Topic
Semigroupoid
About: Semigroupoid is a research topic. Over the lifetime, 54 publications have been published within this topic receiving 1110 citations. The topic is also known as: semicategory & precategory.
Papers published on a yearly basis
Papers
1 Jun 2008
TL;DR: In this paper, the authors describe a special class of representations of an inverse semigroup S on Hilbert's space which they term tight, which are supported on a subset of the spectrum of the idempotent semilattice of S, which is in turn shown to be precisely the closure of the space of ultra-filters, once filters are identified with semicharacters in a natural way.
Abstract: We describe a special class of representations of an inverse semigroup S on Hilbert's space which we term tight. These representations are supported on a subset of the spectrum of the idempotent semilattice of S, called the tight spectrum, which is in turn shown to be precisely the closure of the space of ultra-filters, once filters are identified with semicharacters in a natural way. These representations are moreover shown to correspond to representations of the C*-algebra of the groupoid of germs for the action of S on its tight spectrum. We then treat the case of certain inverse semigroups constructed from semigroupoids, generalizing and inspired by inverse semigroups constructed from ordinary and higher rank graphs. The tight representations of this inverse semigroup are in one-to-one correspondence with representations of the semigroupoid, and consequently the semigroupoid algebra is given a groupoid model. The groupoid which arises from this construction is shown to be the same as the boundary path groupoid of Farthing, Muhly and Yeend, at least in the singly aligned, sourceless case.
369 citations
Posted Content•
TL;DR: In this article, the authors describe a special class of representations of an inverse semigroup S on Hilbert's space which they call tight, and these representations are supported on a subset of the spectrum of the idempotent semilattice of S, which is in turn shown to be precisely the closure of the space of ultra-filters, once filters are identified with semicharacters in a natural way.
Abstract: We describe a special class of representations of an inverse semigroup S on Hilbert's space which we term "tight". These representations are supported on a subset of the spectrum of the idempotent semilattice of S, called the "tight spectrum", which is in turn shown to be precisely the closure of the space of ultra-filters, once filters are identified with semicharacters in a natural way. These representations are moreover shown to correspond to representations of the C*-algebra of the groupoid of germs for the action of S on its tight spectrum.
We then treat the case of certain inverse semigroups constructed from a semigroupoid, generalizing and inspired by inverse semigroups constructed from ordinary and higher rank graphs. The tight representations of this inverse semigroup are in one-to-one correspondence with representations of the semigroupoid, and the semigroupoid algebra is given a groupoid model. The groupoid which arises from this construction is shown to be the same as the boundary path groupoid of Farthing, Muhly and Yeend, at least in the singly aligned, sourceless case.
170 citations
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TL;DR: In this paper, a lattice isomorphism between ideals and invariant subspaces is established, which leads to a complete description of the wot-closed ideal structure for free semigroupoid algebras.
Abstract: A free semigroupoid algebra is the weak operator topol- ogy closed algebra generated by the left regular representation of a directed graph. We establish lattice isomorphisms between ideals and invariant subspaces, and this leads to a complete description of the wot-closed ideal structure for these algebras. We prove a distance formula to ideals, and this gives an appropriate version of the Caratheodory interpolation theorem. Our analysis rests on an investigation of predual properties, specifically the An properties for linear functionals, together with a general Wold Decomposi- tion for n-tuples of partial isometries. A number of our proofs unify proofs for subclasses appearing in the literature.
63 citations
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TL;DR: In this paper, it was shown that the graph is invariant for Banach algebra isomorphisms of tensor algebras of graphs and for algebraic isomorphism of countable directed graphs with no sinks.
Abstract: In this paper we verify that the graph forms a complete invariant for Banach algebra isomorphisms of tensor algebras of graphs. For tensor algebras associated with countable directed graphs having no sinks the graph forms an invariant for algebraic isomorphisms as well. The graph also forms a complete invariant for w*-bicontinuous isomorphisms of free semigroupoid algebras. For free semigroupoid algebras associated with countable directed locally finite graphs with no sinks, the graph also forms an invariant for algebraic isomorphisms.
47 citations
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TL;DR: In this article, the authors introduce the concept of a semigroup coupled cell network and show that the collection of semigroup network vector fields forms a Lie algebra, which implies that near a dynamical equilibrium, the local normal form of a semi-giant network is the same as the original network itself.
Abstract: We introduce the concept of a semigroup coupled cell network and show that the collection of semigroup network vector fields forms a Lie algebra. This implies that near a dynamical equilibrium the local normal form of a semigroup network is a semigroup network itself. Networks without the semigroup property will support normal forms with a more general network architecture, but these normal forms nevertheless possess the same symmetries and synchronous solutions as the original network. We explain how to compute Lie brackets and normal forms of coupled cell networks and we characterize the SN-decomposition that determines the normal form symmetry. This paper concludes with a generalization to nonhomogeneous networks with the structure of a semigroupoid.
33 citations
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Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 2 |
| 2020 | 4 |
| 2019 | 2 |
| 2018 | 2 |
| 2017 | 2 |
| 2016 | 1 |