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The Group of Fractions of the Semigroup of Invertible Nonnegative Matrices of Order Three Over a Field

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Abstract

Let \( \mathbb{F} \) be a linearly ordered field. Consider G n (\( \mathbb{F} \)), which is the subsemigroup of GL n (\( \mathbb{F} \)) consisting of all matrices with nonnegative coefficients. In 1940, A. I. Maltsev introduced the concept of the group of fractions for a semigroup. In the given paper, we prove that the group of fractions of G3(\( \mathbb{F} \)) coincides with GL3(\( \mathbb{F} \)).

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Correspondence to E. I. Bunina.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 18, No. 3, pp. 27–42, 2013.

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Bunina, E.I., Nemiro, V.V. The Group of Fractions of the Semigroup of Invertible Nonnegative Matrices of Order Three Over a Field. J Math Sci 206, 474–485 (2015). https://doi.org/10.1007/s10958-015-2326-4

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Keywords

  • Nonzero Element
  • Minimal Element
  • Elementary Matrice
  • General Linear Group
  • Initial Matrix