In order to characterize additional classes, other means of extending the expressive capacity of first-order logic must be considered such as adding least fixed point or transitive closure operators. Consider, for instance, a formula \(\psi(R,\vec{x})\) in which the \(n\)-ary relation \(R\) only appears positively (i.e. in the scope of an even number of negations) and \(\vec{x}\) is of length \(m\). If \(A\) is the domain of a structure \(\mathcal{A}\), then such a formula will induce a monotone mapping of type \(\Phi_{\psi(R,\vec{x})}\) from the power set of \(A^n\) to the power set of \(A^m\) defined by \(\Phi_{\psi(R,\vec{x})}(B) = \{\vec{a} \in A^m \mid \mathcal{A}^R_B \models \psi(R,\vec{a}) \}\) where \(\mathcal{A}^R_B\) denotes the model in which the symbol \(R\) is interpreted as \(B \subseteq A^n\) and is just like \(\mathcal{A}\) otherwise. In such a case it follows that there will exist a least fixed point for the mapping \(\Phi_{\psi(R,\vec{x})}\) – i.e. a set \(F \subseteq A^n\) such that \(\Phi_{\psi(R,\vec{x})}(F) = F\) and which is contained in all other sets with this property (see, e.g., Moschovakis 1974). Let us denote this set by \(\text{Fix}^{\mathcal{A}}(\psi(R,\vec{x}))\).[46]