Growths of endomorphisms of finitely generated semigroups

Alan J. Cain, Victor Maltcev

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This paper studies the growths of endomorphisms of finitely generated semigroups. The growth is a certain dynamical characteristic describing how iterations of the endomorphism ‘stretch’ balls in the Cayley graph of the semigroup. We make a detailed study of the relation of the growth of an endomorphism of a finitely generated semigroup and the growth of the restrictions of the endomorphism to finitely generated invariant subsemigroups. We also study the possible values endomorphism growths can attain. We show the role of linear algebra in calculating the growths of endomorphisms of homogeneous semigroups. Proofs are a mixture of syntactic algebraic rewriting techniques and analytical tricks. We state various problems and suggestions for future research.

Original languageEnglish
Pages (from-to)163-184
JournalJournal of the Australian Mathematical Society
Publication statusPublished - 1 Apr 2017


  • Cayley graph
  • endomorphism
  • finitely generated semigroups
  • growth


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