### A characterization of linear operators that preserve isolation numbers

#### Abstract

We obtain characterizations of Boolean linear operators that preserve some of the isolation numbers of Boolean matrices. In particular, we show that the following are equivalent: (1) $T$ preserves the isolation number of all matrices; (2) $T$ preserves the set of matrices with isolation number one and the set of those with isolation number $k$ for some $2\leq k\leq \min\{m,n\}$; (3) for $1\leq k\leq \min\{m,n\}-1$, $T$ preserves matrices

with isolation number $k$, and those with isolation number $k+1$, (4) $T$ maps $J$ to $J$ and preserves the set of matrices of isolation number 2; (5) $T$ is a $(P,Q)$-operator, that is, for fixed permutation matrices $P$ and $Q$, $m\times n$ matrix $X,$~ $T(X)=PXQ$ or, $m=n$ and $T(X)=PX^tQ$ where $X^t$ is the transpose of $X$.

with isolation number $k$, and those with isolation number $k+1$, (4) $T$ maps $J$ to $J$ and preserves the set of matrices of isolation number 2; (5) $T$ is a $(P,Q)$-operator, that is, for fixed permutation matrices $P$ and $Q$, $m\times n$ matrix $X,$~ $T(X)=PXQ$ or, $m=n$ and $T(X)=PX^tQ$ where $X^t$ is the transpose of $X$.

#### Keywords

Boolean matrix, Boolean rank, isolation number, Boolean linear opertator

#### Full Text:

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