Re: Preprint: A simple description of Thompson's group F

[Peter Freyd <pjf@saul.cis.upenn.edu>]

Marcelo and Tom write

  We show that Thompson's group F is the symmetry group of the "generic
  idempotent".  That is, take the monoidal category freely generated by an
  object A and an isomorphism A \otimes A --> A; then F is the group of
  automorphisms of A.

Tom has pointed out to me that the review of the old Freyd/Heller I
posted give no hint of its relevance. Therefor this:

  F was defined (40 years ago) as the initial model for a group with
  an endomorphism that's conjugate to its square.

More formally: consider the equational theory that adds to the theory
of groups a constant, s, and a unary operator  e, subject to two
further equations:

    e(xy) = (ex)(ey)             "e is a endomorphism"
    s(ex) = (e(ex))s             "e is a conjugacy-idempotent"

The initial algebra for this theory is the group  F.

(If one insists on removing the type-error in the last sentence, then
try "the initial algebra for this theory when subjected to the
forgetful functor back to groups is  F.")

If one defines a sequence of elements   s_n = e^n(s)  they clearly
generate  F (as a group) and it isn't hard to see that a complete set
of relations for  F (as a group) is the doubly-infinite family

     s_a s_b = s_{b+1} s_a

one such equation for each  a < b. (It took me ten years to find a
proof that just two of these equations imply all the others.)