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If the domain of $B$ is the same as $A$, but you only forget the interpretation of the extra language elements, then $B$ is called a reduct of $A$ to signature $\Sigma'$. But you don't merely have a reduct, since you are taking a substructure in the smaller language. Thus, what you have is that $B$ is a substructure of the reduct of $A$ to $\Sigma'$. Having needed this concept in a recent article, I used the term reduct substructure in exactly this situation, but I haven't seen this terminology elsewhere and I don't think there is an established terminology.