Tight representations of semilattices and inverse semigroups

By R. Exel. 2007

48 citations recorded by the source corpus.

AbstractBy a Boolean inverse semigroup we mean an inverse semigroup whose semilattice of idempotents is a Boolean algebra. We study representations of a given inverse semigroup $\mathcal{S}$ in a Boolean inverse semigroup which are tight in a certain well defined technical sense. These representations are supposed to preserve as much as possible any trace of Booleanness present in the semilattice of idempotents of  $\mathcal{S}$ . After observing that the Vagner–Preston representation is not tight, we exhibit a canonical tight representation for any inverse semigroup with zero, called the regular tight representation. We then tackle the question as to whether this representation is faithful, but it turns out that the answer is often negative. The lack of faithfulness is however completely understood as long as we restrict to continuous inverse semigroups, a class generalizing the E*-unitaries.

Subfields: Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/dd3b7dd0b474a81b78d86c37f1712403a81830a5

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In Computational Theory and Mathematics

1. Jack Spielberg
2. Mark V. Lawson
3. M. Norling
4. Daniel Lenz
5. David G. Jones
6. R. Exel
7. Xin Li
8. Charles Starling

In Mathematical Physics

1. Mark V. Lawson
2. M. Norling
3. David G. Jones
4. Daniel Lenz
5. Jack Spielberg
6. R. Exel
7. Xin Li
8. Ben Steinberg

Citations in the corpus

• Graph inverse semigroups: their characterization and completion (2011, background): “A much more detailed discussion of cover-to-join maps and how they originated from the work of Exel [9] and Lenz [37]can be found in [32] which can be viewed as a substantial generalization of this paper.”
• Partial transformation groupoids attached to graphs and semigroups (2016, unknown): “The boundary quotient in Exel’s sense (see [9, 10, 11, 13]) is given by ∂C λ(S) := C ∗ λ(S)/ 〈I〉.”
• A NONCOMMUTATIVE GENERALIZATION OF STONE DUALITY (2009, background): “[5] A noncommutative generalization of Stone duality 389”
• Pseudogroups and their étale groupoids (2011, background): “This set the stage for [16], where the first author showed that the topological groupoids arising in the case of the Cuntz C∗-algebras could be constructed from a class of inverse monoids, called boolean inverse monoids, in a way generalising the classical Stone duality between unital boolean algebr”
• Groupoids and C * -algebras for categories of paths (2011, methodology): “used to define a boundary space, for directed graphs and their generalizations, in [31], [13], and [6].”
• Inverse Semigroup C*-Algebras Associated with Left Cancellative Semigroups (2012, background): “on of Lif this πis an isomorphism. The fact that πisn’t always an isomorphism is related to the problem of finding the right way to map semilattices into Boolean algebras which is discussed by Exel in [11, 12]. Before we investigate this we want to discuss filters, which is a concept that is important”
• Inverse semigroups and combinatorial C*-algebras (2007, background): “Attempting to understand all of these algebras from a single perspective, I have been interested in the notion of semigroupoid C*-algebras [ 10 ], which includes the Cuntz-Krieger algebras and the higher rank graph C*-algebras in the general infinite case, provided some technical complications are n”