On non-Hopfian groups of fractions

Olga 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 -