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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク abstract elementary class ： ウィキペディア英語版 abstract elementary class In model theory, a discipline within mathematical logic, an abstract elementary class, or AEC for short, is a class of models with a partial order similar to the relation of an elementary substructure of an elementary class in first-order model theory. They were introduced by Saharon Shelah.〔.〕 == Definition == $\backslash langle\; K,\; \backslash prec\_K\backslash rangle$, for $K$ a class of structures in some language $L\; =\; L(K)$, is an AEC if it has the following properties: * $\backslash prec\_K$ is a partial order on $K$. * If $M\backslash prec\_K\; N$ then $M$ is a substructure of $N$. * Isomorphisms: $K$ is closed under isomorphisms, and if $M,N,M\text{'},N\text{'}\backslash in\; K,$ $f\backslash colon\; M\backslash simeq\; M\text{'},$ $g\backslash colon\; N\backslash simeq\; N\text{'},$ $f\backslash subseteq\; g,$ and $M\backslash prec\_K\; N,$ then $M\text{'}\backslash prec\_K\; N\text{'}.$ * Coherence: If $M\_1\backslash prec\_K\; M\_3,$ $M\_2\backslash prec\_K\; M\_3,$ and $M\_1\backslash subseteq\; M\_2,$ then $M\_1\backslash prec\_K\; M\_2.$ * Tarski–Vaught chain axioms: If $\backslash gamma$ is an ordinal and $\backslash \backslash subseteq\; K$ is a chain (i.e. ), then: * * $\backslash bigcup\_\; M\_\backslash alpha\backslash in\; K$ * * If $M\_\backslash alpha\backslash prec\_K\; N$, for all , then $\backslash bigcup\_\; M\_\backslash alpha\backslash prec\_K\; N$ * Löwenheim–Skolem axiom: There exists a cardinal $\backslash mu\; \backslash ge\; |L(K)|\; +\; \backslash aleph\_0$, such that if $A$ is a subset of the universe of $M$, then there is $N$ in $K$ whose universe contains $A$ such that $\backslash |N\backslash |\backslash leq\; |A|+\backslash mu$ and $N\backslash prec\_K\; M$. We let $\backslash operatorname(K)$ denote the least such $\backslash mu$ and call it the Löwenheim–Skolem number of $K$. Note that we usually do not care about the models of size less than the Löwenheim–Skolem number and often assume that there are none (we will adopt this convention in this article). This is justified since we can always remove all such models from an AEC without influencing its structure above the Löwenheim–Skolem number. A $K$-embedding is a map $f:\; M\; \backslash rightarrow\; N$ for $M,\; N\; \backslash in\; K$ such that $f()\; \backslash prec\_K\; N$ and $f$ is an isomorphism from $M$ onto $f()$. If $K$ is clear from context, we omit it. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「abstract elementary class」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.