桔梗, 宏孝

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Title不安定構造のGeneric自己同型写像 (モデル理論とその応用)Author(s)桔梗, 宏孝Citation数理解析研究所講究録 (2001), 1213: 19-27Issue Date2001-06URL http://hdl.handle.net/2433/41159RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityGeneric(Hirotaka Kikyo)Dept. of Math. Sci., Tokai Universitykikyo@ss.u-tokai.ac.jp1$\mathcal{L}$ $T$ ( ) $T_{\sigma}=T\cup$ { $\sigma$ $\mathcal{L}$ }. $T_{\sigma}$ $\mathcal{L}_{\sigma}=\mathcal{L}\cup\{\sigma\}$ . $T$ $M$ $\sigma\in \mathrm{A}\mathrm{u}\mathrm{t}_{\mathcal{L}}(M)$, $\sigma$ ]. generic $(M,\sigma)$ $T_{\sigma}$ (an existentiallyclosed model) .$D$ $U$ , (1) $D$ $U$ , (2) $D$( $D$ ), $D$ $U$ $D$. , ., $T_{\sigma}$ $T$. $T_{\sigma}$, $\mathcal{L}_{\sigma}$, TA . $T_{\sigma}$.$T$ [1], [5], [9].generic Lascar[8] . $T$$ACF$ Chatzid is Hrushovski [2] ,. Chatzidakis Pillay $T$TA [3], TA .Kudaibergenov , TA $T$. , $T$ TA$T$ $\mathrm{f}\mathrm{c}\mathrm{p}$ ., TA ,. TA. Pillay , Kudaibergenov, $\mathrm{f}\mathrm{c}\mathrm{p}$ . , Kudaibergenov$T$ , $T$ TA1213 2001 19-2719-,, TA .,, $T$ independence propertyTA [4]. $T$ independence property$T$ strict order property , Shelah strict order propertyTA [6].independence property ,TA , $T_{\sigma}$(amalgamation property) , $T$ TA[4].TA Hrushovski ,ACFA Psf TA ( )., $\cdot$ . $\mathrm{A}\mathrm{C}\mathrm{F}\mathrm{A}_{\sigma}$, Psf, .$\mathrm{T}$, $a,$ $b,$ $c$ , $x,$ $y$. $a$ $A$ , $a\in A$ $a$ $A$. $\varphi(x,a)$ $M$ , $\varphi(M, a)$$\varphi(x,a)$ $M$ . . $\sigma:Marrow M$ , $p$ $M$. $p$ ( ) $\sigma$$\sigma(p)$ .2generic..2.1 $M$ , $k,$ $M^{k}$ $r(x,y),$ $k$$\in M^{k}(i\in \mathrm{N})$ . $i<j\Leftrightarrow r(a:,a_{j})$ , $M$. $T$ , $T$ .2.2 $T=\mathrm{T}\mathrm{h}(\mathbb{Q}, <)$ , TA .TA . $\sigma_{0}(x)=x+1$ $\sigma_{0}$ $(\mathbb{Q}, <)$. $(\mathbb{Q}, <,\sigma_{0})$ TA $(N,\cdot<,\sigma)$ ., $(N, <,\sigma)$ . ,$p(x)$ .$p(x)=\{n<x:n=0,1,2, \ldots\}$20$\vee \mathrm{t}o k,$ $(N, <, \sigma)|_{\llcorner}^{\vee}k.\mathrm{V}^{\backslash }\vee C*\sigma\supset 2’\supset l^{\mathrm{S}}ffi\mathfrak{h}f\vee\supset$.(1) $p(x)\vdash\exists z(0<z<x\wedge\sigma(z)=z)$ ,(2) $q(x)$ $p(x)$ $q(x)f-\exists z(0<z<x\wedge\sigma(z)=z)$ .(2) . $q(x)$ $p(x)$ , $m$ $q(x)$. , $q(x)\vdash\exists z(0<z<x\wedge\sigma(z)=z)$ , $b\in N$$(0<b<m\wedge\sigma(b)=b)$ $(N, <, \sigma)$ . , $\sigma$ $\sigma_{0}$, $\sigma(0)=1,$ $\sigma^{2}(0)=2,$ $\ldots,$ $\sigma^{m}(0)=m$ . , $b$ $\sigma$, $\sigma^{m}(b)=b$ . $\sigma$ $<$ , $0<b$$m=\sigma^{m}(0)<\sigma^{m}(b)=b<m$ , .