 All the relationships already mentioned are established between structures of the same signature. We can as well define reductions and expansions between structures whose signatures are not the same. Given a many-sorted structure \(\mathcal{A}\), a reduct \(\mathcal{A}^{\prime }\) of \(\mathcal{A}\) is obtained by simply removing some sorts, function or relation symbols from the signature of \(\mathcal{A}\). If \(\mathcal{A}^{\prime }\) is a reduct of \(\mathcal{A}\), then we say that \(\mathcal{A}\) is an expansion of \(\mathcal{A}^{\prime}\).