Continuous monotone maps on matrices for orders induced by the group inverse (Q2790758)

Let \(M_n(F)\) denote the space of square matrices of order \(n\) with coefficients in the field \(F\). For a given \(A \in M_n(F)\) let \(A^\sharp\) denote the well-known group inverse of \(A\) and \(\mathop{\leq}\limits^\sharp\) the partial order relation on matrices induced by \(A^\sharp\). Furthermore, a given core-nilpotent (cn) decomposition of \(A\) induces the partial order relation \(\mathop{\leq}\limits^{\mathrm{cn}}\) on matrices. If \(\leq\) is any partial order on \(M_n(F)\) and \(M \subseteq M_n(F)\), a map \(T: M\to M\) is called monotone if \(A,B \in M\) with \(A \leq B\) implies \(T(A) \leq T(B)\). Such monotone maps have been studied extensively. Both authors have written (joint) papers on monotone maps with respect to \(\mathop{\leq}\limits^\sharp\) and \(\mathop{\leq}\limits^{\mathrm{cn}}\). The main goal of this paper is to characterize continuous injective maps on the set of complex matrices that are monotone with respect to \(\mathop{\leq}\limits^\sharp\) and \(\mathop{\leq}\limits^{\mathrm{cn}}\). It follows as a corollary that all such maps are surjective and \(\mathbb{R}\)-linear. The authors also present counterexamples showing that their assumptions are necessary.