End extension

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

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

The following is an equivalent definition of end extension: [math]\displaystyle{ \mathfrak{A} }[/math] is a substructure of [math]\displaystyle{ \mathfrak{B} }[/math], and [math]\displaystyle{ \{b\in A : b E a\}=\{b\in B : b F a\} }[/math] for all [math]\displaystyle{ a\in A }[/math].

For example, [math]\displaystyle{ \langle B, \in\rangle }[/math] is an end extension of [math]\displaystyle{ \langle A, \in\rangle }[/math] if [math]\displaystyle{ A }[/math] and [math]\displaystyle{ B }[/math] are transitive sets, and [math]\displaystyle{ A\subseteq B }[/math].