Extension (model theory)

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In model theory, a model $\mathfrak{B}$ is an extension of a model $\mathfrak{A}$ (and $\mathfrak{A}$ is a submodel of $\mathfrak{B}$ ) iff the universe A of $\mathfrak{A}$ is a subset of the universe B of $\mathfrak{B}$ , $\mathfrak{A}$ and $\mathfrak{B}$ share the same language $\mathcal{L}$ , and the interpretations in $\mathfrak{A}$ of the nonlogical symbols of $\mathcal{L}$ are the restrictions to A of their interpretations in $\mathfrak{B}$ .