On locally compact semitopological graph inverse semigroups

S. Bardyla

doi:10.15330/ms.49.1.19-28

In this paper we investigate locally compact semitopological graph inverse semigroups. Our main result is the following: if a directed graph $E$ is strongly connected and has finitely many vertices, then any Hausdorff shift-continuous locally compact topology on the graph inverse semigroup $G(E)$ is either compact or discrete. This result generalizes results of Gutik and Bardyla who proved the above dichotomy for Hausdorff locally compact shift-continuous topologies on polycyclic monoids $\mathcal{P}_1$ and $\mathcal{P}_{\lambda}$, respectively.

1. G. Abrams, G. Aranda Pino, The Leavitt path algebra of a graph, J. Algebra, 293 (2005), 319–334.

2. A. Alali, N.D. Gilbert, Closed inverse subsemigroups of graph inverse semigroups, arXiv:1608.04538.

3. P. Ara, M.A. Moreno, E. Pardo, Non-stable K-theory for graph algebras, Algebr. Represent. Theory, 10 (2007), 157–178.

4. S. Bardyla, On universal objects in the class of graph inverse semigroups, preprint, arXiv:1709.01393v2.

5. S. Bardyla, On locally compact topological graph inverse semigroups, preprint, (2017), arXiv:1706.08594v2.

6. S. Bardyla, On locally compact shift-continuous topologies on the K-bicyclic monoid, Topological Algebra and its Applications, 6, (2018), №1, 34–42.

7. S. Bardyla, Classifying locally compact semitopological polycyclic monoids, Math. Bulletin of the Shevchenko Scientific Society, 13, (2016), 21–28.

8. S. Bardyla, O. Gutik, On a semitopological polycyclic monoid, Algebra Discr. Math., 21 (2016), №2, 163–183.

9. S. Bardyla, O. Gutik, On a complete topological inverse polycyclic monoid, Carpathian Math. Publ., 8 (2016), №2, 183–194.

10. M. Bertman, T. West, Conditionally compact bicyclic semitopological semigroups, Proc. Roy. Irish Acad., A76:21–23 (1976), 219–226.

11. A.H. Clifford, G.B. Preston, The Algebraic Theory of Semigroups, V.I and II, Amer. Math. Soc. Surveys, 7, Providence, R.I., 1961 and 1967.

12. J. Cuntz, W. Krieger, A class of $C^*$-algebras and topological Markov chains, Invent. Math., 56 (1980), 251–268.

13. C. Eberhart, J. Selden, On the closure of the bicyclic semigroup, Trans. Amer. Math. Soc., 144 (1969), 115–126.

14. R. Engelking, General Topology, 2nd ed., Heldermann, Berlin, 1989.

15. O. Gutik, On the dichotomy of the locally compact semitopological bicyclic monoid with adjoined zero, Visn. L’viv. Univ., Ser. Mekh.-Mat., 80 (2015), 33–41.

16. D. Jones, Polycyclic monoids and their generalizations, PhD Thesis, Heriot-Watt University, 2011.

17. D. Jones, M. Lawson, Graph inverse semigroups: Their characterization and completion, J. Algebra, 409 (2014), 444–473.

18. A. Kumjian, D. Pask, I. Raeburn, Cuntz-Krieger algebras of directed graphs, Pacific J. Math., 184 (1998), 161–174.

19. M. Lawson, Inverse semigroups. The theory of partial symmetries, Singapore: World Scientific, 1998.

20. M. Lawson, Primitive partial permutation representations of the polycyclic monoids and branching function systems, Period. Math. Hungar., 58 (2009), 189–207.

21. J. Meakin, M. Sapir, Congruences on free monoids and submonoids of polycyclic monoids, J. Austral. Math. Soc. Ser. A, 54 (2009), 236–253.

22. Z. Mesyan, J.D. Mitchell, The structure of a graph inverse semigroup, Semigroup Forum, 93 (2016), 111–130.

23. Z. Mesyan, J.D. Mitchell, M. Morayne, Y.H. Peresse, Topological graph inverse semigroups, Topology and its Applications, 208 (2016), 106–126.

24. M. Nivat, J.-F. Perrot, Une generalisation du monoide bicyclique, C. R. Acad. Sci., Paris, Ser. A, 271 (1970), 824–827.

25. A.L. Paterson, Graph inverse semigroups, groupoids and their $C^*$-algebras, Birkhauser, 1999.

26. W. Ruppert, Compact Semitopological Semigroups: An Intrinsic Theory, Lect. Notes Math., 1079, Springer, Berlin, 1984.

