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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:

[1] L.B. Beasley and N. J. Pullman: Boolean rank preserving operators and Boolean rank-1 spaces. Linear Algebra Appl. 59 (1984), 55–77. · Zbl 0536.20044
[2] J. Denes: Transformations and transformation semigroups. Seminar Report, University of Wisconsin, Madison, Wisconsin, 1976.
[3] K.H. Kim: Boolean Matrix Theory and Applications. Pure and Applied Mathematics, Vol. 70, Marcel Dekker, New York, 1982.
[4] R.D. Luce: A note on Boolean matrix theory. Proc. Amer. Math. Soc. 3 (1952), 382–388. · Zbl 0048.02302
[5] E.H. Moore: General Analysis, Part I. Mem. of Amer. Phil. Soc. 1 (1935).
[6] R. J. Plemmons: Generalized inverses of Boolean relation matrices. SIAM J. Appl. Math. 20 (1971), 426–433. · Zbl 0227.05013
[7] P. S. S.N.V. P. Rao and K.P. S.B. Rao: On generalized inverses of Boolean matrices. Linear Algebra Appl. 11 (1975), 135–153. · Zbl 0322.15011
[8] D.E. Rutherford: Inverses of Boolean matrices. Proc. Glasgow Math. Assoc. 6 (1963), 49–53. · Zbl 0114.01701
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