Homogeneity of ideals

The homogeneity family of the ideal I is a family of subsets such that the restriction of I to this subset is isomorphic to I. We say that an ideal I is homogeneous if all I-positive sets belong to the homogeneity family of I. We investigate basic properties of this notion, give examples of homogeneous ideals and present some applications to topology and ideal convergence. Moreover, we present connections between the homogeneity families and the notion of bi-I-invariant functions introduced by Balcerzak, Głąb and Swaczyna and give answers to several questions related to this topic.