| 翻訳と辞書 |
| Existentially closed model ： ウィキペディア英語版 |
Existentially closed model
In model theory, a branch of mathematical logic, the notion of an existentially closed model (or existentially complete model) of a theory generalizes the notions of algebraically closed fields (for the theory of fields), real closed fields (for the theory of ordered fields), existentially closed groups (for the class of groups), and dense linear orders without endpoints (for the class of linear orders).
==Definition==
A substructure ''M'' of a structure ''N'' is said to be existentially closed in (or existentially complete in) if for every quantifier-free formula φ(''x'',''y''1,…,''y''n) and all elements ''b''1,…,''b''n of ''M'' such that φ(''x'',''b''1,…,''b''n) is realized in ''N'', then φ(''x'',''b''1,…,''b''n) is also realized in ''M''. In other words: If there is an element ''a'' in ''N'' such that φ(''a'',''b''1,…,''b''n) holds in ''N'', then such an element also exists in ''M''. This notion is often denoted .
A model ''M'' of a theory ''T'' is called existentially closed in ''T'' if it is existentially closed in every superstructure ''N'' which is itself a model of ''T''. More generally, a structure ''M'' is called existentially closed in a class ''K'' of structures (in which it is contained as a member) if ''M'' is existentially closed in every superstructure ''N'' which is itself a member of ''K''.
The existential closure in ''K'' of a member ''M'' of ''K'', when it exists, is, up to isomorphism, the least existentially closed superstructure of ''M''. More precisely, it is any extensionally closed superstructure ''M''∗ of ''M'' such that for every existentially closed superstructure ''N'' of ''M'', ''M''∗ is isomorphic to a substructure of ''N'' via an isomorphism that is the identity on ''M''.
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「Existentially closed model」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.
|
|