Extension (model theory)

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In model theory, given two structures $\mathfrak{A}$ and $\mathfrak{B}$ in a language $\mathcal{L}$ , we say that $\mathfrak{B}$ is an extension of $\mathfrak{A}$ (sometimes notated $\mathfrak{A}\subset\mathfrak{B}$ ) if

1. the universe A of $\mathfrak{A}$ is a subset of the universe B of $\mathfrak{B}$ , and

2. the interpretations in $\mathfrak{A}$ of the nonlogical symbols of $\mathcal{L}$ are the restrictions to A of their interpretations in $\mathfrak{B}$ .

We say $\mathfrak{A}$ is a substructure of $\mathfrak{B}$ if and only if $\mathfrak{B}$ is an extension of $\mathfrak{A}$ .

The structure $\mathfrak{B}$ is an extension of $\mathfrak{A}$ precisely when the inclusion map from $\mathfrak{A}$ into $\mathfrak{B}$ is an embedding of $\mathcal{L}$ -structures.

An injective homomorphism (a monomorphism) is also sometimes called an extension.

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Extension (model theory)

From Wikipedia, the free encyclopedia

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