1MP and MP1 inverses and one-sided star orders in a ring with involution

The classes of 1MP-inverses and MP1-inverses are recently introduced classes of generalized inverses of complex matrices. Actually, they coincide with the classes of {1,2,3}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{1,2,3\}$$\end{document} and {1,2,4}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{1,2,4\}$$\end{document} inverses, respectively. We consider these inverses in the context of a ring with involution and prove that their most important characterizations and properties remain true. We show that the binary relations based on these inverses are in fact the well known left-star and right-star partial orders. We extend these relations to the ring case, connect them with the unified theory of partial order relations based on generalized inverses and provide several properties. Finally, we indicate how these results can be applied to bounded Hilbert space operators.