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A topological correspondence between partial actions of groups and inverse semigroup actions

  • Luis Martínez , Héctor Pinedo ORCID logo EMAIL logo and Carlos Uzcátegui
Published/Copyright: March 1, 2022
Forum Mathematicum
From the journal Forum Mathematicum Volume 34 Issue 2

Abstract

We present some generalizations of the well-known correspondence, found by Exel, between partial actions of a group G on a set X and semigroup homomorphism of 𝒮⁡(G) on the semigroup I⁢(X) of partial bijections of X, with 𝒮⁡(G) being an inverse monoid introduced by Exel. We show that any unital premorphism θ:G→S, where S is an inverse monoid, can be extended to a semigroup homomorphism θ*:T→S for any inverse semigroup T with 𝒮⁡(G)⊆T⊆P*⁢(G)×G, with P*⁢(G) being the semigroup of non-empty subsets of G, and such that E⁢(S) satisfies some lattice-theoretical condition. We also consider a topological version of this result. We present a minimal Hausdorff inverse semigroup topology on Γ⁢(X), the inverse semigroup of partial homeomorphisms between open subsets of a locally compact Hausdorff space X.

Keywords: Topological partial action; partial homeomorphism; Birget–Rhodes expansion; compact open topology; Fell topology; small semilattices
MSC 2010: 54H15; 57S99; 20M18

Communicated by Siegfried Echterhoff

Funding source: Universidad Industrial de Santander

Award Identifier / Grant number: VIE-8041

Funding statement: This work was partially supported by grants No. 4.583 of Fundación para la Promoción de la Investigación y la Tecnología del Banco de La República (Colombia) and project VIE-8041 of Universidad Industrial de Santander.

Acknowledgements

We are very thankful to the referee for all their comments that improved the presentation of the paper.

References

[1] F. Abadie, Enveloping actions and Takai duality for partial actions, J. Funct. Anal. 197 (2003), no. 1, 14–67. 10.1016/S0022-1236(02)00032-0Search in Google Scholar

[2] A. M. Abd-Allah and R. Brown, A compact-open topology on partial maps with open domain, J. Lond. Math. Soc. (2) 21 (1980), no. 3, 480–486. 10.1112/jlms/s2-21.3.480Search in Google Scholar

[3] J. Álvarez López and M. F. Moreira Galicia, Topological Molino’s theory, Pacific J. Math. 280 (2016), no. 2, 257–314. 10.2140/pjm.2016.280.257Search in Google Scholar

[4] R. Arens, Topologies for homeomorphism groups, Amer. J. Math. 68 (1946), 593–610. 10.2307/2371787Search in Google Scholar

[5] D. Bagio, D. Flores and A. Paques, Partial actions of ordered groupoids on rings, J. Algebra Appl. 9 (2010), no. 3, 501–517. 10.1142/S021949881000404XSearch in Google Scholar

[6] G. Beer, Topologies on Closed and Closed Convex Sets, Math. Appl. 268, Kluwer Academic, Dordrecht, 1993. 10.1007/978-94-015-8149-3Search in Google Scholar

[7] J.-C. Birget and J. Rhodes, Almost finite expansions of arbitrary semigroups, J. Pure Appl. Algebra 32 (1984), no. 3, 239–287. 10.1016/0022-4049(84)90092-6Search in Google Scholar

[8] A. Buss and R. Exel, Inverse semigroup expansions and their actions on C*-algebras, Illinois J. Math. 56 (2012), no. 4, 1185–1212. 10.1215/ijm/1399395828Search in Google Scholar

[9] K. Choi, Birget–Rhodes expansions of topological groups, Adv. Stud. Contemp. Math. (Kyungshang) 23 (2013), no. 1, 203–211. Search in Google Scholar

[10] A. Di Concilio and S. A. Naimpally, Proximal set-open topologies on partial maps, Acta Math. Hungar. 88 (2000), no. 3, 227–237. 10.1023/A:1006717331197Search in Google Scholar

[11] J. J. Dijkstra, On homeomorphism groups and the compact-open topology, Amer. Math. Monthly 112 (2005), no. 10, 910–912. 10.1080/00029890.2005.11920266Search in Google Scholar

[12] J. J. Dijkstra and R. Tahri, Homeomorphism groups and the topologist’s sine curve, Bull. Pol. Acad. Sci. Math. 58 (2010), no. 3, 269–272. 10.4064/ba58-3-9Search in Google Scholar

[13] M. Dokuchaev, Recent developments around partial actions, São Paulo J. Math. Sci. 13 (2019), no. 1, 195–247. 10.1007/s40863-018-0087-ySearch in Google Scholar

[14] L. Elliott, J. Jonušas, Z. Mesyan, J. Mitchell, M. Morayne and Y. Péresse, Automatic continuity, unique Polish topologies, and Zariski topologies (Part I, monoids), preprint (2020), https://arxiv.org/abs/1912.07029v3. Search in Google Scholar

