The Rank of Matrices Over Positive Cones of Commutative Semirings
Main Article Content
Abstract
In this work, we introduced and studied the PC rank notion of a commutative semiring, and compared the PC rank with the nonnegative rank and the factor ranks of matrices, and gived also some sufficient conditions for these ranks to be the same.
Keywords:
Ring, Commutative semiring, Positive cone, Idempotent matrix, Nonnegative matrix.
References
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[2] L. B. Beasley, T. J. Laffey, Real Rank Versus Nonnegative Rank, Linear Algebra Appl., Vol. 431, No. 12, 2009, pp. 2330-2335, https//doi.org/10.1016/j.laa.2009.02.034.
[3] P. Mohindru, Completely Positive Matrices Over Semirings and Their CP-rank, Diss. University of Guelph, 2014.
[4] P. Mohindru, Pereira, Orderings on Semirings and Completely Positive Matrices, Linear Multilinear Algebr., Vol. 64, No. 5, 2016, pp. 818-833, https//doi.org/10.1080/03081087.2015.1059405.
[5] Y. Shitov, On Tropical and Nonnegative Factorization Ranks of Band Matrices, arXiv:1710.02072v1, 2017.
[6] D. K. Bukovšek, H. Šmigoc, Symmetric Nonnegative Matrix Trifactorization, Linear Algebra Appl., Vol. 665, 2023, pp. 36-60, https//doi.org/10.1016/j.laa.2023.01.027.
[7] L. Q. Tung, T. Hoang, Asymptotic Nonnegative Rank of Matrices, arXiv:2308.07187, 2023.
[8] T. C. Craven, Orderings on Semirings, Semigr, Forum, Vol. 43, No. 1, 1991, pp. 45-52, https//doi.org/10.1007/BF02574250.
[9] J. S. Golan, Semirings and their Applications. Kluwer Academic Publishers, Dordrecht-Boston-London, 1999.
[10] H. C. Cong, Stably Free Rank of Idempotent Matrices on Semirings, Univ. Da Nang, J. Sci. Technol., Vol. 20, 2022, pp. 56-60.
[11] L. B. Beasley, A. E. Guterman, Rank Inequalities Over Semirings, J. Korean Math. Soc., Vol. 42, No. 2, 2005, pp. 223-241.
[12] C. Reutenauer, H. Straubing, Inversion of Matrices Over A Commutative Semiring, J. Algebr, Vol. 88, No. 2, 1984, pp. 350-360.