In this paper, using the Right topology, we introduce three new properties in Banach lattices: the so-called Right orthogonality, the Right WORTH property, and the non-strictly Right Opial condition (and also positive versions of them). Moreover, Banach lattices in which these three properties coincide with order continuity of the norm are characterized. As an application, we give some sufficient conditions under which a Banach lattice has the Right fixed point property (or, positive Right fixed point property). In particular, it is established that for a Banach space X and a suitable Banach lattice F , a Banach lattice M ⊂ K(X; F ) has the Right fixed point property (resp. positive Right fixed point property) if each evaluation operator y∗ on M is a pseudo weakly compact (resp. positive pseudo weakly compact) operator, where y∗ : M ! X∗ is defined by y∗(T ) = T ∗y∗ for y∗ 2 F ∗ and T 2 M.
