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Semiring Isomorphisms and Automorphisms of Matrix Algebras

Abstract

The research shows that each matrix semiring isomorphism over an antinegative commutative semiring R with unity is a composition of an inner automorphism and an automorphism inducted by an automorphism of the semiring R. It follows that every automorphism of such a matrix semiring that preserves scalars is inner. A matrix over an antinegative commutative semiring R with unity is invertible if and only if it is a product of an invertible diagonal matrix and a matrix consisting of idempotent elements such that the product of its elements of one row (column) is 0 and their sum is 1. As a consequence of a theory that was developed for automorphism calculation, the problem of incident semiring isomorphism is solved. Isomorphism of the quasiorders defining these semirings also follows from the isomorphism of incidence semirings over commutative semirings.

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References

  1. 1.

    G. Abrans, J. Haefner, and Á. del Río, “The isomorphism problem for incidence rings,” Pacific J. Math., 187, No. 2, 201–214 (1999).

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    I. M. Isaacs, “Automorphisms of matrix algebras over commutative rings,” Linear Algebra Appl., 31, 215–231 (1980).

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    V. D. Shmatkov, Isomorphisms and Automorphisms of Incidence Rings and Algebras [in Russian], Ph.D. Thesis, Moscow (1994).

  4. 4.

    E. Spiegel and C. O’Donnell, Incidence Algebras, Pure Appl. Math., Vol. 206, Marcel Dekker (1997).

  5. 5.

    Y. Tan, “On invertible matrices over antirings,” Linear Algebra Appl., 423, 428–444 (2007).

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to V. D. Shmatkov.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 19, No. 6, pp. 251–260, 2014.

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Shmatkov, V.D. Semiring Isomorphisms and Automorphisms of Matrix Algebras. J Math Sci 221, 479–485 (2017). https://doi.org/10.1007/s10958-017-3239-1

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