数理解析研究所講究録
第 1708 巻 2010 年 46-49

46

On weak notion of -dividing$\mathfrak{p}$前園 久智 (Hisatomo MAESONO)

早稲田大学メディアネットワークセンター(Media Network Center, Waseda University)
Abstract
I considered the restricted notions of weak dividing. In this note, I
try to define a weak notion of p-dividing (thorn-dividing).

1. Preliminaries
We recall some definitions.

Definition 1

Let$\varphi(x_{0}, x_{1}, \cdots\cdots, x_{n-1})$

be a formula and $p(x)$ be a type.

We denote the type $\{\varphi(x_{0}, x_{1}, \cdots\cdots, x_{n-1})\}\cup p(x_{0})\cup p(x_{1})\cup\cdots\cdots\cup p(x_{n-1})$.
by$A\subset B$ and $p(x)\in S(B)$ .
Let$p(x)$ divides overA if there is a formula $\varphi(x, b)\in p(x)$ and an infinite
sequence $\{b_{i} : i<\omega\}$ with $b\equiv b_{i}(A)$ such that $\{\varphi(x, b_{i}) : i<\omega\}$ is-inconsistent for some $k<\omega$ .
such that
if there is a formula$p(x)$ weakly divides overis inconsistent.
is consistent, while$[\rho]^{\varphi}$$k$$\varphi(\overline{x})\in L_{n}(A)$$A$$|p]^{\varphi}$$[p\lceil A]^{\varphi}$

We can define weak dividing for formulas.
.
Letand a$\psi(x, b)$ weakly divides overA if there is a formula
isis consistent, while
realization of $\psi(x, b)$ such thatinconsistent.
And we can consider weak forking.
if there is a $q(x, y)\in S(A)$ such that $p(x)\cup$$p(x)$ weakly forks over$q(x, y)$ is consistent, and any completion $r(x, y)\in S(B)$ of $p(x)\cup q(x, y)$weakly divides over .
$b\not\in A$$\varphi(\overline{x})\in L_{n}(A)$$[tp(a/A)]^{\varphi}$$a$$[\psi(x, b)]^{\varphi}$

$A$

$A$

If we exchange the role between variables and parametes in the definitionof weak dividing, we could define weak forking naturally.
”in the definition above theIn this note, we call such formula
witness formula of weak dividing for the sake of convenience.
I introduce an example from [3].
“$\varphi(\overline{x})$47Example 2 Let be the theory of an equivalence relation with two infinite.
classes of the language $L=$ {a binary relation $E(x,$ }. And let
Then the type tp $(a/b)$ does not divide over , while tp $(a/b)$ weakly divides
over by the formula $\neg E(x, y)$ .
$T$$\models\neg E(a, b)$$y)$$\emptyset$$\emptyset$

I tried to divide witness formulas into some classes according to theirproperties ago. And I told about the next characterization at the RIMS
meeting last year.
Definition 3 Let $A\subset B$ and $p(x)\in S(B)$ .
$p(x)$ M-weakly divides overif there is a formulasuch that
Morley sequence $I=\{a_{i} : i<n+1\}$ ofwhile the type
is inconsistent.
$A$$\varphi(\overline{x})\in L_{n}(A)$and a$\models\varphi(a_{0}, a_{1}, \cdots\cdots, a_{n-1})$$p\lceil A$$\lceil p]^{\varphi}$

Theorem 4 Let be simple.
Then is stable if and only
$T$

$T$

$if\mathcal{M}$-weak dividing over models is symmetric.

2. Weak notion of p-dividing
In recent years another variant of dividing, “ thorn” -dividing has beencharacterized in rosy theory (see e.g. [4]). I tried to define weak notion ofp-dividing (thorn-dividing). We recall some definitions first.

