Re: Request for more details

Am 11.11.20 um 13:58 schrieb Ben Franksen:
> I should have mentioned perhaps that one of the usual patch laws is that
> commutation is indeed a function i.e. there is at most one solution to
> AB=B'A'.

More formally correct, for given patches AB, the equation

  AB=xy

has at most one non-trivial solution (the trivial one is of course x=A,
y=B).

More generally, for a sequence (A_i) of N patches we have the law of
permutivity which states that

 (A_i)=(X_i)

has at most N! (faculty) solutions, including the trivial one. Note that
N!=|S(N)|, where S(N) is the symmetric group of N elements. The maximum
is attained if all adjacent pairs in the sequence commute. Indeed, the
solutions form a subgroup of S(N), namely the one generated by the
transpositions corresponding to the succeeding pair-commutations.

Cheers
Ben