LINEAR PRESERVERS OF SYMMETRIC ARCTIC RANK OVER THE BINARY BOOLEAN SEMIRING
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A Boolean rank one matrix can be factored as ${\bf u}{\bf v}^t$ for vectors ${\bf u}$ and ${\bf v}$ of appropriate orders. The perimeter of this Boolean rank one matrix is the number of nonzero entries in ${\bf u}$ plus the number of nonzero entries in ${\bf v}$. A Boolean matrix of Boolean rank $k$ is the sum of $k$ Boolean rank one matrices, a rank one decomposition. The perimeter of a Boolean matrix $A$ of Boolean rank $k$ is the minimum over all Boolean rank one decompositions of $A$ of the sums of perimeters of the Boolean rank one matrices. The arctic rank of a Boolean matrix is one half the perimeter. In this article we characterize the linear operators that preserve the symmetric arctic rank of symmetric Boolean matrices.
Abstract
A Boolean rank one matrix can be factored as ${\bf u}{\bf v}^t$ for vectors ${\bf u}$ and ${\bf v}$ of appropriate orders. The perimeter of this Boolean rank one matrix is the number of nonzero entries in ${\bf u}$ plus the number of nonzero entries in ${\bf v}$. A Boolean matrix of Boolean rank $k$ is the sum of $k$ Boolean rank one matrices, a rank one decomposition. The perimeter of a Boolean matrix $A$ of Boolean rank $k$ is the minimum over all Boolean rank one decompositions of $A$ of the sums of perimeters of the Boolean rank one matrices. The arctic rank of a Boolean matrix is one half the perimeter. In this article we characterize the linear operators that preserve the symmetric arctic rank of symmetric Boolean matrices.
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