[15] R. Exel, Circle actions on C*-algebras, partial automorphisms, and a generalized Pimsner–Voiculescu exact sequence, J. Funct. Anal. 122 (1994), no. 2, 361–401. 10.1006/jfan.1994.1073Search in Google Scholar

[16] R. Exel, Partial actions of groups and actions of inverse semigroups, Proc. Amer. Math. Soc. 126 (1998), no. 12, 3481–3494. 10.1090/S0002-9939-98-04575-4Search in Google Scholar

[17] R. Exel, Partial Dynamical Systems, Fell Bundles and Applications, Math. Surveys Monogr. 224, American Mathematical Society, Providence, 2017. 10.1090/surv/224Search in Google Scholar

[18] G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove and D. S. Scott, A Compendium of Continuous Lattices, Springer, Berlin, 1980. 10.1007/978-3-642-67678-9Search in Google Scholar

[19] V. Gould and C. Hollings, Partial actions of inverse and weakly left E-ample semigroups, J. Aust. Math. Soc. 86 (2009), no. 3, 355–377. 10.1017/S1446788708000542Search in Google Scholar

[20] V. Gould and C. Hollings, Actions and partial actions of inductive constellations, Semigroup Forum 82 (2011), no. 1, 35–60. 10.1007/s00233-010-9279-1Search in Google Scholar

[21] J. Kellendonk and M. V. Lawson, Partial actions of groups, Internat. J. Algebra Comput. 14 (2004), no. 1, 87–114. 10.1142/S0218196704001657Search in Google Scholar

[22] J. D. Lawson, Topological semilattices with small semilattices, J. Lond. Math. Soc. (2) 1 (1969), 719–724. 10.1112/jlms/s2-1.1.719Search in Google Scholar

[23] J. D. Lawson, Lattices with no interval homomorphisms, Pacific J. Math. 32 (1970), 459–465. 10.2140/pjm.1970.32.459Search in Google Scholar

[24] M. V. Lawson, Inverse Semigroups, World Scientific, Singapore, 1998. 10.1142/3645Search in Google Scholar

[25] J. Pérez and C. Uzcátegui, Topologies on the symmetric inverse semigroup, preprint (2020), https://arxiv.org/abs/2012.03041. 10.1007/s00233-021-10242-6Search in Google Scholar

[26] M. B. Szendrei, A note on Birget–Rhodes expansion of groups, J. Pure Appl. Algebra 58 (1989), no. 1, 93–99. 10.1016/0022-4049(89)90054-6Search in Google Scholar

Received: 2021-09-10
Revised: 2021-11-30
Published Online: 2022-03-01
Published in Print: 2022-03-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. On the variance of the nodal volume of arithmetic random waves
  3. On length densities
  4. On weighted compactness of commutators of bilinear maximal Calderón–Zygmund singular integral operators
  5. Geometric nilpotent Lie algebras and zero-dimensional simple complete intersection singularities
  6. On cutting blocking sets and their codes
  7. Inertia groups and smooth structures on quaternionic projective spaces
  8. On the number of zeros of diagonal quartic forms over finite fields
  9. Weak martingale Hardy-type spaces associated with quasi-Banach function lattice
  10. A dual version of Huppert's ρ-σ conjecture for character codegrees
  11. A topological correspondence between partial actions of groups and inverse semigroup actions
  12. On homotopy braids
  13. Holomorphic convexity of pseudoconvex spaces in terms of the rank of structural sheaf
  14. The Shintani double zeta functions
  15. Homological dimensions relative to preresolving subcategories II
  16. Simplicity of indecomposable set-theoretic solutions of the Yang–Baxter equation
  17. A converse theorem for quasimodular forms
https://doi.org/10.1515/forum-2021-0232
Keywords for this article
Topological partial action; partial homeomorphism; Birget–Rhodes expansion; compact open topology; Fell topology; small semilattices

Articles in the same Issue

  1. Frontmatter
  2. On the variance of the nodal volume of arithmetic random waves
  3. On length densities
  4. On weighted compactness of commutators of bilinear maximal Calderón–Zygmund singular integral operators
  5. Geometric nilpotent Lie algebras and zero-dimensional simple complete intersection singularities
  6. On cutting blocking sets and their codes
  7. Inertia groups and smooth structures on quaternionic projective spaces
  8. On the number of zeros of diagonal quartic forms over finite fields
  9. Weak martingale Hardy-type spaces associated with quasi-Banach function lattice
  10. A dual version of Huppert's ρ-σ conjecture for character codegrees
  11. A topological correspondence between partial actions of groups and inverse semigroup actions
  12. On homotopy braids
  13. Holomorphic convexity of pseudoconvex spaces in terms of the rank of structural sheaf
  14. The Shintani double zeta functions
  15. Homological dimensions relative to preresolving subcategories II
  16. Simplicity of indecomposable set-theoretic solutions of the Yang–Baxter equation
  17. A converse theorem for quasimodular forms
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