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Lattice-ordered groups having at most two disjoint elements†

Published online by Cambridge University Press:  18 May 2009

P. F. Conrad  and
A. H. Clifford
P. F. Conrad
Affiliation:
Tulane University of Louisiana, New Orleans, Louisiana, U.S.A.
A. H. Clifford
Affiliation:
Tulane University of Louisiana, New Orleans, Louisiana, U.S.A.
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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.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 4 , Issue 3 , January 1960 , pp. 111 - 113
Copyright
Copyright © Glasgow Mathematical Journal Trust 1960

References

1.Birkhoff, Garrott, Lattice theory, Amer. Math. Soc. Colloquium Publication, Rev. Ed. (1948).Google Scholar
2.Jaffard, Paul, Contribution à l'étude dos groupes ordonnés, J. Math. Pures Appl. (9) 32 (1953), 203–280.Google Scholar