On zigzag maps and the path category of an inverse semigroup

Donsig, Allan P.; Gensler, Jennifer; King, Hannah; Milan, David J.; Wdowinski, Ronen

doi:10.48550/arxiv.1811.04124

Cited by 1 publication

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“…Moreover, since E * is singly aligned, then so is E * ⋊ ϕ G by [2, Proposition 3.12(ii)]. Thus, by [5,Theorem 3.2],…”

Section: Zappa-szép Products Of Lcsc Categoriesmentioning

confidence: 92%

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The tight groupoid of the inverse semigroups of left cancellative small categories

Ortega¹,

Pardo²

2020

Trans. Amer. Math. Soc.

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We fix a path model for the space of filters of the inverse semigroup S Λ associated to a left cancellative small category Λ. Then, we compute its tight groupoid, thus giving a representation of its C * -algebra as a (full) groupoid algebra. Using it, we characterize when these algebras are simple. Also, we determine amenability of the tight groupoid under mild, reasonable hypotheses. 2010 Mathematics Subject Classification. Primary: 46L05; Secondary: 46L80, 46L55, 20L05. . α ∨ β := {the minimal extensions of α and β} .Notice that if α ∨ β = ∅ then α ⋓ β, but the converse fails in general.When Λ is a finitely aligned LCSC, we can always assume that α ∨ β = Γ where Γ is a finite set of minimal common extensions of α and β. 1.2. Inverse semigroups. Definition 1.9. A semigroup S is an inverse semigroup if for every s ∈ S there exists a unique s * ∈ S such that s = ss * s and s * = s * ss * . Equivalently, S is an inverse semigroup if and only if the subsemigroup E(S) := {e ∈ S : e 2 = e} of idempotents of S is commutative [12, Theorem 1.1.3].A monoid is a semigroup with unit. We say that a semigroup S has zero if there exists 0 ∈ S such that 0s = s0 = 0 for every s ∈ S.Definition 1.10. Given a set X, we define the (symmetric) inverse semigroup on X as

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“…Moreover, since E * is singly aligned, then so is E * ⋊ ϕ G by [2, Proposition 3.12(ii)]. Thus, by [5,Theorem 3.2],…”

Section: Zappa-szép Products Of Lcsc Categoriesmentioning

confidence: 92%

“…Notice that, if Λ is singly aligned, then every filter enjoy condition ( * ) (see e.g. [5,Proposition 3.5]).…”

Section: Path Modelsmentioning

confidence: 99%

The tight groupoid of the inverse semigroups of left cancellative small categories

Ortega¹,

Pardo²

2020

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

On zigzag maps and the path category of an inverse semigroup

Cited by 1 publication

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The tight groupoid of the inverse semigroups of left cancellative small categories

The tight groupoid of the inverse semigroups of left cancellative small categories

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