15 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
Commutator Leavitt path algebras
For any field K and directed graph E, we completely describe the elements of
the Leavitt path algebra L_K(E) which lie in the commutator subspace
[L_K(E),L_K(E)]. We then use this result to classify all Leavitt path algebras
L_K(E) that satisfy L_K(E)=[L_K(E),L_K(E)]. We also show that these Leavitt
path algebras have the additional (unusual) property that all their Lie ideals
are (ring-theoretic) ideals, and construct examples of such rings with various
ideal structures.Comment: 24 page
Topological Graph Inverse Semigroups
To every directed graph one can associate a \emph{graph inverse semigroup} , where elements roughly correspond to possible paths in . These semigroups generalize polycylic monoids, and they arise in the study of Leavitt path algebras, Cohn path algebras, Cuntz-Krieger -algebras, and Toeplitz -algebras. We investigate topologies that turn into a topological semigroup. For instance, we show that in any such topology that is Hausdorff, must be discrete for any directed graph . On the other hand, need not be discrete in a Hausdorff semigroup topology, and for certain graphs , admits a semigroup topology in which is not discrete. We also describe, in various situations, the algebraic structure and possible cardinality of the closure of in larger topological semigroups.Peer reviewe
\'Etale groupoids and Steinberg algebras, a concise introduction
We give a concise introduction to (discrete) algebras arising from \'etale
groupoids, (aka Steinberg algebras) and describe their close relationship with
groupoid C*-algebras. Their connection to partial group rings via inverse
semigroups also explored
Infinite-dimensional diagonalization and semisimplicity
We characterize the diagonalizable subalgebras of End(V), the full ring of linear operators on a vector space V over a field, in a manner that directly generalizes the classical theory of diagonalizable algebras of operators on a finite-dimensional vector space. Our characterizations are formulated in terms of a natural topology (the “finite topology”) on End(V), which reduces to the discrete topology in the case where V is finite-dimensional. We further investigate when two subalgebras of operators can and cannot be simultaneously diagonalized, as well as the closure of the set of diagonalizable operators within End(V). Motivated by the classical link between diagonalizability and semisimplicity, we also give an infinite-dimensional generalization of the Wedderburn–Artin theorem, providing a number of equivalent characterizations of left pseudocompact, Jacoboson semisimple rings that parallel various characterizations of artinian semisimple rings. This theorem unifies a number of related results in the literature, including the structure of linearly compact, Jacobson semsimple rings and cosemisimple coalgebras over a field
The Bergman-Shelah Preorder on Transformation Semigroups
Let \nat^\nat be the semigroup of all mappings on the natural numbers \nat, and let and be subsets of \nat^\nat. We write if there exists a countable subset of \nat^\nat such that is contained in the subsemigroup generated by and . We give several results about the structure of the preorder . In particular, we show that a certain statement about this preorder is equivalent to the Continuum Hypothesis. The preorder is analogous to one introduced by Bergman and Shelah on subgroups of the symmetric group on \nat. The results in this paper suggest that the preorder on subsemigroups of \nat^\nat is much more complicated than that on subgroups of the symmetric group