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On the coincidence of the factor and Gondran–Minoux rank functions of matrices over a semiring

Abstract

We consider the rank functions of matrices over semirings, functions that generalize the classical notion of the rank of a matrix over a field. We study semirings over which the factor and Gondran–Minoux ranks of any matrix coincide. It is shown that every semiring satisfying that condition is a subsemiring of a field. We provide an example of an integral domain over which the factor and Gondran–Minoux ranks are different.

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Correspondence to Ya. N. Shitov.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 17, No. 6, pp. 223–232, 2011/12.

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Shitov, Y.N. On the coincidence of the factor and Gondran–Minoux rank functions of matrices over a semiring. J Math Sci 193, 802–808 (2013). https://doi.org/10.1007/s10958-013-1498-z

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Keywords

  • Nonzero Element
  • Integral Domain
  • Rank Function
  • Zero Element
  • Zero Divisor