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Fixpoint of a Function

If it can be shown that $ (M\hspace{0.25em}N) =_\beta N$ is true, then $ N$ is called either a `fixed point' or a `fixpoint' of $ M$.

According to H. Curry, there exists a function, that generates such a fixed point of $ M$. This function is the so-called Y-combinator introduced above.

\fbox{
\parbox{12.5cm}{
{\sc Definition 6 (fixpoint of a function)}
\begin{center}$(M N) \underset{\beta}{\rightarrow}N$
\end{center} }
}
According to the above definition a fixpoint of a function can be seen as a lambda expression that, if passed as argument to this function is returned by the function.

domain access counter Georg P. Loczewski 2004-03-05