Inverses and order boundedness of invertible ideal operators

In this paper, we investigate that under which conditions the inverse of an invertible ideal operator becomes an ideal operator. To determine conditions, in the first case, we consider the center of a Riesz space and achieve the solution by using some properties of it. In the second case, we first give an example of an ideal operator which is not order bounded. Then, we research when an ideal operator is order bounded. Thus, we conclude that if an invertible operator is order bounded then being an ideal operator, a band operator, and a disjointness preserving operator are equivalent and the same situation holds for the inverses of these three operators.