scholarly journals Residual finiteness of finitely generated commutative semigroups

1971 ◽  
Vol 36 (1) ◽  
pp. 99-101 ◽  
Author(s):  
W. Carlisle
Keyword(s):  
Residual Finiteness ◽  
Finitely Generated ◽  
Commutative Semigroups

scholarly journals ON SEPARABILITY FINITENESS CONDITIONS IN SEMIGROUPS

2021 ◽  
pp. 1-29
Author(s):  
CRAIG MILLER ◽  
GERARD O’REILLY ◽  
MARTYN QUICK ◽  
NIK RUŠKUC
Keyword(s):  
Commutative Semigroup ◽  
Residual Finiteness ◽  
Finitely Generated ◽  
Finiteness Conditions ◽  
Commutative Semigroups ◽  
Starting Point ◽  
Finiteness Properties

Abstract Taking residual finiteness as a starting point, we consider three related finiteness properties: weak subsemigroup separability, strong subsemigroup separability and complete separability. We investigate whether each of these properties is inherited by Schützenberger groups. The main result of this paper states that for a finitely generated commutative semigroup S, these three separability conditions coincide and are equivalent to every $\mathcal {H}$ -class of S being finite. We also provide examples to show that these properties in general differ for commutative semigroups and finitely generated semigroups. For a semigroup with finitely many $\mathcal {H}$ -classes, we investigate whether it has one of these properties if and only if all its Schützenberger groups have the property.

scholarly journals The residual finiteness of HNN-extensions and generalized free products of nilpotent groups: a characterization

1992 ◽  
Vol 53 (3) ◽  
pp. 408-420 ◽  
Author(s):  
E. Raptis ◽  
D. Varsos
Keyword(s):  
Nilpotent Groups ◽  
Residual Finiteness ◽  
Finitely Generated ◽  
Free Products ◽  
Hnn Extensions ◽  
Base Group ◽  
Generalized Free Products

AbstractWe study the residual finiteness of free products with amalgamations and HNN-extensions of finitely generated nilpotent groups. We give a characterization in terms of certain conditions satisfied by the associated subgroups. In particular the residual finiteness of these groups implies the possibility of extending the isomorphism of the associated subgroups to an isomorphism of their isolated closures in suitable overgroups of the factors (or the base group in case of HNN-extensions).

On the Residual Finiteness of Polygonal Products of Nilpotent Groups

1992 ◽  
Vol 35 (3) ◽  
pp. 390-399 ◽  
Author(s):  
Goansu Kim ◽  
C. Y. Tang
Keyword(s):  
Nilpotent Groups ◽  
Vertex Group ◽  
Residual Finiteness ◽  
Finitely Generated ◽  
Residually Finite ◽  
Cyclic Subgroups ◽  
Isolated Subgroup ◽  
Torsion Free

AbstractIn general polygonal products of finitely generated torsion-free nilpotent groups amalgamating cyclic subgroups need not be residually finite. In this paper we prove that polygonal products of finitely generated torsion-free nilpotent groups amalgamating maximal cyclic subgroups such that the amalgamated cycles generate an isolated subgroup in the vertex group containing them, are residually finite. We also prove that, for finitely generated torsion-free nilpotent groups, if the subgroups generated by the amalgamated cycles have the same nilpotency classes as their respective vertex groups, then their polygonal product is residually finite.

scholarly journals The primitivity index function for a free group, and untangling closed curves on hyperbolic surfaces.With the appendix by Khalid Bou–Rabee

2017 ◽  
Vol 166 (1) ◽  
pp. 83-121
Author(s):  
NEHA GUPTA ◽  
ILYA KAPOVICH
Keyword(s):  
Lower Bounds ◽  
Free Group ◽  
Growth Function ◽  
Residual Finiteness ◽  
Closed Geodesics ◽  
Hyperbolic Surfaces ◽  
Finitely Generated ◽  
Closed Curves ◽  
Index Function ◽  
Finite Covers

AbstractMotivated by the results of Scott and Patel about “untangling” closed geodesics in finite covers of hyperbolic surfaces, we introduce and study primitivity, simplicity and non-filling index functions for finitely generated free groups. We obtain lower bounds for these functions and relate these free group results back to the setting of hyperbolic surfaces. An appendix by Khalid Bou–Rabee connects the primitivity index functionfprim(n,FN) to the residual finiteness growth function forFN.

scholarly journals Markov semigroups, monoids and groups

2014 ◽  
Vol 24 (05) ◽  
pp. 609-653 ◽  
Author(s):  
Alan J. Cain ◽  
Victor Maltcev
Keyword(s):  
Finite Index ◽  
Regular Language ◽  
Minimal Length ◽  
Direct Products ◽  
Markov Semigroups ◽  
Finitely Generated ◽  
Free Products ◽  
Commutative Semigroups ◽  
Generating Set ◽  
Polycyclic Groups

A group is Markov if it admits a prefix-closed regular language of unique representatives with respect to some generating set, and strongly Markov if it admits such a language of unique minimal-length representatives over every generating set. This paper considers the natural generalizations of these concepts to semigroups and monoids. Two distinct potential generalizations to monoids are shown to be equivalent. Various interesting examples are presented, including an example of a non-Markov monoid that nevertheless admits a regular language of unique representatives over any generating set. It is shown that all finitely generated commutative semigroups are strongly Markov, but that finitely generated subsemigroups of virtually abelian or polycyclic groups need not be. Potential connections with word-hyperbolic semigroups are investigated. A study is made of the interaction of the classes of Markov and strongly Markov semigroups with direct products, free products, and finite-index subsemigroups and extensions. Several questions are posed.

ON THE STRUCTURE OF FINITELY GENERATED SEMIGROUPS OF UNARY REGULAR LANGUAGES

2010 ◽  
Vol 21 (05) ◽  
pp. 689-704
Author(s):  
SERGEY AFONIN ◽  
ELENA KHAZOVA
Keyword(s):  
Regular Languages ◽  
Finitely Generated ◽  
Commutative Semigroups ◽  
Rigid Structure ◽  
Letter Alphabet

The set of all regular languages is closed under concatenation and forms a monoid known as the monoid of regular languages. In this paper the structure of finitely generated subsemigroups of this monoid in case of one letter alphabet is investigated. We prove that finitely generated semigroups of regular languages over a one letter alphabet are Kleene, rational and thus automatic. It is already known that not all of finitely generated commutative semigroups are automatic, thus we may conclude that semigroups of unary regular languages have more rigid structure.

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