Let be an r.t. extension of v to defined by a minimal pair for , where and let u be the r.t. extension of v to defined as in (4). Let be an r.t. extension of v to defined by a minimal pair for , where and let be the r.t. extension of v to defined as in (4). A relation between such kind of r.t. extensions of v to can be defined so that if and only if for . This is an order relation, and if , then for each polynomial , is satisfied. Because, for ,
Now, let I be a well-ordered set without the last element and be an ordered system of r.t. extensions of v to , where is defined as in (4), i.e., is the common extension of , where is the r.t. extension of v to defined by the minimal pair , where for all . If for all , then is an ordered system of r.t. extensions of v to . Then the valuation w of defined as
(5)
for all is an extension of v to and it is called a limit of the ordered system .
If is the restriction of w to for , then is the limit of the ordered system of r.t. extensions of v to . Also, w is the common extension of to . Since may not be an r.t. extension of v to , then w may not be an r.t. extension of v to .
If is a residual algebraic torsion extension of v to , then is satisfied. Some other properties of w are studied below.
Denote the extension of to by and the extension of to by for and for all .
Theorem 3.1 Let be an ordered system of r.t. extensions of to , where is defined as in (4), i.e., is the r.t. extension of v to which is the common extension of for and for all . Denote the restriction of to by and the restriction of to by . Then
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1.
For all , , one has , i.e., is an ordered system of r.t. extensions of v to .
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2.
For all , , one has and .
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3.
Suppose that and is not an r.t. extension of to and denote that w is the restriction of to . Then and and .
Proof For every and , denote that and .
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1.
Since for all , , we have . We show that . Assume that . Since is the common extension of and is the common extension of for , we have for . Since is a minimal pair of the definition of , we have for . But it is a contradiction, because is an ordered system of r.t. extensions of to and so , i.e., [3]. Hence for . Since and are common extensions of and respectively for and for all , it is concluded that for all .
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2.
It is enough to study for , where , .
It is seen that by using the [[3], Th. 2.3] and this gives .
Assume that . Then . Since is a minimal pair of the definition of for , we have coincides with the which is the residue of B in . Hence for all , .
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3.
Since , we have and w is not an r.t. extension of v to . Using [[3], Th. 2.3] and the definition of , the proof can be completed. Take such that . Since is a minimal pair of the definition of , . This means that for all and so .
Conversely, let be the restriction of to for and for all . Since , then for every . This gives .
Now, assume that . Then and since , , which is -residue of , coincides with the residue of in . This shows for all , and then .
For the reverse inclusion, let and . For , is equal to and so . Hence .
□
The following theorem can be obtained as a result of Theorem 3.1.
Corollary 3.2 Under the above notations, let w be an r.a.t. extension of v to . Then the following are satisfied:
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1.
and for all , .
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2.
is an ordered system of r.t. extensions of v to and . Moreover, we have and .
Proof If w is an r.a.t. extension of v to , then is an r.a.t. extension to and so is an r.a.t. extension of to for . We can take for as co-final well-ordered subsets of . I has no last element because is not an r.t. extension of . For every , choose the element such that for , and is the smallest possible for . This means that if such that , then . Then is a minimal pair of the definition of with respect to K for . According to [[3], Th. 4.1], if , which means that is an ordered system of r.t. extensions of to for and has a limit which is an r.a.t extension of v to . For all , take as the common extension of to and as the common extension of to for . Denote the restriction of to by and denote the restriction of to by w. In the same way as that in the proof of Theorem 3.1, it is seen that for , and for all and . Moreover, and are satisfied. □