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 \(X\in \mathbb {M}_n\).


15A86 Linear preserver problems
15A09 Theory of matrix inversion and generalized inverses
15B34 Boolean and Hadamard matrices
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