Original scientific paper

A characterization of linear operators that preserve isolation numbers

LeRoy B. Beasley ; Department of Mathematics and Statistics, Utah State University, Logan, Utah, USA
Seok Zun Song ; Department of Mathematics, Jeju National University, Jeju, Korea

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 $X$.

Keywords

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

Hrčak ID:

129584

URI

https://hrcak.srce.hr/129584

Publication date:

5.12.2014.

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