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Title: Linear maps that strongly preserve regular matrices over the Boolean algebra (English)
Author: Kang, Kyung-Tae
Author: Song, Seok-Zun
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 61
Issue: 1
Year: 2011
Pages: 113-125
Summary lang: English
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Category: math
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Summary: 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 $\min \{m,n\}\le 2$, then all operators on ${\mathbb M}_{m,n}$ strongly preserve regular matrices, and if $\min \{m,n\}\ge 3$, 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 $X\in {\mathbb M}_{n}$. (English)
Keyword: Boolean algebra
Keyword: regular matrix
Keyword: $(U,V)$-operator
MSC: 15A09
MSC: 15A86
MSC: 15B34
idZBL: Zbl 1224.15054
idMR: MR2782763
DOI: 10.1007/s10587-011-0001-6
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Date available: 2011-05-23T12:35:09Z
Last updated: 2016-04-07
Stable URL: http://hdl.handle.net/10338.dmlcz/141522
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