FUNDAMENTA MATHEMATICAE 179 (2003) Relatively complete ordered fields without integer parts by Mojtaba Moniri and Jafar S. Eivazloo (Tehran) Abstract. We prove a convenient equivalent criterion for monotone completeness of ordered fields of generalized power series [[F G ]] with exponents in a totally ordered Abelian group G and coefficients in an ordered field F . This enables us to provide examples of such fields (monotone complete or otherwise) with or without integer parts, i.e. discrete subrings approximating each element within 1. We include a new and more straightforward proof that [[F G ]] is always Scott complete. In contrast, the Puiseux series field with coefficients in F always has proper dense field extensions. 1. Introduction and preliminaries. A subset C of an ordered field F is a cut if C<F \ C . A nontrivial cut is a gap whenever it fails to have a least upper bound in the field. A gap G in F is regular whenever (∀ε ∈ F >0 )(G + ε ⊆ G). An ordered field F is Scott complete if it does not have any proper extensions to an ordered field in which it is dense, equivalently it does not have any regular gaps, equivalently all Cauchy nets in the field of length equal to its cofinality converge there. It was proved in [12, Thm. 1] that any ordered field F has a (unique up to an isomorphism of ordered fields which is identity on F ) Scott completion. It is characterized by being Scott complete and having F dense in it. Monotone complete ordered fields were introduced in [6]. They are or- dered fields with no bounded strictly increasing divergent functions. In such ordered fields, those nontrivial cuts which do not have a least upper bound in the field, are also not traversed by strictly increasing nets of length equal to the cofinality of the field. As implied by [11, Cor. 2.7], there are mono- tone complete ordered fields of any uncountable cofinality and so there exist plenty of monotone complete ordered fields not isomorphic to R. On the other hand, it is clear that there are no monotone complete ordered fields of countable cofinality except (those isomorphic to) R. 2000 Mathematics Subject Classification : 12J15, 13J05, 54H13. Key words and phrases : ordered fields, Scott complete, monotone complete, general- ized power series, integer part. [17]