F U N D a M E N T a Mathematicae Bounded Countable Atomic Compactness of Ordered Groups

We show that whenever A is a monotone σ-complete dimension group, then A+ ∪ {∞} is countably equationally compact, and we show how this property can supply the necessary amount of completeness in several kinds of problems. In particular, if A is a countable dimension group and E is a monotone σ-complete dimension group, then the ordered group of all relatively bounded homomorphisms from A to E is a monotone σ-complete dimension group. 0. Introduction. By definition, an ordered abelian group is monotone σ-complete whenever every bounded increasing sequence of elements admits a l.u.b. We will here be concerned with those monotone σ-complete groups that are in addition directed and satisfy the Riesz interpolation property. One can then show that such groups are (strongly) Archimedean, thus they are dimension groups [7, 8]. Monotone σ-complete dimension groups appear naturally as ordered Grothendieck groups K0 of countably continuous regular rings and thus intervene also in the study of Rickart C∗-algebras [7, 8, 9]. Intriguingly, there is another related large class of ordered structures, the class of Tarski’s cardinal algebras, as well as other more general classes, as e.g. refinement algebras [14]. Basically, these objects also appear naturally in a lot of cases as algebras of isomorphism types of various structures such as sets under equipotence or σ-complete Boolean algebras. Many of their properties are already valid in the more general class of weak cardinal algebras, defined simultaneously and independently by K. P. S. Bhaskara Rao and R. M. Shortt on the one hand, and the author on the other hand in [11, 18, 19]. 1991 Mathematics Subject Classification: Primary 06F05, 06F20, 08A45; Secondary 19A49, 19K14.