End Extension

In model theory and set theory, which are disciplines within mathematics, a model $\mathfrak=\langle B, F\rangle$ of some axiom system of set theory $T$ in the language of set theory is an end extension of $\mathfrak=\langle A, E\rangle$, in symbols $\mathfrak\subseteq_\text\mathfrak$, if
# $\mathfrak$ is a substructure of $\mathfrak$, (i.e., $A \subseteq B$ and $E = F, _A$), and
# $b\in A$ whenever $a\in A$ and $bFa$ hold, i.e., no new elements are added by $\mathfrak$ to the elements of $A$.

The second condition can be equivalently written as $\=\$ for all $a\in A$.

For example, $\langle B, \in\rangle$ is an end extension of $\langle A, \in\rangle$ if $A$ and $B$ are transitive sets, and $A\subseteq B$.

A related concept is that of a top extension (also known as rank extension), where a model $\mathfrak=\langle B, F\rangle$ is a top extension of a model $\mathfrak=\langle A, E\rangle$ if $\mathfrak\subseteq_\text\mathfrak$ and for all $a \in A$ and $b \in B\setminus A$, we have $rank(b) > rank(a)$, where $rank(\cdot)$ denotes the rank of a set.

Mathematical logic
Model theory
Set theory

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