Pair analysis of involutive divisions

The main goal of this work is to describe a new approach to the study of involutive divisions using the pairwise property. The paper presents a simple and intuitive method for constructing the Janet division and reveals the deep intrinsic relationship between Janet division and $\mathrm{Lex}$-ordering. A method for constructing some analogues of the Janet division for other orders is described. An example of pairwise, continuous, and nonconstructive involutive division is given.