Some examples of non-existent combinators (Q1314364)

This note discusses some interesting properties that combinators may or may not have. First, the author mentions an open problem: is there a fixed point combinator \(Y\) such that \(Y= Y(SI)\)? Next, he proves that no fixed point combinator can be defined in terms of the combinators \(B\) and \(WI\). He defines hypercreative and hypereasy combinators and shows none exist and also shows that there is no universal generator \(M\) for which there exists an \(N\) such that for all combinators \(P\), \(M\twoheadrightarrow NP\).