In this paper we introduce two new classes of operators that we call strongly order continuous and strongly σ-order continuous operators. An operator T : E → F between two Riesz spaces is said to be strongly order continuous (resp. strongly σ-orderWe give some conditions under which order continuity will be equivalent to strongly order continuity of operators on Riesz spaces. We show that the collection of all so-continuous linear functionals on a Riesz space E is a band of E ∼ .
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