The paper gives a systematic analysis of the convergence conditions used in comparison theorems, proven for a few types of matrix splittings representing a large class of applications. The central idea of this analysis is the scheme of condition implications derived from the properties of regular splittings of a monotone matrix $A=M_1-N_1=M_2-N_2$. Equivalence of some conditions are an autonomous character of the conditions $M^{-1}_1\geq M^{-1}_2\geq0$ and $A^{-1}N_2\geq A^{-1}N_1\geq0$ are pointed out.