Characterization of self-majorizing elements in Archimedean vector lattices

In this paper, new purely topological approaches are furnished in order to characterize self-majorizing elements in an Archimedean vector lattice A. More precisely, it is shown that an element \(0<f\in A\) $0<f\in A$ is a self-majorizing element if and only if every f-maximal order ideal of A is relatively uniformly closed. In addition, it is proved that self-majorizing elements are characterized via the hull–kernel topology on both the set of all proper prime order ideals \({\mathcal {P}}\) $\mathcal{P}$ and on the set of all g-maximal order ideals \({\mathcal {Q}}\) $\mathcal{Q}$ of A,  for all \(g\in A^{+}.\) $g\in A^{+}.$ In fact, the set of all prime order ideals of A not containing f (respectively, the set of all g-maximal order ideals of A not containing f,  for all \(g\in A^{+})\) $g\in A^{+})$ is a closed with respect to the hull–kernel topology on \({\mathcal {P}}\) $\mathcal{P}$ (respectively, on \({\mathcal {Q}})\) $\mathcal{Q})$ if and only if f is a self-majorizing element in A.