	On locally compact semitopological graph inverse semigroups
Author	S. Bardyla sbardyla@yahoo.com Ivan Franko National University of Lviv, Lviv, Ukraine
Abstract	In this paper we investigate locally compact semitopological graph inverse semigroups. Our main result is the following: if a directed graph $E$ is strongly connected and has finitely many vertices, then any Hausdorff shift-continuous locally compact topology on the graph inverse semigroup $G(E)$ is either compact or discrete. This result generalizes results of Gutik and Bardyla who proved the above dichotomy for Hausdorff locally compact shift-continuous topologies on polycyclic monoids $\mathcal{P}_1$ and $\mathcal{P}_{\lambda}$, respectively.
Keywords	locally compact space; semitopological semigroup; polycyclic monoid; graph inverse semigroup
DOI	doi:10.15330/ms.49.1.19-28
Reference	1. G. Abrams, G. Aranda Pino, The Leavitt path algebra of a graph, J. Algebra, 293 (2005), 319–334. 2. A. Alali, N.D. Gilbert, Closed inverse subsemigroups of graph inverse semigroups, arXiv:1608.04538. 3. P. Ara, M.A. Moreno, E. Pardo, Non-stable K-theory for graph algebras, Algebr. Represent. Theory, 10 (2007), 157–178. 4. S. Bardyla, On universal objects in the class of graph inverse semigroups, preprint, arXiv:1709.01393v2. 5. S. Bardyla, On locally compact topological graph inverse semigroups, preprint, (2017), arXiv:1706.08594v2. 6. S. Bardyla, On locally compact shift-continuous topologies on the K-bicyclic monoid, Topological Algebra and its Applications, 6, (2018), №1, 34–42. 7. S. Bardyla, Classifying locally compact semitopological polycyclic monoids, Math. Bulletin of the Shevchenko Scientific Society, 13, (2016), 21–28. 8. S. Bardyla, O. Gutik, On a semitopological polycyclic monoid, Algebra Discr. Math., 21 (2016), №2, 163–183. 9. S. Bardyla, O. Gutik, On a complete topological inverse polycyclic monoid, Carpathian Math. Publ., 8 (2016), №2, 183–194. 10. M. Bertman, T. West, Conditionally compact bicyclic semitopological semigroups, Proc. Roy. Irish Acad., A76:21–23 (1976), 219–226. 11. A.H. Clifford, G.B. Preston, The Algebraic Theory of Semigroups, V.I and II, Amer. Math. Soc. Surveys, 7, Providence, R.I., 1961 and 1967. 12. J. Cuntz, W. Krieger, A class of $C^$-algebras and topological Markov chains, Invent. Math., 56 (1980), 251–268. 13. C. Eberhart, J. Selden, On the closure of the bicyclic semigroup, Trans. Amer. Math. Soc., 144 (1969), 115–126. 14. R. Engelking, General Topology, 2nd ed., Heldermann, Berlin, 1989. 15. O. Gutik, On the dichotomy of the locally compact semitopological bicyclic monoid with adjoined zero, Visn. L’viv. Univ., Ser. Mekh.-Mat., 80 (2015), 33–41. 16. D. Jones, Polycyclic monoids and their generalizations, PhD Thesis, Heriot-Watt University, 2011. 17. D. Jones, M. Lawson, Graph inverse semigroups: Their characterization and completion, J. Algebra, 409 (2014), 444–473. 18. A. Kumjian, D. Pask, I. Raeburn, Cuntz-Krieger algebras of directed graphs, Pacific J. Math., 184 (1998), 161–174. 19. M. Lawson, Inverse semigroups. The theory of partial symmetries, Singapore: World Scientific, 1998. 20. M. Lawson, Primitive partial permutation representations of the polycyclic monoids and branching function systems, Period. Math. Hungar., 58 (2009), 189–207. 21. J. Meakin, M. Sapir, Congruences on free monoids and submonoids of polycyclic monoids, J. Austral. Math. Soc. Ser. A, 54 (2009), 236–253. 22. Z. Mesyan, J.D. Mitchell, The structure of a graph inverse semigroup, Semigroup Forum, 93 (2016), 111–130. 23. Z. Mesyan, J.D. Mitchell, M. Morayne, Y.H. Peresse, Topological graph inverse semigroups, Topology and its Applications, 208 (2016), 106–126. 24. M. Nivat, J.-F. Perrot, Une generalisation du monoide bicyclique, C. R. Acad. Sci., Paris, Ser. A, 271 (1970), 824–827. 25. A.L. Paterson, Graph inverse semigroups, groupoids and their $C^$-algebras, Birkhauser, 1999. 26. W. Ruppert, Compact Semitopological Semigroups: An Intrinsic Theory, Lect. Notes Math., 1079, Springer, Berlin, 1984.
Pages	19-28
Volume	49
Issue	1
Year	2018
Journal	Matematychni Studii
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