On semi-I-regular sets, AB(I)-sets and decompositions of continuity, RIC-continuity, A(I)-continuity

We introduce the notion of a semi-I-regular set and investigate some of its properties. We show that it is weaker than the notion of a regular-I-closed set. Additionally, we also introduce the notion of an AB(I) -set by using the semi-I-regular set and study some of its properties. We conclude that a subset A of an ideal topological space (X,tau,I) is open if and only if it is an AB(I) -set and a pre-I-open set.