In my P.h.d research, I deal (among other things) with non-associative words, which we call monomials, and we need to consider two types of operations with these monomials.
The first one is simply substitution of a word on a variable of another word: for instance, if $w_1(x_1,x_2)=(x_1x_2)x_1$ and $w_2(x_1,x_2)=(x_2^2)x_1$, we can substitute $w_2$ in the variable $x_2$ of $w_1$ to obtain $w_3(x_1,x_2)=(x_1((x_2^2)x_1))x_1$.
The second one, which we call reduction, is to simply reduce a submonomial of a monomial to a variable. That is, $w_2(x_1,x_2)=(x_2^2)x_1$ is a submonomial of $w_3(x_1,x_2)=(x_1((x_2^2)x_1))x_1$, and if we reduce $w_2$ to $x_2$ in $w_3$ we obtain $w_1(x_1,x_2)=(x_1x_2)x_1$.
I need to use in my research some useful properties of these operations, such as the fact that they are "inverses" of each other, in the sense of the example above. Also, often I need to compute the submonomials of the monomials which are results of reductions or substitutions, in terms of the submonomials of the original monomial.
Is there a reference (book, paper, etc.) which offers a formal treatment of these operations and related properties in the non-associative setting? There are plenty of references on the associative case, but I couldn't find any on the non-associative case.
