An Example of Two Nonisomorphic Countable Ordered Abelian Groups with Isomorphic Lexicographical Squares

If M and Nare ordered groups, we denote by Mx N the group MxNequipped with the lexicographical order: (a,b) < (a',b') if and only if a < a' or {a = a' and b < b'). In [1], A. L. S. Corner gives an example of two countable abelian groups A, G such that G and A x A x G are isomorphic while G and A x G are not isomorphic; he also gives an example of two nonisomorphic countable abelian groups G, H which satisfy GxG^HxH. On the other hand, we can easily deduce from [5, Theorem 5.2] that, if A and G are abelian groups and if G and A x A x G are elementarily equivalent, then G and A x G are elementarily equivalent. It follows from the same theorem that two abelian groups G,Hare elementarily equivalent as soon as G x G and HXHSLTQ elementarily equivalent. We showed in [4] that, if A and G are ordered abelian groups and if G and A x A x G are elementarily equivalent, then G and A x G are elementarily equivalent. We also gave an example of two countable ordered abelian groups A, G such that G and A x A x G are isomorphic while G and A x G are not isomorphic. In [2, Corollary 4.7], F. Delon and F. Lucas prove that two ordered abelian groups G,Hare elementarily equivalent as soon as G x G and HxHare elementarily equivalent. The same result is proved independently in [3] by M. Giraudet, who also gives conditions on the ordered abelian group G which imply that any ordered group H such that GxG^HxH is isomorphic to G. In the present paper, we give an example of two nonisomorphic countable ordered abelian groups G, H which satisfy GxG^HxH.