Hulls for various kinds of α-completeness in Archimedean lattice-ordered groups

Within Archimedean l-groups, and with alpha an infinite cardinal or infinity, we consider X-hulls where X stands for any of the following classes of l-groups: alpha-projectable; laterally alpha-complete; boundedly laterally alpha-complete; conditionally alpha-complete; combinations of the preceding, together with divisibility and/or relative uniform completeness. All these hulls exist, and may be obtained by iterated adjunction of the required extra elements, within the essential hull. When the l-groups is relatively alpha-complemented one step in the iteration suffices for several crucial properties. We derive from the above a considerable number of equations involving combinations of these hull operators.