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

Antonio Di Nola, Giacomo Lenzi and Tran Giang Nam


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.

  • [1]

    H. Bass, Algebraic K-theory, W. A. Benjamin, New York, 1968.

  • [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.

  • [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.

  • [4]

    A. Connes and C. Consani, Schemes over 𝔽 1 \mathbb{F}_{1} and zeta functions, Compos. Math. 146 (2010), no. 6, 1383–1415.

  • [5]

    A. Connes and C. Consani, Geometry of the arithmetic site, Adv. Math. 291 (2016), 274–329.

  • [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.

  • [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.

  • [8]

    A. Di Nola and C. Russo, Semiring and semimodule issues in MV-algebras, Comm. Algebra 41 (2013), no. 3, 1017–1048.

  • [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]

    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.

  • [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.

  • [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.

  • [13]

    J. Giansiracusa and N. Giansiracusa, Equations of tropical varieties, Duke Math. J. 165 (2016), no. 18, 3379–3433.

  • [14]

    J. S. Golan, Semirings and Their Applications, Kluwer Academic, Dordrecht, 1999.

  • [15]

    K. R. Goodearl, Von Neumann Regular Rings, Monogr. Stud. Math. 4, Pitman, Boston, 1979.

  • [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.

  • [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.

  • [18]

    Z. Izhakian, M. Johnson and M. Kambites, Pure dimension and projectivity of tropical polytopes, Adv. Math. 303 (2016), 1236–1263.

  • [19]

    Z. Izhakian, M. Knebusch and L. Rowen, Decompositions of modules lacking zero sums, Israel J. Math. 225 (2018), no. 2, 503–524.

  • [20]

    Z. Izhakian and L. Rowen, Supertropical algebra, Adv. Math. 225 (2010), no. 4, 2222–2286.

  • [21]

    Y. Katsov, Tensor products and injective envelopes of semimodules over additively regular semirings, Algebra Colloq. 4 (1997), no. 2, 121–131.

  • [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.

  • [23]

    Y. Katsov and T. G. Nam, Morita equivalence and homological characterization of semirings, J. Algebra Appl. 10 (2011), no. 3, 445–473.

  • [24]

    Y. Katsov, T. G. Nam and J. Zumbrägel, On congruence-semisimple semirings and the K 0 K_{0}-group characterization of ultramatricial algebras over semifields, J. Algebra 508 (2018), 157–195.

  • [25]

    B. Keller, Cluster algebras and derived categories, Derived Categories in Algebraic Geometry, EMS Ser. Congr. Rep., European Mathematical Society, Zürich (2012), 123–183.

  • [26]

    T. Y. Lam, Serre’s Problem on Projective Modules, Springer Monogr. Math., Springer, Berlin, 2006.

  • [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.

  • [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.

  • [29]

    O. Lorscheid, The geometry of blueprints: Part I: Algebraic background and scheme theory, Adv. Math. 229 (2012), no. 3, 1804–1846.

  • [30]

    S. Mac Lane, Categories for the Working Mathematician, Springer, New York, 1971.

  • [31]

    A. W. Macpherson, Projective modules over polyhedral semirings, J. Algebra 518 (2019), 237–271.

  • [32]

    G. Maze, C. Monico and J. Rosenthal, Public key cryptography based on semigroup actions, Adv. Math. Commun. 1 (2007), no. 4, 489–507.

  • [33]

    D. Mundici, Interpretation of AF C C^{\ast}-algebras in Łukasiewicz sentential calculus, J. Funct. Anal. 65 (1986), no. 1, 15–63.

  • [34]

    A. Patchkoria, Projective semimodules over semirings with valuations in nonnegative integers, Semigroup Forum 79 (2009), no. 3, 451–460.

  • [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.

  • [36]

    M. Rørdam, F. Larsen and N. Laustsen, An Introduction to K-theory for C * C^{*}-algebras, London Math. Soc. Stud. Texts 49, Cambridge University, Cambridge, 2000.

Purchase article
Get instant unlimited access to the article.
Price including VAT
Log in
Already have access? Please log in.

Journal + Issues

Forum Mathematicum is a general mathematics journal, which is devoted to the publication of research articles in all fields of pure and applied mathematics, including mathematical physics. Forum Mathematicum belongs to the top 50 journals in pure and applied mathematics, as measured by citation impact.