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Ultramatricial algebras over commutative chain semirings and application to MV-algebras

  • Antonio Di Nola EMAIL logo , Giacomo Lenzi and Tran Giang Nam
Published/Copyright: October 27, 2019
Forum Mathematicum
From the journal Forum Mathematicum Volume 32 Issue 2

Abstract

In this paper, we give a complete description of strongly projective semimodules over a semiring which is a finite direct product of matrix semirings over commutative chain semirings. We then classify ultramatricial algebras over commutative chain semirings by their ordered SK0-groups. Consequently, we get that there is a one-one correspondence between isomorphism classes of ultramatricial algebras A whose SK0⁢(A) is lattice-ordered over a given commutative chain semiring and isomorphism classes of countable MV-algebras.

Keywords: Strongly projective semimodule; ultramatricial algebras over semirings; MV-algebras
MSC 2010: 16Y60; 06D35; 16E20; 18G05

Communicated by Manfred Droste

Funding statement: The third author is partially supported by Vietnam Ministry of Education and Training under the grant number B2018.SPD.02.

Acknowledgements

This work was carried out during the third author’s visit to the University of Salerno. He would like to thank the Department of Mathematics for hospitality. Also, the authors take an opportunity to express their deep gratitude to the anonymous referee for extremely careful reading, highly professional working with our manuscript, and valuable suggestions. Finally, we would like to thank the editor for his suggestion making us aware of the reference [10] which provides an interesting connection between semirings and weighted automata theory.

References

[1] H. Bass, Algebraic K-theory, W. A. Benjamin, New York, 1968. Search in Google Scholar

[2] L. P. Belluce and A. Di Nola, Commutative rings whose ideals form an MV-algebra, MLQ Math. Log. Q. 55 (2009), no. 5, 468–486. 10.1002/malq.200810012Search in Google Scholar

[3] R. L. O. Cignoli, I. M. L. D’Ottaviano and D. Mundici, Algebraic Foundations of Many-valued Reasoning, Trends Log. Stud. Log. Libr. 7, Kluwer Academic, Dordrecht, 2000. 10.1007/978-94-015-9480-6Search in Google Scholar

[4] A. Connes and C. Consani, Schemes over 𝔽1 and zeta functions, Compos. Math. 146 (2010), no. 6, 1383–1415. 10.1112/S0010437X09004692Search in Google Scholar

[5] A. Connes and C. Consani, Geometry of the arithmetic site, Adv. Math. 291 (2016), 274–329. 10.1016/j.aim.2015.11.045Search in Google Scholar

[6] A. Di Nola and B. Gerla, Algebras of Lukasiewicz’s logic and their semiring reducts, Idempotent Mathematics and Mathematical Physics, Contemp. Math. 377, American Mathematical Society, Providence (2005), 131–144. 10.1090/conm/377/06988Search in Google Scholar

[7] A. Di Nola and C. Russo, Łukasiewicz transform and its application to compression and reconstruction of digital images, Inform. Sci. 177 (2007), no. 6, 1481–1498. 10.1016/j.ins.2006.09.002Search in Google Scholar

[8] A. Di Nola and C. Russo, Semiring and semimodule issues in MV-algebras, Comm. Algebra 41 (2013), no. 3, 1017–1048. 10.1080/00927872.2011.610074Search in Google Scholar

[9] A. Di Nola and C. Russo, The semiring-theoretic approach to MV-algebras: A survey, Fuzzy Sets and Systems 281 (2015), 134–154. 10.1016/j.fss.2015.08.026Search in Google Scholar

[10] M. Droste and W. Kuich, Chapter 1: Semirings and formal power series, Handbook of Weighted Automata, Monogr. Theoret. Comput. Sci. EATCS Ser., Springer, Berlin (2009), 3–28. 10.1007/978-3-642-01492-5_1Search in Google Scholar

[11] E. G. Effros, D. E. Handelman and C. L. Shen, Dimension groups and their affine representations, Amer. J. Math. 102 (1980), no. 2, 385–407. 10.2307/2374244Search in Google Scholar

[12] G. A. Elliott, On the classification of inductive limits of sequences of semisimple finite-dimensional algebras, J. Algebra 38 (1976), no. 1, 29–44. 10.1016/0021-8693(76)90242-8Search in Google Scholar

