Poly-Regular Ordered Abelian Groups

Oleg Belegradek

An ordered abelian group is called regular if for any positive integer n every interval of cardinality at least n contains an n-divisible element. A. Robinson and E. Zakon proved that a group is regular iff it is elementarily equivalent to an archimedian group. We call an ordered abelian group G poly-regular if there are n < ω and a series of convex subgroups

0 = G_{0} < G_{1} < … < G_{n} = G

such that G_{i+1}/G_{i} is regular for all i < n. We call such a series poly-regular. The minimal n with this property is called the poly-regular rank (or, for short, just the rank) of G. So the regular groups are exactly the poly-regular groups of rank 1. Clearly, any lexicographically ordered direct sum of finitely many regular groups is a poly-regular group.

Theorem 1 Any poly-regular group G has a unique poly-regular series

0 = G_{0} < G_{1} < … < G_{n} = G

such that G_{i+1}/G_{i} is not divisible for 1 ≤ i < n. The number n is equal to the poly-regular rank of G. All members of the series are 0-definable subgroups.

Corollary 2 Any group elementarily equivalent to a poly-regular group is poly-regular of same rank. The class of poly-regular groups of rank at most n is first order axiomatizable.

Theorem 3 A group is poly-regular iff it is elementarily equivalent to a lexicographically ordered direct sum of finitely many archimedian groups. More precisely, for an ordered abelian group G and a positive integer n the following are eqivalent:

(1) G is poly-regular of rank n,

(2) G has exactly n − 1 definable proper convex subgroups,

(3) G is elementarily equivalent to the lexicographically ordered direct sum of archimedian groups A_{1}, …, A_{n}, where A_{1}, …, A_{n−1} are not divisible.

Address: Department of Mathematics and Computer Science, Istanbul Bilgi University, Kuştepe 80310 Şişli, Istanbul, Turkey

E-mail: olegb@bilgi.edu.tr