For an algebra \({\mathcal A} = (A,F)\), the operations \(f\in F\) can be naturally extended for a power set \({\mathcal P} ({\mathcal A})\) and the resulting algebra \({\mathcal P} ({\mathcal A}) = ({\mathcal P}(A), F^{+})\) is called a power algebra. For a binary relation \(R\) on \(A\), \(a/R = \{b\in A; (b,a) \in R\}\) and \(\epsilon(R)\) is defined by \((a,b) \in \epsilon(R)\) iff \(a/R=b/R\); \(R\) is called good if \(\epsilon(R)\) is a congruence on \(\mathcal A\). Moreover, the authors define binary relations \(R^{\rightarrow}\) and \(R^{\leftarrow}\) on \({\mathcal P}(A)\) as follows:
\(XR^{\rightarrow}Y\) iff \((\forall x\in X) (\exists y\in Y)(x,y)\in R\),
\(XR^{\leftarrow}Y\) iff \((\forall y\in Y) (\exists x\in X)(x,y)\in R\).
A relation \(R\) is Hoare good if \(R^{\rightarrow}\) is good on \({\mathcal P}(A)\) and \(R\) is Smyth good if \(R^{\leftarrow}\) is good on \({\mathcal P}({\mathcal A})\). The authors describe relationships between the mentioned relations. As a consequence, it is proved that every structure-preserving relation is very good.