On self-majorizing elements in Archimedean vector lattices

A finite element in an Archimedean vector lattice is called self-majorizing if its modulus is a majorant. Such elements exist in many vector lattices and naturally occur in different contexts. They are also known as semi-order units as the modulus of a self-majorizing element is an order unit in the band generated by the element. In this paper the properties of self-majorizing elements are studied systematically, and the relations between the sets of finite, totally finite and self-majorizing elements of a vector lattice are provided. In a Banach lattice an element \(\varphi \) is self-majorizing , if and only if the ideal and the band both generated by \(\varphi \) coincide.