(1) . $(N, <, \sigma)$ ,.$p(x)$ $N$ $c$ . 0, 1, 2, ... $c$ $(N, <, \sigma)$$(M, <, \sigma|M)$ . $(M, <)$ 0, 1, 2, .. .$r(z)$ ($r(z)=\{a<z:a\in M, \exists i\in \mathrm{N}(a<i)\}\cup\{z<d$ :$d\in M,$ $\forall i\in \mathrm{N}(i<d)\})$ . , Th$(\mathbb{Q},$ $<)$ la $\mathrm{Q}\mathrm{E}$ $r(z)$ $M$, $0<z<c$ $r(z)$ [ , $\sigma(r)=r$ .$(N, <)$ , $(N, <)$ $(M’, <)$$\tau$ : $r$ $b’$ $\tau(\mathcal{U})=b’$ $M’$ ,$\tau|M=\sigma|M$ . $(M’, <, \tau)$ $(0<b’<c\wedge\tau(b’)=b’)$, $(M, <, \sigma|M)\subset(M’, <, \tau)$ $(M, <, \sigma|M)$ $T_{\sigma}$ ,$(0<z<c\wedge\sigma(z)=z)$ $(M, <, \sigma|M)$ . (1).$T$ independence property () TA . 0, 1, 2, ...[ (indiscernible s $\mathrm{u}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}$) $a_{\dot{1}}$ $(i=0,1,2, \ldots)$ $a_{0}<a_{1}<a_{2}<\ldots$. . , $a$:. $\sigma_{0}(a_{\dot{l}})=a:+1$ $T$ , TA. $r$ $T$ $a:(i=0,1,2, \ldots)$$M$ (average type) . independence propertyH $M$ .23 $(G, R)$ , $T=\mathrm{T}\mathrm{h}(\mathrm{G}, R)$ , TA.TA . $(G, R)$ $\sigma_{0}$ $a\in G$ $a$$\sigma_{0}$ ( ). ,$i\neq j$ $\sigma_{0}^{i}(a)\neq\sigma_{0}^{j}(a)$ . $(G, R,\sigma_{0})$ TA J $(N, R’,\sigma)$21. , $R’$ $R$ . $(N, R, \sigma)$. , $p\Leftarrow$ ) .$p(x)=\{x\neq\sigma^{m}(a) : m\in \mathbb{Z}\}$$p(x)$ $a$ $\sigma$ $x$ . , $(N, R, \sigma)$2 .(1) $p(x)\vdash\exists z(R(a, z)\wedge\neg R(x,z)\wedge\sigma(z)=z)$ ,(2) $q(x)$ $p(x)$ $q(x)r\exists z(R(a, z)\wedge\neg R(x, z)\wedge\sigma(z)=z)$ .(2) . $q(x)$ $p(x)$ , $m$ ,$\sigma^{m}(a.)$ $q(x)$ . (2) , $\sigma$ $z$, $R(a, z)$ $\urcorner R(\sigma^{m}(a), z)$ . $z$ $\sigma$ , $\sigma$, $R(a,z)$ $R(\sigma^{m}(a),z)$ , .(1) . , $(N, R, \sigma)$ ,.$p(x)$ $N$ $c$ . , $a$ $c$ $(N, R,\sigma)$$(M, R|M,\sigma|M)$ . $a$ $\sigma$ $A$ , $M$ $r(z)$$r(z)=\{R(d,z) : d\in A\}\cup\{\neg R(d’,z) : d’\in M^{\cdot}\backslash A\}$$(N, R)$ , $(N, R)$ $(M’, R)$ $\tau$ $b’\in$$M’$ , $\nu$ $r(z)$ $\tau$ , $\tau|M=\sigma|M$ .$(M’, R,\tau)$ $(R(a,b’)\wedge\neg R(c, b’)\wedge\tau(b’)=\nu)$ , $(M, R, \sigma|M)\subset$$(M’, R, \tau)$ $(M, R,\sigma|.M)$ $T_{\sigma}$ , $(R(a, z)\wedge\urcorner R(c, z)\wedge\sigma(z)=$$z)$ $(M, R, \sigma|M)$ . (1) .3.3.1 $T$ . $\varphi(x,y)$ $n$ , $T$$M$ $b\in M$ , $\varphi(x, b)$ $M$ $n$, $T$.3.2 (Kudaibergenov) $T$ TA $T$.22, $T$ . TA ,. , $\varphi(x,y)$, $T$ $M_{0}$ $b_{:}\in M_{0}(i\in \mathrm{N})$ , $i\in \mathrm{N}$$\varphi(x$ , b $M_{0}$ $i$ .