elementary embedding

Let $\tau$ be a signature and $\mathcal{A}$ and $\mathcal{B}$ be two structures for $\tau$ such that $f:\mathcal{A}\to\mathcal{B}$ is an embedding. Then $f$ is said to be elementary if for every first-order formula $\phi\in F(\tau)$, we have

 $\mathcal{A}\vDash\phi\quad\mbox{iff}\quad\mathcal{B}\vDash\phi.$

In the expression above, $\mathcal{A}\vDash\phi$ means: if we write $\phi=\phi(x_{1},\ldots,x_{n})$ where the free variables of $\phi$ are all in $\{x_{1},\ldots,x_{n}\}$, then $\phi(a_{1},\ldots,a_{n})$ holds in $\mathcal{A}$ for any $a_{i}\in\mathcal{A}$ (the underlying universe of $\mathcal{A}$).

If $\mathcal{A}$ is a substructure of $\mathcal{B}$ such that the inclusion homomorphism is an elementary embedding, then we say that $\mathcal{A}$ is an elementary substructure of $\mathcal{B}$, or that $\mathcal{B}$ is an elementary extension of $\mathcal{A}$.

Remark. A chain $\mathcal{A}_{1}\subseteq\mathcal{A}_{2}\subseteq\cdots\subseteq\mathcal{A}_{n}\subseteq\cdots$ of $\tau$-structures is called an elementary chain if $\mathcal{A}_{i}$ is an elementary substructure of $\mathcal{A}_{i+1}$ for each $i=1,2,\ldots$. It can be shown (Tarski and Vaught) that

 $\bigcup_{i<\omega}\mathcal{A}_{i}$

is a $\tau$-structure that is an elementary extension of $\mathcal{A}_{i}$ for every $i$.

Title elementary embedding ElementaryEmbedding 2013-03-22 13:00:29 2013-03-22 13:00:29 CWoo (3771) CWoo (3771) 5 CWoo (3771) Definition msc 03C99 elementary monomorphism elementary substructure elementary extension elementary chain