On (weak) fpc generators

Corrado B\"ohm once observed that if $Y$ is any fixed point combinator (fpc), then $Y(\lambda yx.x(yx))$ is again fpc. He thus discovered the first "fpc generating scheme" -- a generic way to build new fpcs from old. Continuing this idea, define an \emph{fpc generator} to be any sequence of terms $G_1,\dots,G_n$ such that $$Y \text{ is fpc } \Longrightarrow YG_1\cdots G_n \text{ is fpc}$$ In this contribution, we take first steps in studying the structure of (weak) fpc generators. We isolate several classes of such generators, and examine elementary properties like injectivity and constancy. We provide sufficient conditions for existence of fixed points of a given generator $(G_1,..,G_n)$: an fpc $Y$ such that $Y = YG_1\cdots G_n$. We conjecture that weak constancy is a necessary condition for existence of such (higher-order) fixed points. This generalizes Statman's conjecture on the non-existence of ``double fpcs'': fixed points of the generator $(G) = (\lambda yx.x(yx))$ discovered by B\"ohm.