[Haskell-cafe] Some lambda identities

While playing with @pl on #haskell, I noticed some weird and surprising lambda identities. For example: let {c = (.); c4 = c c c c} Then we have: c c4 == c c4 c c c c Proof: Repeatedly apply the identity: (*) x (y z) == c x y z More stuff: c c2 == c4, c c3 == c7, but c cn does not appear to be reducible for n>3. c7 c3 == c3 c7 c c c You get a lot more interesting stuff when you throw flip into the mix. The identity (*) is actually a semi-associativity condition that makes the entire lambda calculus into a semi-monoid. Apparently with very interesting structure. Anyone know more about these things? Thanks, Yitz