End extension

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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

  • \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 \mathfrak{A}.

The following is an equivalent definition of end extension: \mathfrak{A} is a substructure of \mathfrak{B}, and \{b\in A : b E a\}=\{b\in B : b F a\} 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.