TY - JOUR
T1 - κ-bounded exponential-logarithmic power series fields
AU - Kuhlmann, Salma
AU - Shelah, Saharon
PY - 2005/11
Y1 - 2005/11
N2 - In [F.-V. Kuhlmann, S. Kuhlmann, S. Shelah, Exponentiation in power series fields, Proc. Amer. Math. Soc. 125 (1997) 3177-3183] it was shown that fields of generalized power series cannot admit an exponential function. In this paper, we construct fields of generalized power series with bounded support which admit an exponential. We give a natural definition of an exponential, which makes these fields into models of real exponentiation. The method allows us to construct for every κ regular uncountable cardinal, 2κ pairwise non-isomorphic models of real exponentiation (of cardinality κ), but all isomorphic as ordered fields. Indeed, the 2κ exponentials constructed have pairwise distinct growth rates. This method relies on constructing lexicographic chains with many automorphisms.
AB - In [F.-V. Kuhlmann, S. Kuhlmann, S. Shelah, Exponentiation in power series fields, Proc. Amer. Math. Soc. 125 (1997) 3177-3183] it was shown that fields of generalized power series cannot admit an exponential function. In this paper, we construct fields of generalized power series with bounded support which admit an exponential. We give a natural definition of an exponential, which makes these fields into models of real exponentiation. The method allows us to construct for every κ regular uncountable cardinal, 2κ pairwise non-isomorphic models of real exponentiation (of cardinality κ), but all isomorphic as ordered fields. Indeed, the 2κ exponentials constructed have pairwise distinct growth rates. This method relies on constructing lexicographic chains with many automorphisms.
KW - Iterated lexicographic power of a chain
KW - Logarithmic rank
KW - Models of real exponentiation
UR - http://www.scopus.com/inward/record.url?scp=25444528975&partnerID=8YFLogxK
U2 - 10.1016/j.apal.2005.04.001
DO - 10.1016/j.apal.2005.04.001
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AN - SCOPUS:25444528975
SN - 0168-0072
VL - 136
SP - 284
EP - 296
JO - Annals of Pure and Applied Logic
JF - Annals of Pure and Applied Logic
IS - 3
ER -