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On the minimal number of generators of endomorphism monoids of full shifts

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Abstract

For a group G and a finite set A, denote by \(\mathrm{End}(A^G)\) the monoid of all continuous shift commuting self-maps of \(A^G\) and by \(\mathrm{Aut}(A^G)\) its group of units. We study the minimal cardinality of a generating set, known as the rank, of \(\mathrm{End}(A^G)\) and \(\mathrm{Aut}(A^G)\). In the first part, when G is a finite group, we give upper and lower bounds for the rank of \(\mathrm{Aut}(A^G)\) in terms of the number of conjugacy classes of subgroups of G. In the second part, we apply our bounds to show that if G has an infinite descending chain of normal subgroups of finite index, then \(\mathrm{End}(A^G)\) is not finitely generated; such is the case for wide classes of infinite groups, such as infinite residually finite or infinite locally graded groups.

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Acknowledgements

Part of the research presented here was supported by a CONACYT Basic Science Grant (No. A1-S-8013) from the Government of Mexico.

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Correspondence to Alonso Castillo-Ramirez.

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Castillo-Ramirez, A. On the minimal number of generators of endomorphism monoids of full shifts. Nat Comput (2020). https://doi.org/10.1007/s11047-020-09785-4

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Keywords

  • Full shift
  • Endomorphisms
  • Automorphisms
  • Cellular automata
  • Minimal number of generators

Mathematics Subject Classification

  • 37B10
  • 68Q80
  • 05E18
  • 20M20