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Linear maps that strongly preserve regular matrices over the Boolean algebra. (English) Zbl 1224.15054
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 $$\min \{m,n\}\leq 2$$, then all operators on $$\mathbb {M}_{m,n}$$ strongly preserve regular matrices, and, if $$\min \{m,n\}\geq 3$$, 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 $$X\in \mathbb {M}_n$$.
##### MSC:
 15A86 Linear preserver problems 15A09 Theory of matrix inversion and generalized inverses 15B34 Boolean and Hadamard matrices
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##### References:
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