ABSTRACT
We will be dealing with ordered fields of generalized power series and will give a convenient monotone completeness criterion for them. Recall that for an ordered field F and ordered Abelian group G, the ordered field of generalized power series with exponents in G and coefficients in F is:
[{FG}} = | ^ agt9 the support {g \ ag ^ 0} of ]P agtg is well ordered i. ^-n^n ncr: '
It is equipped with the natural addition, multiplication, and order defined by ]T\eG, os£5 > 0 if ago > 0, where go is the minimum of support of X^oeGas^sA well known necessary and sufficient condition for [[-FG]] to be real closed is that G be divisible and F be real closed, see [5, 6.10]. Observe that cf([[FG]]) = cf(G) (where cf denotes cofinality). It is already known, see [2] (by the way we plan to present a general straightforward proof of this fact elsewhere), that all ordered fields of generalized power series satisfy the weaker notion of Cauchy-Scott completeness. Scott complete ordered fields were introduced in [7]. They are ordered fields with no divergent Cauchy nets of length equal to the cofinality of the field.
