There is a difference between the definition of isotopy introduced by Prof. Santilli and adopted by Profs. Valdez and Ganfornina and the definition of isotopy adopted by Prof. Georgiev of which you should perhaps be aware of.

As you know, isotopies originated under the strict conditions of preserving all original axioms and merely introducing broader realizations. This basic requirement lead to the following notion of isofunction $\hat F$ on an isovariable $\hat r$ defined on an isospace over isofield with isounit $\hat I = 1/\hat T > 0$

(1).....$\hat F(\hat r) = [F(r \hat I)] \hat I = F(r/\hat T)/\hat T$

with equivalent isotopic inversion given by the interchange of $\hat I$ with $\hat T$ (which is trivial due to the completely unrestricted character of the isounit (except for being positive-definite)

(2).....$\hat F(\hat r) = [F(r\hat T)]\hat T = [F(r/\hat I)]/\hat I$

Prof. Georgiev did include these two definitions of isofunctions but added two more definitions for which the isounit of the function is not the same as that of the variable, for a total of four types, for which Prof. Georgiev Type III is Prof. Santilli basic type (1) above, Type I is type (3) below and the other two are mixed, e.g.,

(3).....$\hat F(\hat r) = [F(r \hat I) \hat T = F(r/\hat T)/\hat I$

(4).....$\hat F(\hat r) = [F(r\hat T)]\hat I = F(r/\hat I)\hat I$

You should be aware that Prof. Santilli calls the additional two definitions as being "pseudo-isotopies" because they do not necessarilhy preserve the original axioms. To illustrate this point, Prof. Santilli shows that, for the true isotopies or their inversions of the mapped Hilbert product remains inner, thus preserving the original Hilbert axioms, while for the two additional types by Prof. Georgiev the mapped Hilbert product is not necessarily inner. This is acceptable on mathematical grounds for which reasons Prof. Santilli has supported and continues to support Prof. Georgiev as much as possible.

However, it is important to know the physical implications because they are serious. The preservation of the original Hilbert axioms assures the causality of the theory, while the violation of said axioms implies the existence of solutions violating the principle of causality for which reason Prof. Santilli suggested the name of "pseudo-isotopies" for the two additional lifting beautifully studied by Prof. Georgiev.

I though that the above clarification might be of some interest to you.