[13] J. Giansiracusa and N. Giansiracusa, Equations of tropical varieties, Duke Math. J. 165 (2016), no. 18, 3379–3433. 10.1215/00127094-3645544Search in Google Scholar

[14] J. S. Golan, Semirings and Their Applications, Kluwer Academic, Dordrecht, 1999. 10.1007/978-94-015-9333-5Search in Google Scholar

[15] K. R. Goodearl, Von Neumann Regular Rings, Monogr. Stud. Math. 4, Pitman, Boston, 1979. Search in Google Scholar

[16] S. N. Il’in and Y. Katsov, On Serre’s problem on projective semimodules over polynomial semirings, Comm. Algebra 42 (2014), no. 9, 4021–4032. 10.1080/00927872.2013.804529Search in Google Scholar

[17] S. N. Il’in, Y. Katsov and T. G. Nam, Toward homological structure theory of semimodules: On semirings all of whose cyclic semimodules are projective, J. Algebra 476 (2017), 238–266. 10.1016/j.jalgebra.2016.12.013Search in Google Scholar

[18] Z. Izhakian, M. Johnson and M. Kambites, Pure dimension and projectivity of tropical polytopes, Adv. Math. 303 (2016), 1236–1263. 10.1016/j.aim.2016.08.033Search in Google Scholar

[19] Z. Izhakian, M. Knebusch and L. Rowen, Decompositions of modules lacking zero sums, Israel J. Math. 225 (2018), no. 2, 503–524. 10.1007/s11856-018-1661-9Search in Google Scholar

[20] Z. Izhakian and L. Rowen, Supertropical algebra, Adv. Math. 225 (2010), no. 4, 2222–2286. 10.1016/j.aim.2010.04.007Search in Google Scholar

[21] Y. Katsov, Tensor products and injective envelopes of semimodules over additively regular semirings, Algebra Colloq. 4 (1997), no. 2, 121–131. Search in Google Scholar

[22] Y. Katsov, Toward homological characterization of semirings: Serre’s conjecture and Bass’s perfectness in a semiring context, Algebra Universalis 52 (2004), no. 2–3, 197–214. 10.1007/s00012-004-1866-0Search in Google Scholar

[23] Y. Katsov and T. G. Nam, Morita equivalence and homological characterization of semirings, J. Algebra Appl. 10 (2011), no. 3, 445–473. 10.1142/S0219498811004793Search in Google Scholar

[24] Y. Katsov, T. G. Nam and J. Zumbrägel, On congruence-semisimple semirings and the K0-group characterization of ultramatricial algebras over semifields, J. Algebra 508 (2018), 157–195. 10.1016/j.jalgebra.2018.04.024Search in Google Scholar

[25] B. Keller, Cluster algebras and derived categories, Derived Categories in Algebraic Geometry, EMS Ser. Congr. Rep., European Mathematical Society, Zürich (2012), 123–183. 10.4171/115-1/6Search in Google Scholar

[26] T. Y. Lam, Serre’s Problem on Projective Modules, Springer Monogr. Math., Springer, Berlin, 2006. 10.1007/978-3-540-34575-6Search in Google Scholar

[27] E. Leichtnam, A classification of the commutative Banach perfect semi-fields of characteristic 1: Applications, Math. Ann. 369 (2017), no. 1–2, 653–703. 10.1007/s00208-017-1527-1Search in Google Scholar

[28] G. L. Litvinov, The Maslov dequantization, idempotent and tropical mathematics: A very brief introduction, Idempotent Mathematics and Mathematical Physics, Contemp. Math. 377, American Mathematical Society, Providence (2005), 1–17. 10.1090/conm/377/6982Search in Google Scholar

[29] O. Lorscheid, The geometry of blueprints: Part I: Algebraic background and scheme theory, Adv. Math. 229 (2012), no. 3, 1804–1846. 10.1016/j.aim.2011.12.018Search in Google Scholar

[30] S. Mac Lane, Categories for the Working Mathematician, Springer, New York, 1971. 10.1007/978-1-4612-9839-7Search in Google Scholar

[31] A. W. Macpherson, Projective modules over polyhedral semirings, J. Algebra 518 (2019), 237–271. 10.1016/j.jalgebra.2018.09.041Search in Google Scholar

