Characterization of self-majorizing elements in Archimedean vector lattices

Abstract

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 \(0<f\in A\) 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 \({\mathcal {P}}\) and on the set of all g-maximal order ideals \({\mathcal {Q}}\) of A, for all \(g\in A^{+}.\) 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 \(g\in A^{+})\) is a closed with respect to the hull–kernel topology on \({\mathcal {P}}\) (respectively, on \({\mathcal {Q}})\) if and only if f is a self-majorizing element in A.