Hopfian And Co-Hopfian Subsemigroups And Extensions

Demonstratio Mathematica

Cain Alan J.¹  Maltcev Victor² 

[1] CENTRO DE MATEMÁTICA FACULDADE DE CIÊNCIAS UNIVERSIDADE DO PORTO RUA DO CAMPO ALEGRE 687 4169–007 PORTO, PORTUGAL;MATHEMATICS DEPARTMENT TECHNION – ISRAEL INSTITUTE OF TECHNOLOGY HAIFA 32000, ISRAEL;

关键词: and phrases hopfian;    co-hopfian;    subsemigroup;    extension;    finite index;    Rees index;    Green index;   

DOI  :  10.2478/dema-2014-0064

来源: DOAJ

【 摘 要 】

This paper investigates the preservation of hopficity and co-hopficity on passing to finite-index subsemigroups and extensions. It was already known that hopficity is not preserved on passing to finite Rees index subsemigroups, even in the finitely generated case. We give a stronger example to show that it is not preserved even in the finitely presented case. It was also known that hopficity is not preserved in general on passing to finite Rees index extensions, but that it is preserved in the finitely generated case. We show that, in contrast, hopficity is not preserved on passing to finite Green index extensions, even within the class of finitely presented semigroups. Turning to co-hopficity, we prove that within the class of finitely generated semigroups, co-hopficity is preserved on passing to finite Rees index extensions, but is not preserved on passing to finite Rees index subsemigroups, even in the finitely presented case. Finally, by linking co-hopficity for graphs to co-hopficity for semigroups, we show that without the hypothesis of finite generation, co-hopficity is not preserved on passing to finite Rees index extensions.

【 授权许可】

Unknown