Geodesic
Monotone maps on matrices of index one
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXV, Tome 405 (2012), pp. 67-96

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The paper investigates bijective maps on the set of matrices of index one that are monotone with respect to the order induced by the group inverse. 
@article{ZNSL_2012_405_a6,
     author = {A. E. Guterman and M. A. Efimov},
     title = {Monotone maps on matrices of index one},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {67--96},
     publisher = {mathdoc},
     volume = {405},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2012_405_a6/}
}
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A. E. Guterman; M. A. Efimov. Monotone maps on matrices of index one. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXV, Tome 405 (2012), pp. 67-96. http://geodesic.mathdoc.fr/item/ZNSL_2012_405_a6/