Polish topologies on endomorphism monoids of relational structures

Luke Elliott, Julius Jonusas, James D. Mitchell, Yann Peresse, Michael Pinsker

Research output: Contribution to journalArticlepeer-review


In this paper we present general techniques for characterising minimal and maximal semigroup topologies on the endomorphism monoid End(A) of a countable relational structure A. As applications, we show that the endomorphism monoids of several well-known relational structures, including the random graph, the random directed graph, and the random partial order, possess a unique Polish semigroup topology. In every case this unique topology is the subspace topology induced by the usual topology on the Baire space N N. We also show that many of these structures have the property that every homomorphism from their endomorphism monoid to a second countable topological semigroup is continuous; referred to as automatic continuity. Many of the results about endomorphism monoids are extended to clones of polymorphisms on the same structures.

Original languageEnglish
Article number109214
Pages (from-to)1-37
Number of pages37
JournalAdvances in Mathematics
Early online date3 Aug 2023
Publication statusPublished - 15 Oct 2023


  • Semigroups
  • Topology
  • Group Theory
  • Combinatorics
  • Automatic continuity
  • Reconstruction
  • Endomorphism monoid
  • Pointwise convergence topology
  • Polish topology


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