Below we suppose that M is
first-order equivalent to
< N, <, 0, 1,
+, {[/n] | n=1,2,…}, f >
where f is an arbitrary unary function.
Let (M,I) be divided.
Let x be a sequence of elements of
M. Let
Y0 be empty and
X0
be the set of all the elements
of x.
Let Yi+1 be the set of
all the elements of b∈I such that
there exist term t(x),
unbounded terms
t1(y) and
t2(y),
and a∈
(X0∪Yi)
such that
t1(b)<t(a)<t2(b).
(4)
For t and
a, there is no
b∈I or there is the only
b∈I such that (4) holds for
some unbounded
t1 and
t2.
So Yi is countable for any
i. Let
Y be the union of all
Yi. We call
Y the linked part of
I for x and denote
Y by
[I, x].
Let us consider any divided
(M,I). Fix two variable sequences
x and
y. A quantifier-free formula is called divided
iff it is a Boolean of atomic divided formulas. An atomic formula
is called divided iff it has the form
t1(x)+t2(y)<
t3(x)+t4(y).
The theory of
(M,I) is divided iff for any first-order formula
Φ(x,y),
there exist a quantifier-free divided formula
Ψ(x,y)
such that for any
b,
a∈(I−[I,b]), and
c∈([I,b]∪b),
Φ(a,c),
is equivalent to
Ψ(a,c).
A function f
is divided iff there exists a divided
(M,I) such that the theory
of (M,I)
is divided and M
is equivalent to
< N, <, 0, 1,
+, {[/n] | n=1,2,…}, f >.
The next theorem is a trivial corollary of [7], Theorems 3.1
and 3.2.
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