Show Summary Details
More options …

# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

IMPACT FACTOR 2018: 0.867

CiteScore 2018: 0.71

SCImago Journal Rank (SJR) 2018: 0.898
Source Normalized Impact per Paper (SNIP) 2018: 0.964

Mathematical Citation Quotient (MCQ) 2018: 0.71

Online
ISSN
1435-5337
See all formats and pricing
More options …

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

Antonio Di Nola
/ Giacomo Lenzi
/ Tran Giang Nam
Published Online: 2019-10-27 | DOI: https://doi.org/10.1515/forum-2019-0056

## 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 ${\mathrm{SK}}_{0}$-groups. Consequently, we get that there is a one-one correspondence between isomorphism classes of ultramatricial algebras A whose ${\mathrm{SK}}_{0}\left(A\right)$ is lattice-ordered over a given commutative chain semiring and isomorphism classes of countable MV-algebras.

MSC 2010: 16Y60; 06D35; 16E20; 18G05

## References

• [1]

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

• [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. Google Scholar

• [4]

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

• [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. 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.

• [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. 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.

• [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. Google Scholar

• [15]

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

• [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. 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. 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.

• [24]

Y. Katsov, T. G. Nam and J. Zumbrägel, On congruence-semisimple semirings and the ${K}_{0}$-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. Google Scholar

• [26]

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

• [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. Google Scholar

• [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. Google Scholar

• [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 ${C}^{\ast }$-algebras in Łukasiewicz sentential calculus, J. Funct. Anal. 65 (1986), no. 1, 15–63. Google Scholar

• [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. 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. Google Scholar

Revised: 2019-09-20

Published Online: 2019-10-27

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

Citation Information: Forum Mathematicum, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

Export Citation

© 2019 Walter de Gruyter GmbH, Berlin/Boston.