[32] G. Maze, C. Monico and J. Rosenthal, Public key cryptography based on semigroup actions, Adv. Math. Commun. 1 (2007), no. 4, 489–507. 10.3934/amc.2007.1.489Search in Google Scholar

[33] D. Mundici, Interpretation of AF C∗-algebras in Łukasiewicz sentential calculus, J. Funct. Anal. 65 (1986), no. 1, 15–63. 10.1016/0022-1236(86)90015-7Search in Google Scholar

[34] A. Patchkoria, Projective semimodules over semirings with valuations in nonnegative integers, Semigroup Forum 79 (2009), no. 3, 451–460. 10.1007/s00233-009-9147-zSearch in Google Scholar

[35] J. Richter-Gebert, B. Sturmfels and T. Theobald, First steps in tropical geometry, Idempotent Mathematics and Mathematical Physics, Contemp. Math. 377, American Mathematical Society, Providence (2005), 289–317. 10.1090/conm/377/06998Search in Google Scholar

[36] M. Rørdam, F. Larsen and N. Laustsen, An Introduction to K-theory for C*-algebras, London Math. Soc. Stud. Texts 49, Cambridge University, Cambridge, 2000. 10.1017/CBO9780511623806Search in Google Scholar

Received: 2019-02-27
Revised: 2019-09-20
Published Online: 2019-10-27
Published in Print: 2020-03-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Operational rules and generalized special polynomials
  3. Linearity of some low-complexity mapping class groups
  4. Ultramatricial algebras over commutative chain semirings and application to MV-algebras
  5. On a group analogue of the Heyde theorem
  6. Explicit presentation of an Iwasawa algebra: The case of pro-p Iwahori subgroup of SLn(ℤp)
  7. Generalized Riesz potentials of functions in Morrey spaces L(1,ϕ;κ)(G) over non-doubling measure spaces
  8. On rough singular integrals along real-analytic submanifolds
  9. Dimension-free estimates for the vector-valued variational operators
  10. Upper bound of multiplicity in prime characteristic
  11. Asymptotic behaviour for elliptic operators with second-order discontinuous coefficients
  12. Irreducible and permutative representations of ultragraph Leavitt path algebras
  13. Twisted tensor products of φ-coordinated modules for nonlocal vertex algebras
  14. Explicit decomposition theorem for special Schubert varieties
  15. On the Hirzebruch–Kobayashi–Ono proportionality principle and the non-existence of compact solvable Clifford–Klein forms of certain homogeneous spaces
  16. The Lie group of vertical bisections of a regular Lie groupoid
  17. Chain conditions for graph C*-algebras
  18. Congruences in Hermitian Jacobi and Hermitian modular forms
  19. The Jensen–Pólya program for various L-functions
https://doi.org/10.1515/forum-2019-0056
Keywords for this article
Strongly projective semimodule; ultramatricial algebras over semirings; MV-algebras

Articles in the same Issue

  1. Frontmatter
  2. Operational rules and generalized special polynomials
  3. Linearity of some low-complexity mapping class groups
  4. Ultramatricial algebras over commutative chain semirings and application to MV-algebras
  5. On a group analogue of the Heyde theorem
  6. Explicit presentation of an Iwasawa algebra: The case of pro-p Iwahori subgroup of SLn(ℤp)
  7. Generalized Riesz potentials of functions in Morrey spaces L(1,ϕ;κ)(G) over non-doubling measure spaces
  8. On rough singular integrals along real-analytic submanifolds
  9. Dimension-free estimates for the vector-valued variational operators
  10. Upper bound of multiplicity in prime characteristic
  11. Asymptotic behaviour for elliptic operators with second-order discontinuous coefficients
  12. Irreducible and permutative representations of ultragraph Leavitt path algebras
  13. Twisted tensor products of φ-coordinated modules for nonlocal vertex algebras
  14. Explicit decomposition theorem for special Schubert varieties
  15. On the Hirzebruch–Kobayashi–Ono proportionality principle and the non-existence of compact solvable Clifford–Klein forms of certain homogeneous spaces
  16. The Lie group of vertical bisections of a regular Lie groupoid
  17. Chain conditions for graph C*-algebras
  18. Congruences in Hermitian Jacobi and Hermitian modular forms
  19. The Jensen–Pólya program for various L-functions
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