The Amalgamation Property for G-metric Spaces

Let G be a (totally) ordered (abelian) group. A Gmetric space (X, p) consists of a nonempty set A"and a G-metric />: XxX->-G (satisfying the usual axioms of a metric, with G replacing the ordered group of real numbers). That the amalgamation property holds for the class of all metric spaces is attributed, by Morley and Vaught, to Sierpiñski. The following theorem is proved. Theorem. The class of all G-metric spaces has the amalgamation property if, and only if, G is either the ordered group of the integers or the ordered group of the reals. The purpose of this paper is to prove that the amalgamation property fails for the class of all G-metric spaces, for every (totally) ordered (abelian) group G which is not equal to the additive group of the integers or that of the real numbers. The amalgamation property, in its abstract form, was first formulated by Fraïssé [3] in connection with embedding problems. It has been studied by Jónsson [6], [7], [8] and [9], and Morley and Vaught [12] in connection with the general theory of homogeneous-universal structures in Jónsson classes. Not too many examples of classes of relational systems are known for which the amalgamation property fails. Among them the following are included: (a) the class of semigroups and hence of rings (Kimura [11], Jónsson [6], [7], Howie [5]); (b) the class of modular lattices (Jónsson [10]); (c) the class of /-groups (Pierce [13]). To these, because of our theorem, we can now add a whole family of such examples, among which is the class of all metric spaces on which the metric takes on only rational values. Classes of metric spaces were studied by Urysohn [17], [18] and Sierpiñski [14], [15] from the point of view of the existence of universal Presented to the Society, July 6, 1971 ; received by the editors April 25, 1972. AMS (MOS) subject classifications (1970). Primary 06A55, 08A05, 54E35.