There is a second, related problem where lack of effectiveness comes in. In our discussion of denotation we considered a set \(E\) of equations \eqref{equationsE} and a functional \([E]\) associated with them. Recall that \([E]\) mapped \(k\)-tuples of partial relations to \(k\)-tuples of partial relations. We noted that \([E]\) would be monotone, and by very general results such functionals always have least fixed points. There is more than one way of showing this. One well-known argument has a decidedly algorithmic flavor to it. It goes as follows. Start with the smallest \(k\)-tuple of partial relations—this is the one where every partial relation is always undefined. Call this \(T_0\). Apply the functional \([E]\) to \(T_0\), getting \(T_1\). Apply the functional \([E]\) to \(T_1\) getting \(T_2\), and so on. It is easy to show that \(T_0\subseteq T_1\subseteq T_2\subseteq\ldots\). We have that \(T_0\subseteq T_1\) because \(T_0\) is in the \(\subseteq\) relation to every \(k\)-tuple. By monotonicity we then have \([E](T_0)\subseteq [E](T_1)\), but this says \(T_1 \subseteq T_2\). And so on. Continue with this increasing sequence and eventually the least fixed point of \([E]\) will be reached.