Linear maps that strongly preserve regular matrices over the Boolean algebra

Kang, Kyung-Tae, and Song, Seok-Zun. "Linear maps that strongly preserve regular matrices over the Boolean algebra." Czechoslovak Mathematical Journal 61.1 (2011): 113-125. <http://eudml.org/doc/196636>.

@article{Kang2011,
abstract = {The set of all $m\times n$ Boolean matrices is denoted by $\{\mathbb \{M\}\}_\{m,n\}$. We call a matrix $A\in \{\mathbb \{M\}\}_\{m,n\}$ regular if there is a matrix $G\in \{\mathbb \{M\}\}_\{n,m\}$ such that $AGA=A$. In this paper, we study the problem of characterizing linear operators on $\{\mathbb \{M\}\}_\{m,n\}$ that strongly preserve regular matrices. Consequently, we obtain that if $\min \lbrace m,n\rbrace \le 2$, then all operators on $\{\mathbb \{M\}\}_\{m,n\}$ strongly preserve regular matrices, and if $\min \lbrace m,n\rbrace \ge 3$, then an operator $T$ on $\{\mathbb \{M\}\}_\{m,n\}$ strongly preserves regular matrices if and only if there are invertible matrices $U$ and $V$ such that $T(X)=UXV$ for all $X\in \{\mathbb \{M\}\}_\{m,n\}$, or $m=n$ and $T(X)=UX^TV$ for all $X\in \{\mathbb \{M\}\}_\{n\}$.},
author = {Kang, Kyung-Tae, Song, Seok-Zun},
journal = {Czechoslovak Mathematical Journal},
keywords = {Boolean algebra; regular matrix; $(U,V)$-operator; Boolean algebra; regular matrix; -operator; regular matrix preservers; Boolean matrices; invertible matrices},
language = {eng},
number = {1},
pages = {113-125},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Linear maps that strongly preserve regular matrices over the Boolean algebra},
url = {http://eudml.org/doc/196636},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Kang, Kyung-Tae
AU - Song, Seok-Zun
TI - Linear maps that strongly preserve regular matrices over the Boolean algebra
JO - Czechoslovak Mathematical Journal
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 61
IS - 1
SP - 113
EP - 125
AB - The set of all $m\times n$ Boolean matrices is denoted by ${\mathbb {M}}_{m,n}$. We call a matrix $A\in {\mathbb {M}}_{m,n}$ regular if there is a matrix $G\in {\mathbb {M}}_{n,m}$ such that $AGA=A$. In this paper, we study the problem of characterizing linear operators on ${\mathbb {M}}_{m,n}$ that strongly preserve regular matrices. Consequently, we obtain that if $\min \lbrace m,n\rbrace \le 2$, then all operators on ${\mathbb {M}}_{m,n}$ strongly preserve regular matrices, and if $\min \lbrace m,n\rbrace \ge 3$, then an operator $T$ on ${\mathbb {M}}_{m,n}$ strongly preserves regular matrices if and only if there are invertible matrices $U$ and $V$ such that $T(X)=UXV$ for all $X\in {\mathbb {M}}_{m,n}$, or $m=n$ and $T(X)=UX^TV$ for all $X\in {\mathbb {M}}_{n}$.
LA - eng
KW - Boolean algebra; regular matrix; $(U,V)$-operator; Boolean algebra; regular matrix; -operator; regular matrix preservers; Boolean matrices; invertible matrices
UR - http://eudml.org/doc/196636
ER -