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The solvable conditions of linear system over commutative semirings

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Abstract

This paper deals with the solution of linear system over commutative semirings. It uses the bideterminant of a matrix to investigate the solvable conditions of a system of linear equations and gives some necessary and sufficient conditions that a system of linear equations is solvable.

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References

  • Di Nola A, Lettieri A, Perfilieva I, Novák V (2007) Algebraic analysis of fuzzy systems. Fuzzy Sets Syst 158:1–22

    Article  MathSciNet  MATH  Google Scholar 

  • Golan JS (1999) Semirings and their applications. Kluwer Academic Publishers, Dordrecht

    Book  MATH  Google Scholar 

  • Gondran M, Minoux M (2008) Graphs, dioids and semirings. Springer, New York

    MATH  Google Scholar 

  • Hermida JA, Sánchez-Giralda T (1986) Linear equations over commutative rings and determinantal ideals. J Algebra 99:72–79

    Article  MathSciNet  MATH  Google Scholar 

  • Kuntzman J (1972) Théorie des réseaaux graphes [Theory of network graphs]. Dunod (Libraire), Paris

    Google Scholar 

  • McDonald BR (1984) Linear algebra over commutative rings. Marcel Dekker, New York

    MATH  Google Scholar 

  • Perfilieva I, Kupka J (2010) Kronecker-capelli theorem in semilinear spaces. In: Ruan D, Li TR, Xu Y, Chen GQ, Kerre EE (eds) Computational intelligence: foundations and applications. World Scientific, Emei, pp 43–51

    Chapter  Google Scholar 

  • Poplin PL, Hartwig RE (2007) Determinantal identities over commutative semirings. Linear Algebra Appl 387:99–132

    Article  MathSciNet  MATH  Google Scholar 

  • Shu Q, Wang X (2011) Bases in semilinear spaces over zerosumfree semirings. Linear Algebra Appl 435:2681–2692

    Article  MathSciNet  MATH  Google Scholar 

  • Shu Q, Wang X (2012) Standard orthogonal vectors in semilinear spaces and their applications. Linear Algebra Appl 437:2733–2754

    Article  MathSciNet  MATH  Google Scholar 

  • Shu Q, Wang X (2017) The applications of the bideterminant of a matrix over commutative semirings. Linear Multilinear Algebra 65:1462–1478

    Article  MathSciNet  MATH  Google Scholar 

  • Wang X, Shu Q (2014) Bideterminant and rank of matrix. Soft Comput 18:729–742

    Article  Google Scholar 

  • Zimmermann U (1981) Linear and combinatorial optimization in ordered algebraic structures, annals of discrete mathematics, vol 10. North Holland, Amsterdam

    Google Scholar 

Download references

Acknowledgements

The authors thank the referees for their valuable comments and suggestions. Supported by National Natural Science Foundation of China (No. 61573240), Postdoctoral Science Foundation of Jiangsu Province (No. 2018K031A) and Postdoctoral Science Foundation of China (No. 2018M642203).

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Authors and Affiliations

  1. College of Mathematics and Software Science, Sichuan Normal University, Chengdu, 610066, People’s Republic of China

    Qian-yu Shu & Xue-ping Wang

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  1. Qian-yu Shu
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  2. Xue-ping Wang
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Correspondence to Xue-ping Wang.

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Communicated by A. Di Nola.

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Shu, Qy., Wang, Xp. The solvable conditions of linear system over commutative semirings. Soft Comput 23, 7591–7602 (2019). https://doi.org/10.1007/s00500-019-03758-1

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  • DOI: https://doi.org/10.1007/s00500-019-03758-1

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