In this paper, a companion preorder \( \prec _G^\textrm{comp}\,\) to G-majorization \( \prec _G \) of Eaton type is introduced and studied. Attention is paid to the case of effective groups G. A criterion for G-majorization inequalities to hold is established by utilizing that companion preorder. A characterization of Gateaux differentiable \( \prec _G^\textrm{comp}\,\)-increasing functions is provided using their gradients. Next, some G-majorization relations are derived for gradients of some functions. New classes of c-strongly (weakly) \( \prec _G \)-increasing functions and c-strongly (weakly) \( \prec _G^\textrm{comp}\,\)-increasing functions are introduced. A Tarski like theorem is established on fixed points of the gradients maps of weakly \( \prec _G^\textrm{comp}\,\)-increasing functions. Some interpretations for the weak-majorization preorder and singular values of matrices are shown.

本文介绍并研究了Eaton类型的G -majorization \( \prec _G \)的伴随预序\( \ prec _G^\textrm{comp}\,\) 。注意有效组G的情况。G -majorization 不等式成立的标准是通过利用该伴随预序建立的。Gateaux 可微\( \prec _G^\textrm{comp}\,\)增加函数的特征是使用它们的梯度提供的。接下来，针对某些函数的梯度推导了一些G -majorization 关系。c的新类- 强（弱）\( \prec _G \) - 增加函数和c- 强（弱）\( \prec _G^\textrm{comp}\,\) -增加功能被引入。Tarski 类定理建立在弱\( \prec _G^\textrm{comp}\,\)梯度图的固定点上- 增加函数。显示了对矩阵的弱优先化预序和奇异值的一些解释。