End extension

In model theory and set theory, which are disciplines within mathematics, a model ${\mathfrak {B}}=\langle B,F\rangle$ of some axiom system of set theory $T$ in the language of set theory is an end extension of ${\mathfrak {A}}=\langle A,E\rangle$, in symbols ${\mathfrak {A}}\subseteq _{\text{end}}{\mathfrak {B}}$, if

1. ${\mathfrak {A}}$ is a substructure of ${\mathfrak {B}}$, (i.e., $A\subseteq B$ and $E=F|_{A}$), and
2. $b\in A$ whenever $a\in A$ and $bFa$ hold, i.e., no new elements are added by ${\mathfrak {B}}$ to the elements of $A$.

The second condition can be equivalently written as $\{b\in A:bEa\}=\{b\in B:bFa\}$ 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 ${\mathfrak {B}}=\langle B,F\rangle$ is a top extension of a model ${\mathfrak {A}}=\langle A,E\rangle$ if ${\mathfrak {A}}\subseteq _{\text{end}}{\mathfrak {B}}$ 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.