坪井, 明人

桔梗, 宏孝

Kyoto University Research Information Repository

TitlePAPAをもたない理論 (モデル理論と代数幾何の交流)Author(s)桔梗, 宏孝; 坪井, 明人Citation数理解析研究所講究録 (2003), 1344: 11-15Issue Date2003-10URL http://hdl.handle.net/2433/43512RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityPAPA(Hirotaka Kikyo)Dept. of Math. Sci., Tokai Universitykikyo@ss.u-tokai.ac.jp(Akito Tsuboi)Dept. of Math., The Tsukuba Universitytsuboi@math.tsukuba.ac.jpEhud HrushovskiThe Hebrew University at Jerusalemehud@math.huji.ac.il1$\mathcal{L}$ $T$ , $\sigma$ , $T$ $\mathrm{r}_{\sigma}$ $\mathcal{L}$$T_{\sigma}$ . $T$ PAPA$T_{\sigma}$ [2]. , $T$ $\mathrm{R}\mathrm{U}$ A(la Propri\’et\’ed’Amalgamation Pour les Automorphismae)[4] , $T_{\sigma}$. . $T$ 3 $(M0,\sigma 0),$ $(M_{1},\sigma_{1})$ ,$(M_{2},\sigma_{2})$ , $M_{1}$ $M_{2}$ $M_{0}$ . $\sigma_{1}$ $\sigma_{2}$ $\sigma_{0}$, $T$ $M_{3}$ $\sigma_{3}$ , $(M_{1}, \sigma_{1})$$(M_{2}, \sigma_{2})$ $(M_{3},\sigma_{3})$ , $M\mathit{0}$.$T$ $T_{\sigma}$ ,. $T$ strict order property $T_{\sigma}$[3].1344 2003 11-1511$T\hslash^{\mathrm{i}}$ PAPA $\epsilon\not\in)^{\prime\supset \mathrm{g}-}1^{\backslash }\overline{\mathcal{D}}arrow k\ ,$ $T\hslash^{\theta}\mathrm{l}$ strict order property $k\mathrm{t}’\supset k\mathrm{V}^{\backslash }\overline{2}-\sim k\mathrm{t}\subset 1\mathrm{Z}$. , PAPA.$T$ , $T_{\sigma}$$T$ . ,,.Hrushovski $T=\mathrm{A}\mathrm{C}\mathrm{F}\mathrm{A}$ $T_{\sigma}$( ). , .ACFA generic . generic, 1 . ACFAPAPA . , ACFA PAPA.PAPA . , PAPA. PAPA Ziegler, PAPA .PAPA ,$T$ $T_{\sigma}$ .PAPA , .$T$ $T_{\sigma}$ .2PAPA$\mathcal{L}=\{R(x, y), f(x)\}$ , 1 2 1 1 .$T$ 5 .1. $f(x)\neq x$ $f^{2}(x)=x$ ($f$ involution).2. $\neg R(x,x)$ .3. $R(x,y)$ $R(y,x)$ . ( 2, 3 $R$ )4. 4 $a,$ $a’,$ $b,$ $b’$ , $f(a)=a’$ $f(b)=b’$ , 4 $R$2 , $R(a, b)\wedge R(a,b’)$ $R(a’,b)\wedge R(a’,b’)$$R(a, b)\wedge R(a’, b)$ $R(a, b’)\wedge R(a’, b’)$ 1 .$|^{}.$ , $\neg R(a, a’)$ $\neg R(b, b’)$ .125. $x_{1},$ $\ldots,$ $x_{m},$ $y_{1},$ $\ldots,$ $y_{n}$ I $f$ , $z$, $R(z,x:)\wedge R(z$, f(x ) $1\leq i\leq m$ [ , $R(z,y_{i})\wedge R(f(z)$ ,y$1\leq i\leq n$ [ .2.1 $T$ . .$T$ , .badc-and-forth method( ) .. (1) (4) .$T_{0}$ . $T_{0}$ , $\mathrm{H}\mathrm{P}$ (Hereditary Property), JEP(Joint Embdding Property), AP (Amalgamation property) . ,. onffiy locally finite , . Hodges ModelTheoIy [1] 7.1.4 , $T_{0}$ “generic ”( $\mathrm{R}\dot{\mathrm{a}}\mathrm{i}\mathrm{s}\mathrm{s}\acute{\mathrm{e}}$ limit) $M_{0}$ , $\mathrm{T}\mathrm{h}(\mathrm{M}_{0})$ . $\mathrm{T}\mathrm{h}(M_{0})$$T_{0}$ model completion . (5) $M_{0}$ , $T$T=Th(Af\rightarrow ., $f$ $T$ . $T$ , $SU$ 1.$f$ $f$ $E$(interdefinable ) . , f() .2.2 $T$ PAPA . , PAPA.