# Article

 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 . Category: math . 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 . Date available: 2011-05-23T12:35:09Z Last updated: 2016-04-07 Stable URL: http://hdl.handle.net/10338.dmlcz/141522 . Reference: [1] Beasley, L. B., Pullman, N. J.: Boolean rank preserving operators and Boolean rank-1 spaces.Linear Algebra Appl. 59 (1984), 55-77. Zbl 0536.20044, MR 0743045, 10.1016/0024-3795(84)90158-7 Reference: [2] Denes, J.: Transformations and transformation semigroups.Seminar Report, University of Wisconsin, Madison, Wisconsin (1976). Reference: [3] Kim, K. H.: Boolean Matrix Theory and Applications.Pure and Applied Mathematics, Vol. 70, Marcel Dekker, New York (1982). Zbl 0495.15003, MR 0655414 Reference: [4] Luce, R. D.: A note on Boolean matrix theory.Proc. Amer. Math. Soc. 3 (1952), 382-388. Zbl 0048.02302, MR 0050559, 10.1090/S0002-9939-1952-0050559-1 Reference: [5] Moore, E. H.: General Analysis, Part I.Mem. of Amer. Phil. Soc. 1 (1935). Reference: [6] Plemmons, R. J.: Generalized inverses of Boolean relation matrices.SIAM J. Appl. Math. 20 (1971), 426-433. Zbl 0227.05013, MR 0286806, 10.1137/0120046 Reference: [7] Rao, P. S. S. N. V. P., Rao, K. P. S. B.: On generalized inverses of Boolean matrices.Linear Algebra Appl. 11 (1975), 135-153. Zbl 0322.15011, MR 0376706, 10.1016/0024-3795(75)90054-3 Reference: [8] Rutherford, D. E.: Inverses of Boolean matrices.Proc. Glasgow Math. Assoc. 6 (1963), 49-53. Zbl 0114.01701, MR 0148585 .

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