Lattice-ordered groups having at most two disjoint elements

Abstract

Let L = L( +, v, ＾) be a lattice-ordered group, or l-group (Birkhoff [1, p. 214]). Two elements a and b of L will be called disjoint if a > 0, b > 0, and a ＾; b = 0. It is easily seen that if L does not contain two disjoint elements, then it is linearly ordered (and, of course, conversely). What can we say about Z-groups containing two but not more than two mutually disjoint elements?Let Aand B be linearly ordered groups (o-groups), and let A ⋏ B be the cardinal sum of A and B. That is, A ⋏ B is the direct sum of A and B, and (a, b) is positive in A + B if and only if a is positive in A and b is positive in B. An l-group L containing A ⋏ B as a convex normal subgroup (or Z-ideal) is called a lexico-extension of A ⋏ B if every positive element of L not in A ⋏ B exceeds every element of A ⋏ B. It then follows (subsection 2.9 below) that L/(A ⋏ B) is an o-group. Such an l-group L is easily seen to satisfy the following condition: (D)There exists a pair of disjoint elements in L, but no triple of pairwise disjoint elements exists in L.