Endomorphisms of semigroups of invertible nonnegative matrices over ordered rings (Q376159)

Let \(R\) be a linearly ordered commutative ring with \(1/2\) and \(G_n(R)\) be the subsemigroup of \(\mathrm{GL}_n(R)\) consisting of all matrices with nonnegative elements. In this paper, the author describes all endomorphisms of this semigroup for \(n>2\). Some subsets of \(G_n(R)\) play an important role in the main result of this paper. Let \(I=I_n\), \(\Gamma_n(R)\) be the group of all invertible matrices from \(G_n(R)\), \(S_{\sigma}\) be the matrix of \(\sigma \in S_n\) (symmetric group of order \(n\)), \(D_n(R)\) be the group of all invertible diagonal matrices from \(G_n(R)\), and \(P\) be the subsemigroup of \(G_n(R)\), generated by all \(S_{\sigma}\) (\(\sigma \in \Sigma_n\)), \(B_{i,j}(x)=I+xE_{i,j}\) (\(x \in R_+,~ i \neq j\)) and \(\text{diag}(\alpha_1, \alpha_2, \dots, \alpha_n) \in D_n(R)\). Finally, \(GE_n^+(R)\) denotes the subsemigroup of \(G_n(R)\) generated by all matrices that are \({\mathcal P}\)-equivalent to matrices from \(P\). If \(G\) is some semigroup, then a homomorphism \(\lambda(\cdot): G \rightarrow G\) is called a central homomorphism of \(G\) if \(\lambda(G)\subset Z(G)\). The mapping \(\Omega(\cdot): G \rightarrow G\) such that for all \(X \in G\) \[ \Omega(X)=\lambda(X) \cdot X, \] where \(\lambda(\cdot)\) is a central homomorphism, is called a central homothety. For every matrix \(M \in \Gamma_n(R)\), \(\Phi_M\) denotes an automorphism of \(G_n(R)\) such that \(\Phi_M(X)=MXM^{-1}\) for all \(X \in G_n(R)\). For every \(y(\cdot) \in \mathrm{End}(R_+)\), \(\Phi^y\) denotes the endomorphism of \(G_n(R)\) such that \(\Phi^y(X)=\Phi^y((x_{ij}))=(y(x_{ij}))\) for all \(X=(x_{ij}) \in G_n(R)\). From different technical lemmas, the author establishes the main result, which affirms that if \(\Phi\) is an endomorphism such that \(\Phi(B_{i,j}(1)) \neq I_n\), then there exist \(M \in \Gamma_n(R)\), \(b \in \mathrm{End}(R_+)\) and a central homothety \(\Omega\) such that \(\Phi\) coincides with \(\Phi_M \circ \Phi^b \circ \Omega\) on the semigroup \(GE_n^+(R)\).