Skip to main content
SpringerLink
Log in

The bideterminants of matrices over semirings

  • Foundations
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

This paper deals with the relationships between the concepts of linear dependence and independence of vectors, and the bideterminant of a matrix including the links between the bideterminant and rank of a square matrix in semilinear spaces of \(n\)-dimensional vectors over commutative zerosumfree semirings. First, it discusses some properties of bideterminant of a matrix, then introduces the concepts of semi-linear dependence and strong linear independence of a set of vectors, respectively, and gives a necessary and sufficient condition that the bideterminant of a matrix is equal to \(0\). In the end, it shows a necessary and sufficient condition that the rank of an \(n\)-square matrix is equal to \(n\).

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

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. Some authors defined a linear dependence of vectors in \(\mathcal {V}_n\) as that: vectors \(\mathbf a _1\),\(\ldots , \mathbf a _n\), \( n\geqslant 2\), are linearly dependent if and only if there exist two nonempty disjoint subsets of indices \(J_{1}\subset \underline{n}\) and \(J_{2}\subset \underline{n}\) together with \(0\ne \lambda _{i}\in L, i\in J_{1}\cup J_{2}\), such that \(\sum _{j\in J_{1}}\lambda _{j}{\mathbf{a}}_j=\sum _{j\in J_{2}}\lambda _{j}{\mathbf{a}}_j\) (see e.g. Cechlárová and Plávka 1996; Golan 1999; Gondran and Minoux 2008). This definition means that a set of vectors is linearly dependent if and only if it is either linearly dependent in the sense of Definition 2.5 or semi-linearly dependent in the sense of Definition 4.1.

References

  • Baccelli FL, Cohen G, Olsder GJ, Quadrat JP (1992) Synchronization and linearity. Wiley, Chichester, New York

    MATH  Google Scholar 

  • Butkovič P (2003) Max-algebra: the linear algebra of combinatorics? Linear Algebra Appl 367:313–335

    Google Scholar 

  • Cao ZQ, Kim KH, Roush FW (1984) Incline algebra and applications. Wiley, New York

  • Cechlárová K, Plávka J (1996) Linear independence in bottleneck algebras. Fuzzy Sets Syst 77:337–348

    Article  MATH  Google Scholar 

  • Cuninghame-Green RA (1979) Minimax algebra, vol 166. In: Lecture notes in economics and mathematical systems. Springer, Berlin

  • Cuninghame-Green RA, Butkovič P (2004) Bases in max-algebra. Linear Algebra Appl 389:107–120

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  • Gondran M, Minoux M (1984) Linear algebra in dioïds: a survey of recent results. Ann Discret Math 19:147–164

    MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  • Kim KH, Roush FW (1980) Generalized fuzzy matrices. Fuzzy Sets Syst 4:293–315

    Article  MATH  MathSciNet  Google Scholar 

  • Kuntzman J (1972) Théorie des réseaaux graphes. Dunod (Libraire), Paris

  • Perfilieva I (2004) Semi-linear spaces. In: Noguchi H, Ishii H et al (eds) Proceedings of the seventh Czech-Japanese seminar on data analysis and decision making under uncertainty. Hyogo, Japan, pp 127–130

  • 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. China, World Scientific, Emei, Chengdu, pp 43–51

  • Phillip L (2007) Poplin, Robert E. Hartwig, Determinantal identities over commutative semirings. Linear Algebra Appl 387:99–132

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  • Tan YJ (2007) On invertible matrices over antirings. Linear Algebra Appl 423:428–444

    Article  MATH  MathSciNet  Google Scholar 

  • Zhao S, Wang X (2010) Invertible matrices and semilinear spaces over commutative semirings. Inform Sci 180:5115–5124

    Article  MATH  MathSciNet  Google Scholar 

  • Zhao S, Wang X (2011) Bases in semilinear spaces over join-semirings. Fuzzy Sets Syst 182:93–100

    Article  MATH  Google Scholar 

  • Zimmermann U (1981) Linear and combinatorial optimization in ordered algebrac structures. North Holland, Amsterdam

    Google Scholar 

Download references

Acknowledgments

The authors thank the referees for their valuable comments and suggestions.

Author information

Authors and Affiliations

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

    Xue-ping Wang & Qian-yu Shu

Authors
  1. Xue-ping Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qian-yu Shu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xue-ping Wang.

Additional information

Communicated by L. Spada.

Supported by National Natural Science Foundation of China (No. 11171242), Doctoral Fund of Ministry of Education of China (No. 20105134110002), Sichuan Youth Fund (No. 2011JQ0055) and Sichuan Education Department Scientific Research Fund (No. 13ZB0165).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, Xp., Shu, Qy. The bideterminants of matrices over semirings. Soft Comput 18, 729–742 (2014). https://doi.org/10.1007/s00500-013-1163-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-013-1163-y

Keywords

Search

Navigation