Vol. 27, No. 2, 1968

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On abelian pseudo lattice ordered groups

J. Roger Teller

Vol. 27 (1968), No. 2, 411–419

Throughout this paper po-group will mean partially ordered abelian group. A subgroup H of a po-group G is an o-ideal if H is a convex, directed subgroup of G. A subgroup M of G is a value of 0g G if M is an o-ideal of G that is maximal without g. Let (g) = {M GM is a value of g} and (g) = ∩ℳ(g). Two positive elements a,b G are pseudo disjoint (p-disjoint) if a ∈ℳ(b) and b ∈ℳ(a), and G is a pseudo-lattice ordered group (pl-group) if each g G can be written g = ab where a and b are p-disjoint.

The main result of §2 shows that every pl-group G is a Riesz group. That is, G is semiclosed (ng 0 implies g 0 for all g G and all positive integers n), and G satisfies the Riesz interpolation property; if, whenever x1,,xm, y1,,yn are elements of G and xi yj for 1 i m, 1 j n, then there is an element z G such that xi z yj.

Mathematical Subject Classification
Primary: 06.78
Received: 3 July 1967
Published: 1 November 1968
J. Roger Teller