End extension
In model theory and set theory, which are disciplines within mathematics, a model B = ⟨ B , F ⟩ {\displaystyle {\mathfrak {B}}=\langle B,F\rangle } of some axiom system of set theory T {\displaystyle T} in the language of set theory is an end extension of A = ⟨ A , E ⟩ {\displaystyle {\mathfrak {A}}=\langle A,E\rangle } , in symbols A ⊆ end B {\displaystyle {\mathfrak {A}}\subseteq _{\text{end}}{\mathfrak {B}}} , if A {\displaystyle {\mathfrak {A}}} is a substructure of B {\displaystyle {\mathfrak {B}}} , (i.e., A ⊆ B {\displaystyle A\subseteq B} and E = F | A {\displaystyle E=F|_{A}} ), and b ∈ A {\displaystyle b\in A} whenever a ∈ A {\displaystyle a\in A} and b F a {\displaystyle bFa} hold, i.e., no new elements are added by B {\displaystyle {\mathfrak {B}}} to the elements of A {\displaystyle A} . The second condition can be equivalently written as { b ∈ A : b E a } = { b ∈ B : b F a } {\displaystyle \{b\in A:bEa\}=\{b\in B:bFa\}} for all a ∈ A {\displaystyle a\in A} .