Study of some asymptotic properties of the axis-quasi-graduations of the same ring, particular case of quasi-graduations

This article generally studies a new class of families not necessarily nested with subgroups obeying a certain compatibility, or from two axis-quasi-graduations of an arbitrarily chosen ring we give properties of binary relations and classical operations obtained from two rings axis-quasi-graduations. The main results obtained will be used to study and characterize certain concepts of commutative algebra.