An alternative look at the structure of graph inverse semigroups

S. Bardyla

doi:10.15330/ms.51.1.3-11

For any graph inverse semigroup $G(E)$ we describe subsemigroups $D^0=D\cup\{0\}$ and $J^0=J\cup\{0\}$ of $G(E)$ where $D$ and $J$ are arbitrary $\mathcal{D}$-class and $\mathcal{J}$-class of $G(E)$, respectively. In particular, we prove that for each $\mathcal{D}$-class $D$ of a graph inverse semigroup over an acyclic graph the semigroup $D^0$ is isomorphic to a semigroup of matrix units. Also we show that for any elements $a,b$ of a graph inverse semigroup $G(E)$, $J_a\cdot J_b\cup J_b\cdot J_a\subset J_b^0$ if there exists a path $w$ such that $s(w)\in J_a$ and $r(w)\in J_b$.

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

2. Amal Alali, N.D. Gilbert, Closed inverse subsemigroups of graph inverse semigroups, Communications in Algebra, 45 (11), (2017), 4667–4678.

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

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

5. S. Bardyla, On universal objects in the class of graph inverse semigroups, European Journal of Mathematics, in press, DOI: 10.1007/s40879-018-0300-7.

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

7. S. Bardyla, Embeddings of graph inverse semigroups into compact-like topological semigroups, preprint, (2019), arXiv:1810.09169.

8. S. Bardyla, On locally compact semitopological graph inverse semigroups, Mat. Stud., 49 (2018), №1, 19-28.

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

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

11. A.H. Clifford, G.B. Preston, The Algebraic Theory of Semigroups, Vol. 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. O. Gutik, On the dichotomy of the locally compact semitopological bicyclic monoid with adjoined zero, Visn. Lviv. Univ., Ser. Mekh.-Mat., 80 (2015), 33-41.

14. O. Gutik, K. Pavlyk, On Brandt $\lambda^0$-extensions of semigroups with zero, Mat. Metody Fiz.-Mech. Polya, 49 (2006), №3, 26-40.

15. O. Gutik, D. Repovs, On the Brandt $\lambda^0$-extensions of monoids with zero, Semigroup Forum, 80, (2010), 8-32.

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

17. D.G. Jones, M.V. 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.V. 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. Paterson, Graph inverse semigroups, groupoids and their $C^*$-algebras, J. Operator Theory, 48 (2002), №3, suppl., 645-662.

	An alternative look at the structure of graph inverse semigroups
Author	S. Bardyla sbardyla@yahoo.com Ivan Franko State Pedagogical University of Drohobych; Institute of Mathematics, Kurt Godel Research Center, Vienna, Austria
Abstract	For any graph inverse semigroup $G(E)$ we describe subsemigroups $D^0=D\cup\{0\}$ and $J^0=J\cup\{0\}$ of $G(E)$ where $D$ and $J$ are arbitrary $\mathcal{D}$-class and $\mathcal{J}$-class of $G(E)$, respectively. In particular, we prove that for each $\mathcal{D}$-class $D$ of a graph inverse semigroup over an acyclic graph the semigroup $D^0$ is isomorphic to a semigroup of matrix units. Also we show that for any elements $a,b$ of a graph inverse semigroup $G(E)$, $J_a\cdot J_b\cup J_b\cdot J_a\subset J_b^0$ if there exists a path $w$ such that $s(w)\in J_a$ and $r(w)\in J_b$.
Keywords	polycyclic monoid; graph inverse semigroup; Brandt $\lambda^0$-extension
DOI	doi:10.15330/ms.51.1.3-11
Reference	1. G. Abrams and G. Aranda Pino, The Leavitt path algebra of a graph, J. Algebra 293, (2005), 319–334. 2. Amal Alali, N.D. Gilbert, Closed inverse subsemigroups of graph inverse semigroups, Communications in Algebra, 45 (11), (2017), 4667–4678. 3. P. Ara, M. Moreno, E. Pardo, Non-stable K-theory for graph algebras, Algebr. Represent. Th., 10 (2007), 157–178. 4. S. Bardyla, Classifying locally compact semitopological polycyclic monoids, Math. Bulletin of the Shevchenko Scientific Society, 13 (2016), 21-28. 5. S. Bardyla, On universal objects in the class of graph inverse semigroups, European Journal of Mathematics, in press, DOI: 10.1007/s40879-018-0300-7. 6. S. Bardyla, On locally compact topological graph inverse semigroups, preprint, (2017), arXiv:1706.08594. 7. S. Bardyla, Embeddings of graph inverse semigroups into compact-like topological semigroups, preprint, (2019), arXiv:1810.09169. 8. S. Bardyla, On locally compact semitopological graph inverse semigroups, Mat. Stud., 49 (2018), №1, 19-28. 9. S. Bardyla, O. Gutik, On a semitopological polycyclic monoid, Algebra Discr. Math., 21 (2016), №2, 163-183. 10. S. Bardyla, O. Gutik, On a complete topological inverse polycyclic monoid, Carpathian Math. Publ., 8 (2016), №2, 183-194. 11. A.H. Clifford, G.B. Preston, The Algebraic Theory of Semigroups, Vol. 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. O. Gutik, On the dichotomy of the locally compact semitopological bicyclic monoid with adjoined zero, Visn. Lviv. Univ., Ser. Mekh.-Mat., 80 (2015), 33-41. 14. O. Gutik, K. Pavlyk, On Brandt $\lambda^0$-extensions of semigroups with zero, Mat. Metody Fiz.-Mech. Polya, 49 (2006), №3, 26-40. 15. O. Gutik, D. Repovs, On the Brandt $\lambda^0$-extensions of monoids with zero, Semigroup Forum, 80, (2010), 8-32. 16. D.G. Jones, Polycyclic monoids and their generalizations, PhD Thesis, Heriot-Watt University, 2011. 17. D.G. Jones, M.V. 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.V. 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. Paterson, Graph inverse semigroups, groupoids and their $C^$-algebras, J. Operator Theory, 48 (2002), №3, suppl., 645-662.
Pages	3-11
Volume	51
Issue	1
Year	2019
Journal	Matematychni Studii
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Table of content of issue	html