Kappa-bounded exponential groups and exponential-logarithmic power series fields without log-atomic elements

A divisible ordered abelian group is an exponential group if its rank as an ordered set is isomorphic to its negative cone. Exponential groups appear as the value groups of ordered exponential fields, and were studied in [1]. In [2] we gave an explicit construction of exponential groups as Hahn groups of series with support bounded in cardinality by an uncountable regular cardinal kappa. An exp-log series s is said to be log atomic if the nth-iterate of log(s) is a monomial for all n in N. In this talk I will present a modified construction of kappa-bounded Hahn groups and exploit it to construct kappa bounded Hahn fields without log-atomic elements. This is ongoing joint work with Berarducci, Mantova and Matusinski.[1] S. Kuhlmann, Ordered exponential fields, The Fields Institute Monograph Series, vol 12. Amer. Math. Soc. (2000) [2] S. Kuhlmann and S. Shelah, Kappa-bounded Exponential-Logarithmic power series fields, Annals Pure and Applied Logic, 136, 284-296 (2005)