Some characterizations of Riesz spaces in the sense of strongly order bounded operators

We investigate some properties of strongly order bounded operators. For example, we prove that if a Riesz space E is an ideal in E-similar to similar to and F is a Dedekind complete Riesz space then for each ideal A of E, T is strongly order bounded on A if and only if T-A is strongly order bounded. We show that the class of strongly order bounded operators satisfies the domination problem. On the other hand, we present two ways for decomposition of strongly order bounded operators, and we give some of their properties. Also, it is shown that E has order continuous norm or F has the b-property whenever each pre-regular operator form E into F is order bounded.