Existentially Complete Abelian Lattice-ordered Groups
چکیده
The theory of abelian totally ordered groups has a model completion. We show that the theory of abelian lattice-ordered groups has no model companion. Indeed, the Archimedean property can be captured by a first order V3V sentence for existentially complete abelian lattice-ordered groups, and distinguishes between finitely generic abelian lattice-ordered groups and infinitely generic ones. We then construct (by sheaf techniques) the model companions of certain classes of discrete abelian lattice-ordered groups. The class of abelian groups has a model companion [7]. The slightly larger class of nilpotent class 2 groups, however, is already very misbehaved; there is no model companion and, indeed, the finitely generic and infinitely generic theories share no models in common ([17] and [18]). It is natural to wonder, therefore, what happens if, instead of enlarging the class of groups, we enlarge the language by adding a binary relation < compatible with the group operation (cf. passing from fields to ordered fields). A. Robinson has shown the following: Theorem A [15, p. 36]. The theory of abelian totally ordered groups has a model completion; viz. the theory of divisible abelian totally ordered groups having at least two different elements. When total order is replaced by lattice-order, the conclusion changes radically. Our main theorem is: Theorem B. (1) The theory of abelian lattice-ordered groups has no model companion. Moreover, (2) the Archimedean property is equivalent to a V3V sentence for existentially complete abelian lattice-ordered groups, and (3) finitely generic abelian lattice-ordered groups satisfy the sentence (i.e., are Archimedean) but infinitely generic ones do not (i.e., are not Archimedean). Of course, the Archimedean property is not expressible by a first order sentence in general; it is only when we restrict to existentially complete abelian latticeordered groups that we can capture it by a first order sentence. This expressibility phenomenon is frequently used to distinguish between finitely generic structures Received by the editors April 6, 1979 and, in revised form, October 12, 1979. AMS (MOS) subject classifications (1970). Primary 03C25, 03C60, 06F20.