Abstract
We define a subclass of separated graphs, the class of adaptable separated graphs, and study their associated monoids. We show that these monoids are primely generated conical refinement monoids, and we explicitly determine their associated I-systems. We also show that any finitely generated conical refinement monoid can be represented as the monoid of an adaptable separated graph. These results provide the first step toward an affirmative answer to the Realization Problem for von Neumann regular rings, in the finitely generated case.
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References
Abrams, G., Ara, P., Molina, M.S.: Leavitt Path Algebras, Springer Lecture Notes in Mathematics, vol. 2191. Springer, London (2017)
Ara, P.: The realization problem for von Neumann regular rings. In: Marubayashi, H., Masaike, K., Oshiro, K., Sato, M. (eds.) Ring Theory 2007. Proceedings of the Fifth China–Japan–Korea Conference, pp. 21–37. World Scientific, Hackensack (2009)
Ara, P.: The regular algebra of a poset. Trans. Am. Math. Soc. 362(3), 1505–1546 (2010)
Ara, P., Bosa, J., Pardo, E.: The realization problem for finitely generated refinement monoids, arXiv:1907.03648 [math. OA]
Ara, P., Bosa, J., Pardo, E., Sims, A.: The groupoids of adaptable separated graphs and their type semigroups, arXiv:1904.05197 [math.OA]
Ara, P., Brustenga, M.: The regular algebra of a quiver. J. Algebra 309, 207–235 (2007)
Ara, P., Exel, R.: Dynamical systems associated to separated graphs, graph algebras, and paradoxical decompositions. Adv. Math. 252, 748–804 (2014)
Ara, P., Goodearl, K.R.: Leavitt path algebras of separated graphs. J. Reine Angew. Math. 669, 165–224 (2012)
Ara, P., Goodearl, K.R., O’Meara, K.C., Pardo, E.: Separative cancellation for projective modules over exchange rings. Isr. J. Math. 105, 105–137 (1998)
Ara, P., Moreno, M.A., Pardo, E.: Nonstable K-theory for graph algebras. Algebra Represent. Theory 10, 157–178 (2007)
Ara, P., Pardo, E.: Primely generated refinement monoids. Isr. J. Math. 214, 379–419 (2016)
Ara, P., Pardo, E.: Representing finitely generated refinement monoids as graph monoids. J. Algebra 480, 79–123 (2017)
Ara, P., Perera, F., Wehrung, F.: Finitely generated antisymmetric graph monoids. J. Algebra 320, 1963–1982 (2008)
Brookfield, G.: Cancellation in primely generated refinement monoids. Algebra Universalis 46, 343–371 (2001)
Dobbertin, H.: Primely generated regular refinement monoids. J. Algebra 91, 166–175 (1984)
Exel, R.: Inverse semigroups and combinatorial \(C^*\)-algebras. Bull. Braz. Math. Soc. (N.S.) 39, 191–313 (2008)
Goodearl, K.R.: Von Neumann regular rings and direct sum decomposition problems. In: “Abelian groups and modules” (Padova, 1994), Mathematics and its Applications. vol. 343, pp. 249–255. Kluwer Academic Publishers, Dordrecht (1995)
Ketonen, J.: The structure of countable Boolean algebras. Ann. Math. 108, 41–89 (1978)
Ortega, E., Perera, F., Rørdam, M.: The corona factorization property and refinement monoids. Trans. Am. Math. Soc. 363, 4505–4525 (2011)
Paterson, A.L.T.: Groupoids, Inverse Semigroups, and Their Operator Algebras, Progress in Mathematics, vol. 170. Birkhäuser Boston Inc, Boston (1999)
Pierce, R.S.: Countable Boolean algebras. In: Monk, J.D., Bonnet, R. (eds.) Handbook of Boolean Algebras, vol. 3, pp. 775–876. North-Holland, Amsterdam (1989)
Rainone, T., Sims, A.: A dichotomy for groupoid C*-algebras. Ergod. Theor. Dyn. Syst. (2018). https://doi.org/10.1017/etds.2018.52
Steinberg, B.: A groupoid approach to discrete inverse semigroup algebras. Adv. Math. 223, 689–727 (2010)
Wehrung, F.: Refinement Monoids, Equidecomposable Types, and Boolean Inverse Semigroups Springer Lecture Notes in Mathematics, vol. 2188. Springer, Cham (2017)
Acknowledgements
We are very grateful to the anonymous referee, whose helpful comments have significantly improved the exposition of the paper. This research project was initiated when the authors were at the Centre de Recerca Matemàtica as part of the Intensive Research Program Operator algebras: dynamics and interactions in 2017, and the work was significantly supported by the research environment and facilities provided there. The authors thank the Centre de Recerca Matemàtica for its support.
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Communicated by Benjamin Steinberg.
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The three authors were partially supported by the DGI-MINECO and European Regional Development Fund, jointly, through Grants MTM2014-53644-P and MTM2017-83487-P. The first- and second-named authors acknowledge support from the Spanish Ministry of Economy and Competitiveness, through the María de Maeztu Programme for Units of Excellence in R&D (MDM-2014-0445) and from the Generalitat de Catalunya through the Grant 2017-SGR-1725. The third-named author was partially supported by PAI III Grant FQM-298 of the Junta de Andalucía.
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Ara, P., Bosa, J. & Pardo, E. Refinement monoids and adaptable separated graphs. Semigroup Forum 101, 19–36 (2020). https://doi.org/10.1007/s00233-019-10077-2
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DOI: https://doi.org/10.1007/s00233-019-10077-2