10 research outputs found
Automatic continuity, unique Polish topologies, and Zariski topologies on monoids and clones
In this paper we explore the extent to which the algebraic structure of a
monoid determines the topologies on that are compatible with its
multiplication. Specifically we study the notions of automatic continuity;
minimal Hausdorff or Polish semigroup topologies; and we formulate a notion of
the Zariski topology for monoids.
If is a topological monoid such that every homomorphism from to a
second countable topological monoid is continuous, then we say that has
\emph{automatic continuity}. We show that many well-known monoids have
automatic continuity with respect to a natural semigroup topology, namely: the
full transformation monoid ; the full binary relation
monoid ; the partial transformation monoid ;
the symmetric inverse monoid ; the monoid Inj
consisting of the injective functions on ; and the monoid
of continuous functions on the Cantor set.
We show that the pointwise topology on , and its
analogue on , are the unique Polish semigroup topologies on
these monoids. The compact-open topology is the unique Polish semigroup
topology on and . There are at least 3
Polish semigroup topologies on , but a unique Polish inverse
semigroup topology. There are no Polish semigroup topologies
nor on the partitions monoids. At the other extreme, Inj and the
monoid Surj of all surjective functions on each have
infinitely many distinct Polish semigroup topologies. We prove that the Zariski
topologies on , , and Inj
coincide with the pointwise topology; and we characterise the Zariski topology
on . In Section 7: clones.Comment: 51 pages (Section 7 about clones was added in version 4
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