$T$ $\mathcal{M}$ . $M\prec \mathcal{M}$ ., $M$ . , $M$., $f$ involution , $M=A\cup B,$ $A\cap B=\emptyset,$ $B=\{f(a) : a\in A\}$. $e\in \mathcal{M}$ :$a\in A$$R(e,a)\wedge R(f(e),a)\wedge\neg R(e,f(a))\wedge\neg R(f(e), f(a))$ . (1), $e$ $f(e)$ $M$ . , $N\supset$$M\cup\{e, f(e)\}$ $T$ $N$ $N$ $\mathcal{L}$ $\sigma$ , $\sigma(e)=f(e),$ $\sigma|M=\mathrm{i}\mathrm{d}_{M}$13. , $\sigma(f(e))=f(\sigma(e))=f(f(e))=e$ ., $e’\in \mathcal{M}$ , $b\in B$ ,$R(e’,b)\wedge R(f(e’),\overline{b})\wedge\neg R(e’,f(b))\wedge\neg R(f(e’),f(b))$ (2). $e’$ , $N’\supset M\cup\{e’, f(e’)\}$ $T$ $N’$ $N’$$\mathcal{L}$ ’ , $o^{J}(e’)=f(e’),$ $\sigma’(f(e’))=e’,$ $d|M=\mathrm{i}\mathrm{d}_{M}$ .$(N,\sigma)$ $(N’,d)$ $(M, \mathrm{i}\mathrm{d}_{M})$ ., $(N^{*},\sigma^{*})$ $T_{\sigma}$ . $N,$ $N’$ $N^{*}$$e,$ $e’$ , $e,$ $e’$ .(1), (2) , $f(e)\neq e’$ $e\neq e’$ . $T$ 4 ,$R(e,e’)\wedge R(e, f(e’))\wedge\neg R(f(e),e’)\wedge\neg R(f(e), f(e’))$. , $\sigma^{*}(e)=f(e)$ $\sigma^{*}(e’)=f(e’)$ $\sigma^{*}$ ,$R(e,e’)$ , $R(f(e), f(e’))$ . , .2.3 $T_{\sigma}$ .$T_{\sigma}$ TA .$T$ 4 5 , $R(x,y)\wedge R(x, f(y))$ order property. , $T$ $M$ $a_{\dot{*}}(i\in \mathbb{Z})$ ,$i<j\Leftrightarrow R(a:, a_{j})$ $R(a:, f(aj))$. Ramsey , $M$ , $a:(i\in \mathbb{Z})$ (indis-cernible sequence) .$M$ , $M$ $\mathcal{L}$ $\sigma$ , $a_{*}$. $=\sigma^{:}(a\mathrm{o})(i\in \mathbb{Z})$. , $(M,\sigma)$ TA , $(M,\sigma)$TA . TA, $(M,\sigma)$ $\aleph_{1}$ .$a=$ , $:<j$ ,$i<j\Leftrightarrow R(\sigma^{:}(a),\sigma^{j}(a))$ $R(\sigma^{:}(a), f(\sigma^{j}(a)))$.$\Psi(y)=\{R(\sigma^{:}(a),y)\wedge R(\sigma^{:}(a), f(y)):i\in \mathbb{Z}\}$ .14$(M,\sigma)$ ,$\Psi\cdot\langle y)\vdash\exists x[R(a,x)\wedge R(a, f(x))\wedge R(x,y)\wedge R(x, f(y))\wedge\sigma(x)=x]$.$b$ $\Psi(y)$ $(M,\sigma)$ . , $T$ 4 . $n\in \mathbb{Z}$[ $\sigma^{n}(a)\neq b,$ $f(b)$ .$M$ $\mathcal{L}$ $p(x)$ .$\bullet$ $R(\sigma^{n}(a),x)$ $R(\sigma^{n}(a), f(x))$ ( $n$ [ )$\bullet$ $R(x, c)$ $R(x, f(c))(c\in M$ , $n$ , $\sigma^{n}(a)\neq c,$ $f(c)$)$f$ $(\{x, f(x)\})$ ., $p$ $M$ $\mathcal{L}$ , $\sigma(p)=p$ . ,$x$ $p$ $(M,\sigma)$ $T_{\sigma}$ . $(M,\sigma)$TA (generic ) , $x$ $M$ .$(M, \sigma)$ , $\psi(y)$. $\psi(y)$ $\sigma^{n}(a)$ $n$ $k$ . , $\sigma^{k}(a)$$\psi(y)$ $(M,\sigma)$ . , $(M,\sigma)$$R(a,c)\wedge R(a, f(c))\wedge R(c,\sigma^{k}(a))\wedge R(c, f(\sigma^{k}(a)))\wedge\sigma(c)=c$$c\in M$ . $\sigma$ $\mathcal{L}$ , $(M,\sigma)$$R(\sigma^{k}(a),c)\wedge R(\sigma^{k}(a), f(c))$. $R(c, f(\sigma^{k}(a)))$ , $T$ 4 .[1] W. Hodges, Model Theory, Cambridge University Press, 1993.[2] H. Kikyo, Model companions of theories with an automorphism, J. Symbolic Logic65 (2000), No. 3, 1215-1222.[3] H. Kikyo, S. Shelah, The strict order property and generic automorphisms, J.Symbolic Logic 67 (2002), No. 1, 214-216.[4] D. Lascar, Autour de la propri\’et\’e du petit indice, Proc. London Math. Soc. 62(1991) 25-53.15

PAPAをもたない理論 (モデル理論と代数幾何の交流)

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PAPAをもたない理論 (モデル理論と代数幾何の交流)

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