Semiring fank versus column rank

Beasley, LeRoy B.; Pullman, Norman J.

doi:10.1016/0024-3795(88)90141-3

Cited by 41 publications

(14 citation statements)

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“…A semiring ( [2]) is a set S equipped with two binary operations + and · such that (S, +) is a commutative monoid with identity element 0 and (S, ·) is a monoid with identity element 1. In addition, operations + and · are connected by distributivity and 0 annihilates S. Thus all rings with identity are semirings.…”

Section: Introductionmentioning

confidence: 99%

Characterizations of Zero-Term Rank Preservers of Matrices over Semirings

Kang¹,

Shi²,

Beasley³

et al. 2014

Kyungpook mathematical journal

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Abstract. Let M(S) denote the set of all m×n matrices over a semiring S. For A ∈ M(S), zero-term rank of A is the minimal number of lines (rows or columns) needed to cover all zero entries in A. In [5], the authors obtained that a linear operator on M(S) preserves zero-term rank if and only if it preserves zero-term ranks 0 and 1. In this paper, we obtain new characterizations of linear operators on M(S) that preserve zero-term rank. Consequently we obtain that a linear operator on M(S) preserves zero-term rank if and only if it preserves two consecutive zero-term ranks k and k + 1, where 0 ≤ k ≤ min{m, n} − 1 if and only if it strongly preserves zero-term rank h, where 1 ≤ h ≤ min{m, n}.

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Section: Introductionmentioning

confidence: 99%

Characterizations of Zero-Term Rank Preservers of Matrices over Semirings

Kang¹,

Shi²,

Beasley³

et al. 2014

Kyungpook mathematical journal

Self Cite

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“…A set G of vectors over Z + is linearly dependent [2] if for some g ∈ G, g is a linear combination of elements in G − {g}. Otherwise G is linearly independent.…”

Section: Of a χ(A)mentioning

confidence: 99%

“…The inequality in (1.1) may be strict over Z + . For example, we consider the matrix A = [1,2,3] over Z + . Then the column rank of A is one, while the maximal column rank of it is two since the last two columns of A are linearly independent over Z + .…”

Section: Of a χ(A)mentioning

confidence: 99%

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Linear operators that preserve maximal column ranks of nonnegative integer matrices

Shi

1998

Proc. Amer. Math. Soc.

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“…Beasley and Pullman [2] showed the following: Lemma 1.1 [2]. If the columns of A £ Mm<n(B) are linearly independent, then c(A) = n. Rank preservers are defined in a manner similar to (iii).…”

Section: Introductionmentioning

confidence: 99%