The logic \(\textsf{FO}(\texttt{LFP})\) can now be defined as the extension of first-order logic with the new relation symbols \(\texttt{LFP}_{\psi({R,\vec{x}})}\) for each formula \(\psi(R,\vec{x})\) in which the relation variable appears only positively and with new atomic formulas of the form \(\texttt{LFP}_{\psi({R,\vec{x}})}(\vec{t})\). Such formulas are interpreted as follows: \(\mathcal{A} \models \texttt{LFP}_{\psi({R,\vec{x}})}(\vec{t})\) if and only if \(\vec{t}^{\mathcal{A}} \in \text{Fix}^{\mathcal{A}}(\psi(R,\vec{x}))\). The logic \(\textsf{FO}(\texttt{TC})\)is similarly defined by adding a transitive closure operator \(\texttt{TC}_{\psi(\vec{x})}(\vec{x})\) which holds of a term \(\vec{t}\) just in case \(\vec{t}^{\mathcal{A}}\) is in the transitive closure of the relation denoted by \(\psi(\vec{x})\) in \(\mathcal{A}\).  The logic \(\textsf{SO}(\texttt{LFP})\) and \(\textsf{SO}(\texttt{TC})\) are defined analogously by adding these operators to \(\textsf{SO}\) and allowing them to apply to formulas containing second-order variables.