Skip to main content
SpringerLink
Log in

The realization problem for finitely generated refinement monoids

  • Published:
Selecta Mathematica Aims and scope Submit manuscript

Abstract

We show that every finitely generated conical refinement monoid can be represented as the monoid \(\mathcal V (R)\) of isomorphism classes of finitely generated projective modules over a von Neumann regular ring R. To this end, we use the representation of these monoids provided by adaptable separated graphs. Given an adaptable separated graph (E, C) and a field K, we build a von Neumann regular K-algebra \(Q_K(E,C)\) and show that there is a natural isomorphism between the separated graph monoid M(E, C) and the monoid \(\mathcal V (Q_K(E,C))\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abrams, G., Ara, P., Siles Molina, M.: Leavitt Path Algebras. Lecture Notes in Mathematics, vol. 2191. Springer, London (2017)

    Book  Google Scholar 

  2. Alahmadi, A., Alsulami, H., Jain, S.K., Zelmanov, E.: Leavitt path algebras of finite Gelfand–Kirillov dimension. J. Algebra Appl. 11(6), 1250225 (2012)

    Article  MathSciNet  Google Scholar 

  3. Ánh, P.N., Márki, L.: Morita equivalence for rings withoug identity. Tsukuba J. Math 11, 1–16 (1987)

    Article  MathSciNet  Google Scholar 

  4. 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. Hackensack, NJ World Scientific, pp. 21–37 (2009)

  5. Ara, P.: The regular algebra of a poset. Trans. Am. Math. Soc. 362(3), 1505–1546 (2010)

    Article  MathSciNet  Google Scholar 

  6. Ara, P.: Leavitt path algebras over a poset of fields. In: Dani, S.G., Jain, S.K., Verma, J.K., Wasadikar, M.P. (eds.) Contributions in Algebra and Algebraic Geometry. Contemporary Mathematics, vol. 738, pp. 1–12. American Mathematical Society, Providence (2019)

    Chapter  Google Scholar 

  7. Ara, P., Bosa, J., Pardo, E.: Refinement monoids and adaptable separated graphs. Semigroup Forum (2019). https://doi.org/10.1007/s00233-019-10077-2

    Article  Google Scholar 

  8. Ara, P., Bosa, J., Pardo, E., Sims, A.: The groupoids of adaptable separated graphs and their type semigroups. Int. Math. Res. Not. IMRN (2020). https://doi.org/10.1093/imrn/rnaa022

    Article  Google Scholar 

  9. Ara, P., Brustenga, M.: The regular algebra of a quiver. J. Algebra 309, 207–235 (2007)

    Article  MathSciNet  Google Scholar 

  10. Ara, P., Exel, R.: Dynamical systems associated to separated graphs, graph algebras, and paradoxical decompositions. Adv. Math. 252, 748–804 (2014)

    Article  MathSciNet  Google Scholar 

  11. Ara, P., Goodearl, K.R.: Leavitt path algebras of separated graphs. J. Reine Angew. Math. 669, 165–224 (2012)

    MathSciNet  MATH  Google Scholar 

  12. Ara, P., Goodearl, K.R.: Tame and wild refinement monoids. Semigroup Forum 91, 1–27 (2015)

    Article  MathSciNet  Google Scholar 

  13. Ara, P., Goodearl, K.R.: The realization problem for some wild monoids and the Atiyah problem. Trans. Am. Math. Soc. 369, 5665–5710 (2017)

    Article  MathSciNet  Google Scholar 

  14. Ara, P., Goodearl, K.R., O’Meara, K.C., Pardo, E.: Separative cancellation for projective modules over exchange rings. Israel J. Math. 105, 105–137 (1998)

    Article  MathSciNet  Google Scholar 

  15. Ara, P., Moreno, M.A., Pardo, E.: Nonstable K-Theory for graph algebras. Algebr. Represent. Theory 10, 157–178 (2007)

    Article  MathSciNet  Google Scholar 

  16. Ara, P., Pardo, E.: Primely generated refinement monoids. Israel J. Math. 214, 379–419 (2016)

    Article  MathSciNet  Google Scholar 

  17. Ara, P., Pardo, E.: Representing finitely generated refinement monoids as graph monoids. J. Algebra 480, 79–123 (2017)

    Article  MathSciNet  Google Scholar 

  18. Ara, P., Perera, F., Wehrung, F.: Finitely generated antisymmetric graph monoids. J. Algebra 320, 1963–1982 (2008)

    Article  MathSciNet  Google Scholar 

  19. Baccella, G., Spinosa, L.: \(K_0\) of semiartinian von Neumann regular rings. Direct finiteness versus unit-regularity. Algebr. Represent. Theory 20, 1189–1213 (2017)

    Article  MathSciNet  Google Scholar 

  20. Bergman, G.M.: Coproducts and some universal ring constructions. Trans. Am. Math. Soc. 200, 33–88 (1974)

    Article  MathSciNet  Google Scholar 

  21. Bergman, G.M., Dicks, W.: Universal derivations and universal ring constructions. Pacific J. Math. 79, 293–337 (1978)

