End extension

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

• $\mathfrak{𝔄}$ is a substructure of $\mathfrak{𝔅}$, and
• $b\in A$ whenever $a\in A$ and $bFa$ hold, i.e., no new elements are added by $\mathfrak{𝔅}$ to the elements of $A$.

The following is an equivalent definition of end extension: $\mathfrak{𝔄}$ is a substructure of $\mathfrak{𝔅}$, and $\{b\in A:bEa\}=\{b\in B:bFa\}$ for all $a\in A$.

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