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Discussiones Mathematicae General Algebra and Applications 23(2) (2003) 125-137
DOI: https://doi.org/10.7151/dmgaa.1068

RANK AND PERIMETER PRESERVER OF RANK-1 MATRICES OVER MAX ALGEBRA

 Seok-Zun Song  and Kyung-Tae Kang

Department of Mathematics, Cheju National University
Jeju 690-756, Republic of Korea
e-mail: szsong@cheju.ac.kr
e-mail: kangkt@cheju.ac.kr

Abstract

For a rank-1 matrix A =  a ⊗ b^t over max algebra, we define the perimeter of A as the number of nonzero entries in both a and b. We characterize the linear operators which preserve the rank and perimeter of rank-1 matrices over max algebra. That is, a linear operator T preserves the rank and perimeter of rank-1 matrices if and only if it has the form T(A) = U ⊗ A ⊗ V , or T(A) = U⊗ A^t ⊗ V with some monomial matrices U and V.

Keywords: max algebra; semiring; linear operator; monomial; rank; dominate; perimeter; (U,V)-operator.

2000 Mathematics Subject Classification: 15A03, 15A04, 12K10, 16Y60.

References

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[3]	L.B. Beasley and N.J. Pullman, Boolean rank-preserving operators and Boolean rank-1 spaces, Linear Algebra Appl. 59 (1984), 55-77.
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[5]	S.-Z. Song and S.-R. Park, Maximal column rank preservers of fuzzy matrices, Discuss. Math. - Gen. Algebra Appl. 21 (2001), 207-218.

Received 24 April 2003