The Inverse Hull of 0-Left Cancellative Semigroups

Exel, Ruy; Steinberg, Benjamin

doi:10.1142/9789813272880_0109

Cited by 8 publications

(13 citation statements)

References 41 publications

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“…Note that since intersection distributes over union, a semigroup is right (left) ideal Howson if and only if the intersection of principal right (left) ideals is finitely generated; we use this fact throughout this article. We remark that a monoid is right ideal Howson if and only if it is finitely aligned [7]; for semigroups being finitely aligned is a stronger condition, as we demonstrate. From this point we will explicitly refer to and give results for right ideal Howson semigroups; clearly, the dual results hold for left ideal Howson semigroups.…”

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confidence: 55%

“…The notion of being finitely aligned [7] is closely connected with that of being right ideal Howson, being a stronger condition (Lemma 2.1); and coincides for many semigroups, including monoids. Indeed, it is noted in [7] that finitely aligned semigroups may be called right Howson. However, the situation for semigroups is more subtle; as we show in Remark 5.8, a right ideal Howson semigroup need not be finitely aligned.…”

mentioning

confidence: 99%

“…This connection has been developed by a number of authors such as Lawson [19], McAlister [21] and Reilly [24]. The notion of being finitely aligned [7] is closely connected with that of being right ideal Howson, being a stronger condition (Lemma 2.1); and coincides for many semigroups, including monoids. Indeed, it is noted in [7] that finitely aligned semigroups may be called right Howson.…”

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confidence: 99%

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Right ideal Howson semigroups

Carson

Gould

2020

Semigroup Forum

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We define a semigroup S to be right ideal Howson if the intersection of any two finitely generated right ideals, or, equivalently, any two principal right ideals, is again finitely generated. We give many examples of such semigroups, including right coherent monoids, finitely aligned semigroups, and inverse semigroups. We investigate the closure of the class of right ideal Howson semigroups under algebraic constructions. For any n 2 N 0 we give a presentation of a right ideal Howson semigroup possessing an intersection of principal right ideals that requires exactly n generators that is, in a particular sense, universal. We give analogous presentations for commutative, and for commutative cancellative, (right) ideal Howson semigroups. Keywords Semigroup Á Monoid Á One-sided ideal Á Presentations 1 Introduction An algebra exhibits the Howson property if the intersection of two finitely generated subalgebras is also finitely generated. This term is in honour of the author of [15], who showed that the intersection of finitely generated subgroups of free groups is finitely generated. There have been a number of investigations of the Howson property for other classes of algebras. In particular, the Howson property for inverse semigroups has been studied by several authors such as Jones and Trotter [16, 17], Lawson and Vdovina [20] and Silva and Soares [25]. By contrast to the situation for Communicated by Mark V. Lawson. The first author acknowledges the support of EPSRC in the form of a Ph.D. studentship.

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confidence: 99%

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confidence: 99%

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Right ideal Howson semigroups

Carson

Gould

2020

Semigroup Forum

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“…Matsumoto then studied orbit equivalence, eventual conjugacy and two-sided conjugacy of these λ-graphs and how they are reflected in the C * -algebras [37,40]. Recently, Exel and Steinberg have further investigated semigroups of shift spaces and shown that there is a universal groupoid which can be suitably restricted to model either Matsumoto's C * -algebras or O X , [23,Theorem 10.3].…”

Section: Introductionmentioning

confidence: 99%

𝐶*-algebras, groupoids and covers of shift spaces

Brix¹,

Carlsen²

2020

Trans. Amer. Math. Soc. Ser. B

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To every one-sided shift space X we associate a cover X, a groupoid G X and a C *-algebra O X. We characterize one-sided conjugacy, eventual conjugacy and (stabilizer-preserving) continuous orbit equivalence between X and Y in terms of isomorphism of G X and G Y , and diagonal-preserving *-isomorphism of O X and O Y. We also characterize two-sided conjugacy and flow equivalence of the associated two-sided shift spaces Λ X and Λ Y in terms of isomorphism of the stabilized groupoids G X × R and G Y × R, and diagonal-preserving *-isomorphism of the stabilized C *-algebras O X ⊗ K and O Y ⊗ K. Our strategy is to lift relations on the shift spaces to similar relations on the covers. Restricting to the class of sofic shifts whose groupoids are effective, we show that it is possible to recover the continuous orbit equivalence class of X from the pair (O X , C(X)), and the flow equivalence class of Λ X from the pair (O X ⊗ K, C(X) ⊗ c 0). In particular, continuous orbit equivalence implies flow equivalence for this class of shift spaces. Contents 154 7. Two-sided conjugacy 166 8. Flow equivalence 170 References 182

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“…Zigzag inverse semigroups appeared in Spielberg's construction of the C *algebra of a category of paths [10] and more recently in the construction due to Bédos, Kaliszewski, Quigg, and Spielberg of the C * -algebra of a left cancellative small category [3]. Exel and Steinberg [6] have recently studied a related and even more general class of semigroups, though that construction is not considered in this paper.…”

Section: Introductionmentioning

confidence: 99%

On zigzag maps and the path category of an inverse semigroup

et al. 2019

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We study the path category of an inverse semigroup admitting unique maximal idempotents and give an abstract characterization of the inverse semigroups arising from zigzag maps on a left cancellative category. As applications we show that every inverse semigroup is Morita equivalent to an inverse semigroup of zigzag maps and hence the class of Cuntz-Krieger C * -algebras of singly aligned categories include the tight C * -algebras of all countable inverse semigroups, up to Morita equivalence.

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The Inverse Hull of 0-Left Cancellative Semigroups

Cited by 8 publications

References 41 publications

Right ideal Howson semigroups

Right ideal Howson semigroups

𝐶*-algebras, groupoids and covers of shift spaces

On zigzag maps and the path category of an inverse semigroup

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