicable Group

AbstractIn this paper we look at presentations of subgroups of finitely presented groups with infinite cyclic quotients. We prove that if H is a finitely generated normal subgroup of a finitely presented group G with G/H cyclic, then H has ascending finite endomorphic presentation. It follows that any finitely presented indicable group without free semigroups has the structure of a semidirect product H ⋊ ℤ, where H has finite ascending endomorphic presentation.

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On Cogrowth, Amenability, and the Spectral Radius of a Random Walk on a Semigroup

International Mathematics Research Notices ◽

10.1093/imrn/rny125 ◽

2018 ◽

Vol 2020 (12) ◽

pp. 3753-3793

Author(s):

Robert D Gray ◽

Mark Kambites

Keyword(s):

Random Walks ◽

Spectral Radius ◽

Broad Class ◽

Sufficient Conditions ◽

Inverse Semigroups ◽

Finitely Generated ◽

Regular Semigroups ◽

Markov Operators ◽

Free Semigroups ◽

The Relationship

Abstract We introduce two natural notions of cogrowth for finitely generated semigroups —one local and one global — and study their relationship with amenability and random walks. We establish the minimal and maximal possible values for cogrowth rates, and show that nonmonogenic-free semigroups are exactly characterised by minimal global cogrowth. We consider the relationship with cogrowth for groups and with amenability of semigroups. We also study the relationship with random walks on finitely generated semigroups, and in particular the spectral radius of the associated Markov operators (when defined) on $\ell _2$-spaces. We show that either of maximal global cogrowth or the weak Følner condition suffices for the spectral radius to be at least $1$; since left amenability implies the weak Følner condition, this represents a generalisation to semigroups of one implication of Kesten’s Theorem for groups. By combining with known results about amenability, we are able to establish a number of new sufficient conditions for (left or right) amenability in broad classes of semigroups. In particular, maximal local cogrowth left implies amenability in any left reversible semigroup, while maximal global cogrowth (which is a much weaker property) suffices for left amenability in an extremely broad class of semigroups encompassing all inverse semigroups, left reversible left cancellative semigroups and left reversible regular semigroups.

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Automorphisms of finite order of nilpotent groups II

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.51.2014.4.1297 ◽

2014 ◽

Vol 51 (4) ◽

pp. 547-555 ◽

Cited By ~ 1

Author(s):

B. Wehrfritz

Keyword(s):

Nilpotent Group ◽

Finite Order ◽

Finite Index ◽

Nilpotent Groups ◽

Finitely Generated

Let G be a nilpotent group with finite abelian ranks (e.g. let G be a finitely generated nilpotent group) and suppose φ is an automorphism of G of finite order m. If γ and ψ denote the associated maps of G given by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\gamma :g \mapsto g^{ - 1} \cdot g\phi and \psi :g \mapsto g \cdot g\phi \cdot g\phi ^2 \cdots \cdot \cdot g\phi ^{m - 1} for g \in G,$$ \end{document} then Gγ · kerγ and Gψ · ker ψ are both very large in that they contain subgroups of finite index in G.

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