    Article  MathSciNet  Google Scholar 

  22. Brookfield, G.: Cancellation in primely generated refinement monoids. Algebra Universalis 46, 343–371 (2001)

    Article  MathSciNet  Google Scholar 

  23. Cohn, P.M.: Free Ideal Rings and Localization in General Rings, New Mathematical Monographs, vol. 3. Cambridge University Press, Cambridge (2006)

    Book  Google Scholar 

  24. Dobbertin, H.: Primely generated regular refinement monoids. J. Algebra 91, 166–175 (1984)

    Article  MathSciNet  Google Scholar 

  25. Facchini, A.: Semilocal Categories and Modules with Semilocal Endomorphism Rings, Progress in Mathematics, vol. 331. Birkhäuser/Springer, Cham (2019)

    Book  Google Scholar 

  26. Facchini, A., Finocchiaro, C.A.: Pretorsion theories, stable category and preordered sets. Annali di Matematica Pura ed Applicata (2019). https://doi.org/10.1007/s10231-019-00912-2

    Article  MATH  Google Scholar 

  27. Geroldinger, A., Zhong, Q.: Factorization theory in commutative monoids. Semigroup Forum 100, 22–51 (2020)

    Article  MathSciNet  Google Scholar 

  28. Goodearl, K.R.: Von Neumann Regular Rings. Pitman, London 1979; 2nd Ed., Krieger, Malabar (1991)

  29. Goodearl, K.R.: Von Neumann regular rings and direct sum decomposition problems. In: Facchini, A., Menini, C. (eds.) Abelian groups and modules, Math. Appl., vol. 343, pp. 249–255. Kluwer Acad. Publ., Dordrecht (1995)

  30. Goodearl, K.R.: Leavitt path algebras and direct limits. In: Dung, N.V., Guerriero, F., Hammoudi, L., Kanwar, P. (eds.) Rings, Modules and Representations. Contemporary Mathematics, vol. 480, pp. 165–187. American Mathematical Society, Providence (2009)

    Chapter  Google Scholar 

  31. Goodearl, K.R., Wehrung, F.: Representations of distributive semilattices in ideal lattices of various algebraic structures. Algebra Universalis 45, 71–102 (2001)

    Article  MathSciNet  Google Scholar 

  32. Menal, P., Moncasi, J.: Lifting units in self-injective rings and an index theory for Rickart \(C^*\)-algebras. Pacific J. Math. 126, 295–329 (1987)

    Article  MathSciNet  Google Scholar 

  33. 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)

    Google Scholar 

  34. Råžička, P.: On the construction and the realization of wild monoids. Arch. Math. (Brno) 54, 33–64 (2018)

    Article  MathSciNet  Google Scholar 

  35. Schofield, A.H.: Representations of Rings over Skew Fields, LMS Lecture Notes Series 92. Cambridge University Press, Cambridge (1985)

    Book  Google Scholar 

  36. Vas, L.: Simplicial and dimension groups with group action and their realization. arXiv:1805.07636 [math.KT]

  37. Wehrung, F.: Non-measurability properties of interpolation vector spaces. Israel J. Math. 103, 177–206 (1998)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

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 continued at the Universidad de Cádiz due to an invitation of the third author to the first two authors in 2018. The work developed was significantly supported by the research environment and facilities provided in both centers. We would also like to thank the anonymous referee for very useful comments and suggestions.

Author information

Authors and Affiliations

  1. Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, 08193, Barcelona, Spain

    Pere Ara & Joan Bosa

  2. Departamento de Matemáticas, Facultad de Ciencias, Universidad de Cádiz, Campus de Puerto Real, 11510, Puerto Real, Cádiz, Spain

    Enrique Pardo

Authors
  1. Pere Ara
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Joan Bosa
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Enrique Pardo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pere Ara.

Additional information

Dedicated to the memory of Antonio Rosado Pérez

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The three authors were partially supported by the DGI-MINECO and European Regional Development Fund, jointly, through the Grant MTM2017-83487-P. The first and second authors were partially supported by the Generalitat de Catalunya through the Grant 2017-SGR-1725. The second author was partially supported by the Beatriu de Pinós postdoctoral programme of the Government of Catalonia’s Secretariat for Universities and Research of the Ministry of Economy and Knowledge (BP2017-0079). The third author was partially supported by PAI III Grant FQM-298 of the Junta de Andalucía.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ara, P., Bosa, J. & Pardo, E. The realization problem for finitely generated refinement monoids. Sel. Math. New Ser. 26, 33 (2020). https://doi.org/10.1007/s00029-020-00559-5

Download citation

  • Published:

  • DOI: https://doi.org/10.1007/s00029-020-00559-5

Keywords

Mathematics Subject Classification

Search

Navigation