Unbounded Order-Order Continuous Operators

In this paper, we define two new classes of operators, named unbounded order-order continuous (uoo-continuous, for short) operators and order-unbounded order continuous operators (ouo-continuous, for short). We show that the collection of all uoo-continuous operators (respectively ouo-continuous operators) on a Riesz space $\mathbf{E}$ is an ideal. We give a condition under which if T belongs to this class, then $|T|$ also belongs to this class. And we present the relation between these classes and other classes of operators (the class of order continuous operators and the class of unbounded order continuous operators). we give a condition for the classes uoo-continuous, ouo-continuous, order continuous and unbounded order continuous operators to be equal to each other.