Polish topologies on endomorphism monoids of relational structures

Abstract

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.

Pettis property for Polish inverse semigroups

“…We show next that the previous result does not occur for a finite partition of N, namely, the generated inverse semigroup has automatic continuity. Automatic continuity for semigroups is obtained by lifting the same property from a subgroup of the semigroup ( [2,3,9]). Thus, we use the following property of the symmetric group (see also [11, § 5]).…”

“…Recently, there have been some works about automatic continuity for Polish semigroup topologies (see [2,3,9,10]). In this paper we explore a possible generalization of Pettis theorem now for Polish inverse semigroups.…”