End extension

In model theory and set theory, which are disciplines within mathematics, a model B = ⟨ B , F ⟩ of some axiom system of set theory T in the language of set theory is an end extension of A = ⟨ A , E ⟩ , in symbols A ⊆ end B , if

The following is an equivalent definition of end extension: A is a substructure of B , and { b ∈ A : b E a } = { b ∈ B : b F a } for all a ∈ A .

For example, ⟨ B , ∈ ⟩ is an end extension of ⟨ A , ∈ ⟩ if A and B are transitive sets, and A ⊆ B .