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Compatible tight Riesz orders and prime subgroups

N. R. Reilly
1973 Glasgow Mathematical Journal  
Introduction. A tight Riesz group is a partially ordered group which satisfies a strengthened form of the Riesz interpolation property. The term " tight" was introduced by Miller in [8] , and the tight interpolation property has been considered in papers by Fuchs [3], Miller [8, 9] , Loy and Miller [7] and Wirth [12]. If the closure P of the cone P, in the interval topology, of such a partially ordered group G contains no pseudozeros, then P is itself the cone of a partial order on G. Loy and
more » ... der on G. Loy and Miller found of particular interest the case in which this associated partial order is a lattice order. This situation was then considered in reverse by A. Wirth [12] who investigated under what circumstances a lattice ordered group would permit the existence of a tight Riesz order (called a compatible tight Riesz order) for which the initial lattice order is the order defined by the closure of the cone of the tight Riesz order. Wirth gave two fundamental and useful characterizations of those subsets of the cone of a lattice ordered group that can be the strict cone of a compatible tight Riesz order; one is in terms of archimedean classes and the other is an elementwise characterization. Although Loy, Miller and Wirth restricted their attention to abelian groups, much of what they do carries over verbatim to nonabelian groups. In the main result of this paper (Theorem 2.6) a description of the strict cone of a compatible tight Riesz order on a lattice ordered group G is given in terms of the prime subgroups of G. This is particularly useful when one is attempting to identify the compatible tight Riesz orders on some particular lattice ordered group or class of lattice ordered groups, since it narrows down to a convenient family of subsets the possible candidates for strict cones of compatible tight Riesz orders. These can then be tested under Wirth's criteria. This technique is illustrated in § 5, where the compatible tight Riesz orders are determined on a lattice ordered group of the type V(T, G y ), where T is of finite width, and in § 6, where two examples are considered. 1. By a. partially ordered group (G, ^) we mean a group G and a partial ordering, ^, of G which is compatible with the group operation. We shall usually denote the group operation by +, even for nonabelian groups. A partially ordered group (G, ^) is called a tight Riesz group if £ is a directed partial ordering and if, for any elements a, b, c, d in G such that a, b < c, d there exists an element xin G with The order relation is then called a tight Riesz order. This interpolation property, which is a refinement of the Riesz interpolation property [11], has been extensively considered for abelian partially ordered groups by Miller and Loy in [7], [8] and [9], by Wirth in [12] and has been
doi:10.1017/s0017089500001890 fatcat:kxtyfppl4zfajp3aozakpsaxdi