Geodesic
Linear matrix transformations that are monotone with respect to the $\overset\sharp\leq$- or $\overset{\mathrm{cn}}\leq$-order
Fundamentalʹnaâ i prikladnaâ matematika, Tome 13 (2007) no. 4, pp. 53-66.

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We characterize linear transformations on the matrix algebra over an arbitrary field with characteristic not equal to 2 that are monotone with respect to the $\overset\sharp\leq$- or $\overset{\mathrm{cn}}\leq$-order. 
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M. A. Efimov. Linear matrix transformations that are monotone with respect to the $\overset\sharp\leq$- or $\overset{\mathrm{cn}}\leq$-order. Fundamentalʹnaâ i prikladnaâ matematika, Tome 13 (2007) no. 4, pp. 53-66. http://geodesic.mathdoc.fr/item/FPM_2007_13_4_a2/

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