Restricted pseudofinite homogeneity and isolation

In [8], it was presented a relativize version of results from [2] as to the collapse theorem. In the paper, I propose improved and more accurate presentation of the version. The properties of $(M,I)$-Pseudo-finite Homogeneity and $(M,I)$-Isolation are in the focus of the paper. They both imply the collapse theorem. It is investigated so called $P$-reducible theories. It is proved that, for the $P$-reducible theories, a version of $(M,I)$-Isolation Propery holds. So the collapse theorem holds for $P$-reducible theories.
In [5], it was proposed and expansion of Presburger's arithmetics by a unary function such that the first-order theory of the expansion is decidable and the expansion has an independent formula. I prove that the $(M,I)$-Isolation Property does not hold for the expansion.