Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

Article

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J. Korean Math. Soc. 2015; 52(3): 625-636

Printed May 1, 2015

https://doi.org/10.4134/JKMS.2015.52.3.625

Copyright © The Korean Mathematical Society.

Linear preservers of Boolean rank between different matrix spaces

LeRoy B. Beasley, Kyung-Tae Kang, and Seok-Zun Song

Utah State University, Jeju National University, Jeju National University

Abstract

The Boolean rank of a nonzero $m\times n$ Boolean matrix $A$ is the least integer $k$ such that there are an $m\times k$ Boolean matrix $B$ and a $k\times n$ Boolean matrix $C$ with $A=BC$. We investigate the structure of linear transformations $\tmnpq$ which preserve Boolean rank. We also show that if a linear transformation preserves the set of Boolean rank $1$ matrices and the set of Boolean rank $k$ matrices for any $k$, $2\le k\le \min\{m,n\}$ (or if $T$ strongly preserves the set of Boolean rank $1$ matrices), then $T$ preserves all Boolean ranks.

Keywords: Boolean matrix, Boolean rank, linear transformation

MSC numbers: 15A86, 15A04, 15B34