Definition 5 Let $A\subset B$ and $p(x)\in S(B)$ .
$p(x)$ strongly divides over
if there is a formula $\varphi(x, b)\in p(x)$ such that$b\not\in acl(A)$ andis -inconsistent for some $k<\omega$ .
$p(x)p$ -divides over
if $p(x)$ strongly divides over $Ac$ for some parameter$A$$\{\varphi(x, b_{i}) :b_{i}\models tp(b/A)\}$$k$$A$$c$.

if there is a formula $\varphi(x, b)\in p(x)$ such that
implies a finite disjunction of formulas which p-divides over A.
Given a formula , a set of formulas in variables $x,$ , a set of formulasin variables $y,$ , and a number , we define $p(\varphi, \triangle, \Pi, k)(thorn$ -rank
inductively as follows:(1)
for limit ordinal is defined as usual.
(2) $p(\varphi, \Delta, \Pi, k)\geq\alpha+1$ if and only if there is a, some $\pi(y, z)\in\Pi$and parameters such that(a)for infinitely many $a\models\pi(y, c)$(b)is -inconsistent.
For a type , we define $p(p, \triangle, \Pi, k)=\min\{p(\varphi, \triangle, \Pi, k)|\varphi\in p\}$ .
A theory is rosy if for any type $p(x)$ , any finite sets of formulas and, and any finite
is finite.
$p(x)\mathfrak{p}$-forks over$A$$\varphi(x, b)$$\Delta$$y$$\varphi$

$\Pi$

$k$

$z$

$p(\varphi, \triangle, \Pi, k)\geq 0,$$\lambda$$)$$\lambda$$\infty,$$\delta\in\triangle$$c$$\mathfrak{p}(\varphi\wedge\delta(x, a), \triangle, \Pi, k)\geq\alpha$$\{\delta(x, a)\}_{a\models\pi(y,c)}$

$k$

$p$

$T$

$\Pi$

Remark 6
ioms.

$\Delta$$k,$$\mathfrak{p}(\varphi, \triangle, \Pi, k)$(1) In rosy theories, p-forking satisfies the independence ax-,

48
and(2) If
acl $(Cda)-$ acl $(Cd)$ .
then$\varphi(x, b)$$a\models\varphi(x, b)$

p-divides over$C$by the set$\{b_{i}\models\theta(y, d)\}$,$b\in$

Weak notions of p-dividing could be defined in many ways. By thedefinition, p-dividing implies dividing. So we expect that weak p-dividing
implies weak dividing.

.
Let$\psi(x, b)$ weakly-divides overA if there is a formula

Definition 7$b\not\in A$$p$$\varphi(\overline{x})=\exists y\wedge\theta(x_{i}, y)$$i<n$is consistent,such thatand a realization ofis inconsistent.
whileWe define weak p-dividing(p-forking) for types just like weak divid$\in L_{n}(A)$$[tp(a/A)]^{\varphi}$$\psi(x, b)$$a$$[\psi(x, b)]^{\varphi}$ing(forking).

We can check the next fact easily.

Fact 8 Let$T$

be rosy. Then -forking implies weak -forking.
$p$$\mathfrak{p}$3. Weak p-dividing and NIP theories
has the independence property if forA formula
every $n<\omega$ , there are sequences $a_{l}(l<n)$ such that for every $w\subset n$ ,Definition 9$\models($$\varphi(x, y)$ョ$x)[ \bigwedge_{l<n}\varphi(x, a_{l})^{if(l\in w)}]$

A theory$T$.

is NIP if no formula$\varphi(x, y)$

has the independence property.

Weak p-dividing is a kind of algebraic extension.
( $T$ is any theory. ) $A\subset B$ .
Then tp $(a/B)$ does not weakly -divide over
if and only if

Lemma 10$\mathfrak{p}$for
if$an\tau/n<\omega,$$A$$am/C$ and $am/extensionq(x, C, A)$$\bigcup_{i<n}q(x_{i}, C, A)$

is consistent, then $\bigcup_{i<n}q(x_{i}, Z, A)\cup\bigcup_{i<n}r(x_{i}, Y, A)$is consistent where tp $(a/B);=r(x, B, A)$ .

By the lemma above, we can prove the next fact.
Proposition 11 Let be NIP and unstable.
Then weak -dividing is not symmetric.
$T$$\mathfrak{p}$of tp $(a/A)$ over $AC$ ,49
References[1] S.Shelah, Simple unstable theories, Annals of Pure and Applied Logic 19(1980) 177-203[2] A.Dolich, Weak dividing, chain conditions, and simplicity, Archive forMathematical Logic 43 (2004) 265-283[3] B.Kim and N.Shi, A note on weak dividing, preprint[4] A.Onshuus, Properties and consequences of Thorn-independence, Journal of Symbolic Logic 71 (2006) 1-21
[5] E.Hrushovski and A.Pillay, On NIP and invariant measures, preprint[6] F.O.Wagner, Simple theories, Kluwer Academic Publishers (2000)