# On non-Hopfian groups of fractions

Open Mathematics (2017)

- Volume: 15, Issue: 1, page 398-403
- ISSN: 2391-5455

## Access Full Article

top## Abstract

top## How to cite

topOlga Macedońska. "On non-Hopfian groups of fractions." Open Mathematics 15.1 (2017): 398-403. <http://eudml.org/doc/288121>.

@article{OlgaMacedońska2017,

abstract = {The group of fractions of a semigroup S, if exists, can be written as G = SS−1. If S is abelian, then G must be abelian. We say that a semigroup identity is transferable if being satisfied in S it must be satisfied in G = SS−1. One of problems posed by G.Bergman in 1981 asks whether the group G must satisfy every semigroup identity which is satisfied in S, that is whether every semigroup identity is transferable. The first non-transferable identities were constructed in 2005 by S.V.Ivanov and A.M. Storozhev. A group G is called Hopfian if each epimorphizm G → G is the automorphism. The residually finite groups are Hopfian, however there are many problems concerning the Hopfian property e.g. of infinite Burnside groups, of finitely generated relatively free groups [11, Problem 15]. We prove here that if G = SS−1 is an n-generator group of fractions of a relatively free semigroup S, satisfying m-variable (m < n) non-transferable identity, then G is the non-Hopfian group.},

author = {Olga Macedońska},

journal = {Open Mathematics},

keywords = {Positive law; Hopfian group; positive law},

language = {eng},

number = {1},

pages = {398-403},

title = {On non-Hopfian groups of fractions},

url = {http://eudml.org/doc/288121},

volume = {15},

year = {2017},

}

TY - JOUR

AU - Olga Macedońska

TI - On non-Hopfian groups of fractions

JO - Open Mathematics

PY - 2017

VL - 15

IS - 1

SP - 398

EP - 403

AB - The group of fractions of a semigroup S, if exists, can be written as G = SS−1. If S is abelian, then G must be abelian. We say that a semigroup identity is transferable if being satisfied in S it must be satisfied in G = SS−1. One of problems posed by G.Bergman in 1981 asks whether the group G must satisfy every semigroup identity which is satisfied in S, that is whether every semigroup identity is transferable. The first non-transferable identities were constructed in 2005 by S.V.Ivanov and A.M. Storozhev. A group G is called Hopfian if each epimorphizm G → G is the automorphism. The residually finite groups are Hopfian, however there are many problems concerning the Hopfian property e.g. of infinite Burnside groups, of finitely generated relatively free groups [11, Problem 15]. We prove here that if G = SS−1 is an n-generator group of fractions of a relatively free semigroup S, satisfying m-variable (m < n) non-transferable identity, then G is the non-Hopfian group.

LA - eng

KW - Positive law; Hopfian group; positive law

UR - http://eudml.org/doc/288121

ER -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.