$(M_{0},id)$ ($id$ ) TA $(M, \sigma)$ . $T$$M_{0}$ $\prec M$ . , $i\in \mathrm{N}$ , $|\varphi(M_{0}, b_{1}.)|$ ,$|\varphi(M, b_{\dot{l}})|=|\varphi(M_{0}, b:)|$ , $\varphi(M, b:)=\varphi(M_{0}$ , b .$\sigma$ $\varphi(M, b:)$ ., $\mathrm{N}$ $\mathcal{U}$ , $(M, \sigma)$ $\mathcal{U}$ $(\tilde{M},\tilde{\sigma})$. $\tilde{b}=\langle b: : i\in \mathrm{N}\rangle/\mathcal{U}$ , $(\tilde{M},\tilde{\sigma})$ $\tilde{\sigma}$ $\varphi(x,\tilde{b})$., $\varphi(x,\tilde{b})$ $\tilde{M}$ . , $\varphi(x,\tilde{b})$ $\tilde{M}$$p(x)$ . $c\models p(x)$ $d\neq c$ $d\models\tilde{\sigma}(p)$. $\tilde{M}$ $N\models T$ $N$ $\tau$ , $\tau$ } $\tilde{M}=\tilde{\sigma}$ $\tau(c)=d\neq c$. $(\tilde{M},\tilde{\sigma})$ $T_{\sigma}$ , $\varphi(x, \tilde{b})$ , $\tilde{\sigma}$. .33 $M$ strict order property , $M$ $a:(i\in \mathrm{N})$$\varphi(x,y)$ ,$i<j$ $\Leftrightarrow$ $\varphi(M,a:)\subset\varphi$ ($M$, a ). $\varphi(M, x)\subset\varphiarrow(M,y)$ $x<y$ , $x<y$ $T$( ) , . $T$strict order property , $T$ strict order property.34(with Shelah) $T$ $T$ strict order propertyTA .$M_{0}$ strict order property $T$ . , $T$ $\mathcal{L}$. , $k$ $M_{0}^{k}$ $k$$\{a: : i\in \mathrm{N}\}\subset M_{0}^{k}$ $i<j$ $a:<a_{j}$ ., $M_{0}$ , $\langle a_{-} : i<\omega\rangle$ $M_{0}$ $\mathcal{L}$. $M_{0}$ , $\sigma_{0}(a:)=a_{\dot{\iota}+1}$ $M_{0}$ $\sigma_{0}$. , $(M_{0}, \sigma_{0})$ $T_{\sigma}$ ., TA . $(M_{0}, \sigma_{0})$ TA $(N, \sigma)$ ffi.. $T$ , $M_{0}\prec N$ . $(N, \sigma)$. , $(N,\sigma)$ .$p(x)=\{a:<x:i<\omega\}$ $\psi(x)\equiv\exists y(a_{0}.<.\sigma(y)<y<x)$$(N, \sigma)$ .23(1) $p(x)\vdash\psi(x)$ ,(2) $p(x)$ $q(x)$ , $q(x)\mu\psi(x)$ .$(N, \sigma)$ .(2) . $p(x)$ $q(x)$ [ , $q(x)\subset\{a:<x : i<n^{*}\}$$n^{*}$ . . $q(x)$ . $q(x)\vdash\psi(x)$ , $a_{n}*$ $\psi(x)$. $b\in N$ $\sigma(b)<b<a_{n}*$ . $a_{0}<\sigma(b)$ ,$a_{n}$. =\sigma 1( ) $<\sigma^{n^{*}+1}(b)$ . $\sigma(b)<b<a_{\mathrm{n}}$. ,$\sigma^{n+1}.(b)<\sigma^{n}.(b)<\cdots<\sigma(b)<b<a_{n}\cdot$ .’ $a_{n}\cdot<a_{\mathrm{n}}*$ ., (1) . $c\in p(N)$ . $a_{0},$ $c\in M,$ $|M|=|T|$ ,$(N, \sigma)$ $(M,\sigma|M)$ .$d\in p(N)$ , $d’\in p(N)$ $d’<d$ $d’$ $N$$\mathcal{L}(M)$ $\Psi(d)$ ., $d_{1},$ $d_{2}\in p(N)$ $d_{2}<d_{1}$ $\Psi(d_{2})\subseteq\Psi(d_{1})$ , ,, $d_{1},$ $d_{2}\in p(N)$ [ , $d_{3}<d_{1}$ $d_{3}<d_{2}$ $d_{3}\in p(N)$.