Endomorphisms of semigroups: growth and interactions with subsemigroups
- Prelegent(ci)
- Alan Cain
- Afiliacja
- Universidade Nova de Lisboa
- Termin
- 2 grudnia 2021 12:15
- Informacje na temat wydarzenia
- Zoom
- Seminarium
- Seminarium „Algebra”
This talk is divided into two parts: the first will cover the concept of the growth of an endomorphism of a subsemigroup. Informally, this
is a measure of how much balls in the Cayley graph of a semigroup are stretched by iterations of the endomorphism. I will describe how every real number $r > 1$ arises as the growth of some semigroup endomorphism, but the main focus will be on how the growth of an
endomorphism of a semigroup interacts with the growth of its restriction to suitable subsemigroups (which must be preserved by the
endomorphism).
The second part of the talk examines hopficity and co-hopficity. A semigroup is hopfian if every surjective endomorphism is also
injective (equivalently, the semigroup is not a proper homomorphic image of itself). A semigroup is co-hopfian if every injective
endomorphism is also surjective (equivalently, the semigroup is not isomorphic to any proper subsemigroup of itself). Maltcev and Ruškuc
proved that if $S$ is a finite Rees index extension of a finitely generated hopfian semigroup $T$, then $S$ is itself hopfian, but that
the converse does not hold, and that the result does not hold without the hypothesis of finite generation. I will describe the corresponding results for co-hopficity.
This is joint work with Victor Maltcev.