Reconstructing the Topology of the Elementary Self-embedding Monoids of Countable Saturated Structures

Abstract

Abstract. Every transformation monoid comes equipped with a canonical topology-the topology of pointwise convergence. For some structures, the topology of the endomorphism monoid can be reconstructed from its underlying abstract monoid. This phenomenon is called automatic homeomorphicity.In this paper we show that whenever the automorphism group of a countable saturated structure has automatic homeomorphicity and a trivial center, then the monoid of elementary self-embeddings has automatic homeomorphicity, too.As a second result we strengthen a result by Lascar by showing that whenever A is a countable ℵ 0 -categorical G-finite structure whose automorphism group has a trivial center and if B is any other countable structure, then every isomorphism between the monoids of elementary self-embeddings is a homeomorphism. Automatic homeomorphicityIf we consider a set A as a discrete topological space, then the set of self-mappings A A of A is naturally equipped with the product topology. A sub-basis of this topology is given by {Φ a,b | a, b ∈ A}, where Φ a,b is given by Φ a,b = {f ∈ A A | f (a) = b}. This topology is also known as the Tychonoff-topology or topology of pointwise convergence.The set A A , together with the composition operation is called the full transformation monoid on A. We denote it by T A . Every submonoid M of T A is naturally equipped with the subspace-topology (in particular, the composition on M is continuous with respect to this topology).When using the predicates open or closed in connection with transformation monoids we make an implicit distinction between transformation monoids in general and the special case of permutation groups. A transformation monoid on A is called closed (open) whenever it is Date: March 23, 2017.

On a stronger reconstruction notion for monoids and clones

“…In this setting, as a by-product, we prove a new characterization of automatic homeomorphicity for transformation monoids on arbitrary carrier sets. It involves a weakening of the technical assumption that the only injective monoid endomorphism fixing every group member be the identical one, which has featured in several earlier reconstruction results [10,25,30,3]. We believe that our weakened version, so, by our characterization, automatic homeomorphicity, would also have been sufficient in any of these previous cases.…”

Polish topologies on endomorphism monoids of relational structures

“…It follows that the pointwise topology is the unique Polish group topology on Sym(N). Uniqueness of compatible topologies has been studied for many further groups and semigroups, and more general objects such as clones; see for example [2,4,6,11,12,14,17,22,23,24,25,26,27,28,29,31,40,41,42,43,45,47,48,49].…”