Topological stability and shadowing in uniform transformation semigroups modulo an ideal

In this paper, we introduce and analyse several key dynamical properties – namely shadowing modulo an ideal, expansivity modulo an ideal, and topological stability modulo an ideal – within the framework of uniform transformation semigroups. Given an ideal I on semigroup T, we investigate the interplay between these properties in a compact Hausdorff transformation semigroup (T,X,X). Our main result establishes that if a compact Hausdorff transformation semigroup exhibits the shadowing property modulo I and is expansive modulo I then it is also topologically stable modulo I. This extends known stability theorems in classical topological dynamics to the setting of ideal-constrained dynamics. Additionally, we explore the relationship between shadowing modulo I and the conventional shadowing property.