# A characterization of linear operators that preserve isolation numbers

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Mathematical Communications, Vol.19 No.2 December 2014. -

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 $X,$~ $TX=PXQ$ or, $m=n$ and $TX=PX^tQ$ where $X^t$ is the transpose of $X$.

Boolean matrix; Boolean rank; isolation number; Boolean linear opertator

Autor: LeRoy B. Beasley - ; Department of Mathematics and Statistics, Utah State University, Logan, Utah, USA Seok Zun Song - ; Departme

Fuente: http://hrcak.srce.hr/

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