CHARACTERIZATIONS OF BOOLEAN RANK PRESERVERS OVER BOOLEAN MATRICES
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The Boolean rank of a nonzero m × n Boolean matrix A is the leastinteger k such that there are an m×k Boolean matrix B and a k×n Boolean matrix C with A = BC. In 1984, Beasley and Pullman showed that a linear operator preservesthe Boolean rank of any Boolean matrix if and only if it preserves Boolean ranks1 and 2. In this paper, we extend this characterization of linear operators thatpreserve the Boolean ranks of Boolean matrices. We show that a linear operatorpreserves all Boolean ranks if and only if it preserves two consecutive Boolean ranksif and only if it strongly preserves a Boolean rank k with 1≤ k ≤ min{m,n}.
Abstract
The Boolean rank of a nonzero m × n Boolean matrix A is the leastinteger k such that there are an m×k Boolean matrix B and a k×n Boolean matrix C with A = BC. In 1984, Beasley and Pullman showed that a linear operator preservesthe Boolean rank of any Boolean matrix if and only if it preserves Boolean ranks1 and 2. In this paper, we extend this characterization of linear operators thatpreserve the Boolean ranks of Boolean matrices. We show that a linear operatorpreserves all Boolean ranks if and only if it preserves two consecutive Boolean ranksif and only if it strongly preserves a Boolean rank k with 1≤ k ≤ min{m,n}.
참고문헌 (7)
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