$\Psi=\bigcap_{a\epsilon p(N)}\Psi(d)$ .$\Psi$ $\mathcal{L}(M)$ $\{\varphi:(x) : i<|M|\}$ . $\Psi$ ,$:<|M|$ , $\varphi:(x)\not\in\Psi(dt)$ $d_{t}\in p(N)$ . $N$, $c$. $\in p(N)$ , $i<|M|$ c’ $<d_{\mathrm{t}}$ .$\Psi(c^{*})\subseteq\Psi(d_{t})$ , $\varphi:(x)$ $\Psi(c^{*})$ . , $\Psi(c^{*})\subseteq\Psi$ ,$\Psi(c.)$ $=\Psi$ .$d\in p(N)$ $d$ $\Psi(d)=\Psi(c^{*})$ . , $c$. $p(N)$ $M$ $\sigma$ , $\Psi(c^{*})$ $\sigma$ . ,$\mathcal{L}$ $\varphi(x,y)$ $a\in M$ , $\varphi(x,a)\in\Psi(c^{*})$ $\varphi(x, \sigma(a))\in\Psi(c^{*})$., $b_{1}\in p(N)$ $b_{1}<$ , $q_{1}(x)=\mathrm{t}\mathrm{p}_{\mathcal{L}}(b_{1}/M)$ ., $q_{1}(x)\subseteq\Psi(c^{*})$ . $\Psi(c.)$ $\sigma$ , $\sigma(q_{1}(x))\subseteq\Psi(c^{*})$. $\Psi(c^{*})$ $=\Psi(b_{1})$ , $\sigma(q_{1}(x))\subseteq\Psi(b_{1})$ . $\Psi(b_{1})$. $b_{2}\in p(N)$ $b_{2}<b_{1}$ $h$ $\sigma(q_{1}(x))$.$\sigma(q_{1}(x))$ $M$ $\mathcal{L}$ , $Mb_{1}b_{2}\subset M’,$ $\tau(b_{1})=b_{2}$ $\tau|M=$$\sigma|M$ $N$ $\mathcal{L}$ $M’$ $M’$ $\mathcal{L}$ $\tau$ ., .$(M’,\tau)\models a_{0}<\tau(b_{1})<b_{1}<c$ .24$(M, \sigma|M)$ TA , $T_{\sigma}$ . $T$, $<$ $\mathcal{L}$, $a_{0}<\sigma(y)<y<c$ $(M, \sigma|M)$ ., $(M, \sigma|M)\models\psi(c)$ . (1) , .strict order property , TA . $T_{\sigma}$$T$ TA .35 $U$ (amalgamation property) , $U$$M_{0}$ , $M_{1},$ $M_{2}$ , $M_{0}\subset M_{1}$ $M_{0}\subset M_{2}$ , $M_{1}\subset M_{3}$ $U$$M_{3}$ , $M_{2}$ $M_{0}$ $M_{3}$ .$T_{\sigma}$ Lascar PAPA . , PAPA.3.6 $T$ , $T_{\sigma}$ TA.$T$ $\mathcal{L}$ , $Ts$ , $\mathcal{L}_{S}$. , . $T$, $T$ $D$ $\mathcal{L}$ $D^{k}$ ( $k$ ) 2$r(x, y)$ $a:\in D^{k}(i\in \mathrm{N})$ $i<j\Leftrightarrow r(a_{\dot{l}}, a_{j})$ . $r(x,y)$. $D$ $T_{S}$. $D$ . ,, $\langle a_{\dot{\iota}} : i\in \mathbb{Z}\rangle^{\wedge}\langle b_{1}. : i\in \mathbb{Z}\rangle$ $D$ $\mathcal{L}_{S}$. $M_{0}$ , $M_{0}$ $Ts$ . ,$\sigma_{0}(a_{i})=a_{i+1},$ $\sigma_{0}(b:)=b_{i+1}$ , $\sigma_{0}$ $M_{0}$ $\mathcal{L}s$ [. $(M_{0}, \sigma_{0})$ $\mathcal{L}(\sigma)$ $T_{\sigma}$ ., TA . . , TA$(M_{1}, \sigma_{1})$ $(M_{0}, \sigma_{0})$ . $M_{1}$. , $M_{1}$ $M_{0}$$P$ , $\mathcal{L}_{S}(P,\sigma)$ $(M_{1}, P, \sigma_{1})$ . , $P(M_{1})=M_{0}$ ,$P$ . , $\sigma_{1}$ $P(M_{1})$ $Ls$.$(M_{1}, P, \sigma_{1})$ , .$p(x)$ $\psi\equiv\exists z(r(a, z)\wedge r(z,x)\wedge\sigma(z)=z)$.