Theorem 13
For any structure
< N, <, 0, 1,
+, {[/n] | n=1,2,…}, f >
where f
is a divided function, any
locally generic extended query is equivalent to a restricted one.
Theorem 14 (Dudakov, [6])
The theory of divided (M,I),
where M is equivalent to
< N, <, 0, 1,
+, {[/n] | n=1,2,…}, f >
and f is a concordant with
addition monotone function, is divided.
Now I propose a sufficient condition for f
to be divided. The
condition is a generalization of the Dudakov's proof of the last
theorem. The functions (1) and (3) support the condition. So the
functions are divided, and the collapse theorem holds for
the expansion of the Presburger's arithmetics by any one of
the functions.
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