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