elementary embedding
Let be a signature and and be two structures for such that is an embedding. Then is said to be elementary if for every first-order formula , we have
In the expression above, means: if we write where the free variables of are all in , then holds in for any (the underlying universe of ).
If is a substructure of such that the inclusion homomorphism is an elementary embedding, then we say that is an elementary substructure of , or that is an elementary extension of .
Remark. A chain of -structures is called an elementary chain if is an elementary substructure of for each . It can be shown (Tarski and Vaught) that
is a -structure that is an elementary extension of for every .
Title | elementary embedding |
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Canonical name | ElementaryEmbedding |
Date of creation | 2013-03-22 13:00:29 |
Last modified on | 2013-03-22 13:00:29 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 5 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 03C99 |
Synonym | elementary monomorphism |
Defines | elementary substructure |
Defines | elementary extension |
Defines | elementary chain |