In this paper we show the equality of the order continuity of the norm and weak orthogonality in discrete Banach lattices. Then we consider several fundamental concepts in Banach lattices, including weak orthogonality, the WORTH property, and the non-strictly Opial condition. We characterize Banach lattices in which these properties are equivalent to the order continuity of the norm. Also, we check out the relationship of these properties with each other and the disjoint (positive) version of those. Indeed, we show the equality of the weak orthogonality, the WORTH property, and the non-strictly Opial condition with order continuity of the norm in σ-Dedekind complete discrete Banach lattices.