$p(x)=\{P(x)\}\cup$ { $\langle a_{\dot{\iota}}$ : $i\in \mathbb{Z}\rangle^{\wedge}\langle\sigma^{:}(x)$ : $i\in \mathbb{Z}\rangle$ $\mathcal{L}_{S}$ }, $(M_{1}, P,\sigma_{1})$ 2 .25(1) $p(x)\vdash\psi(x)$ ,(2) $p(x)$ $q(x)$ , $q(x)r\psi(x)$ .2 , $(M_{1}, P, \sigma_{1})$ , TA.(2) , $q(x)$ $p(x)$ $m$ $a_{m}$ $q(x)$[ . $q(x)\vdash\psi(x)$ $r(a_{m}, z)$ $r(z, a_{m})$, $r$ .(1) . $M_{1}$ $p(x)$ $c$ . , $c\in P(M_{1})$$\langle$$:i\in \mathbb{Z}\rangle^{\wedge}\langle\sigma \mathrm{i}(c):i\in \mathbb{Z}\rangle$ $\mathcal{L}s$ . $N_{0}$$N_{0}$ $T_{S}$ . Nl $\langle$ $1\in \mathrm{N}\rangle$ $\mathcal{L}$ $s(z)$, $\langle a: : i\in \mathrm{N}\rangle$ $s(z)$ $\mathcal{L}$ ., , r( , $z$) $r(z, c)$ $s(z)$ , $\sigma_{1}(s)=s$ ., $(N_{0}, \sigma_{1}|N_{0})$ $(N_{2},\tau)$ , $N_{2}\models T,$ $\tau$ $N_{2}$ $\mathcal{L}$, $s$ $\nu\in N_{2}$ $\tau(y)=b’$ . , $(N_{2},\tau)$ $\psi(c)$. $T_{\sigma}$ , $(M_{1}, \sigma_{1})$ $(M_{3}, \sigma_{3})$ , $(N_{2}, \tau)$$(N_{0},\sigma_{1}|N_{0})$ . $T$ , $r$$T$ , $\psi(c)$ $(M_{3}, \sigma_{3})$ ,$(M_{1}, \sigma_{1})$ $T_{\sigma}$ , $(M_{1},\sigma_{1})$ $\psi(c)$.4TA . , TA., TA . ACFA Psf, Hrushovski TA , ,.[1] Z. Chatzidakis, Generic automorphisms of separably closed fields, to appearin the Illinois J. of Math.[2] Z. Chatzidakis and E. Hrushovski, The model theory of difference fields,Thansactions of AMS, 351 (1999), 2997-3071.[3] Z. Chatzidakis and A. Pillay, Generic structures and simple theories, Annalsof Pure and Applied Logic 95 (1998) 71-92.26[4] H. Kikyo, Model companions of theories with an automorphism, J. Symb.Logic 65 (2000) $1215\ovalbox{\tt\small REJECT} 222$.[5] H. Kikyo and A. Pillay, The definable multiplicity property and generic automorphism, to appear in Annals of Pure and Applied Logic.[6] H. Kikyo and S. Shelah, The strict order property and generic automorphism,to appear in J. Symb. Logic.[7] D. Lascar, Autour de la pr0pri\’et6 du petit indice, Proc. London Math. Soc.62 (1991) 25-53.[8] D. Lascar, Les beaux automorphismes, Arch. Math. Logic 31 (1991) 5568.[9] A. Macintyre, Generic automorphisms of fields, Annals of Pure and AppliedLogic 88 (1997), 165-180.[10] S. Shelah, Classffication theory and the number of non-isomorphic models,North Holland, Amsterdam, 1978.27

不安定構造のGeneric自己同型写像 (モデル理論とその応用)

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不安定構造のGeneric自己同型写像 (モデル理論とその応用)

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