Geodesic
Lattice-ordered groups having at most two disjoint elements†
Glasgow mathematical journal, Tome 4 (1960) no. 3, pp. 111-113

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Conrad, P. F.; Clifford, A. H. Lattice-ordered groups having at most two disjoint elements†. Glasgow mathematical journal, Tome 4 (1960) no. 3, pp. 111-113. doi: 10.1017/S2040618500034018
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[1] 1.Birkhoff, Garrott, Lattice theory, Amer. Math. Soc. Colloquium Publication, Rev. Ed. (1948). Google Scholar

[2] 2.Jaffard, Paul, Contribution à l'étude dos groupes ordonnés, J. Math. Pures Appl. (9) 32 (1953), 203–280. Google Scholar

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