[r-t] What's the meaning of a method having aparticularfalsecourse head

[Richard Smith]

Don Morrison wrote:

> On Apr 25, 2005, at 6:00 AM, Richard Smith wrote:
>
> > They don't form a group, but the set of all combinations of
> > falseness "groups" form a monoid -- a mathematical structure
> > similar to a group, but without the requirement for elements
> > to have inverses.  The A falseness "group" forms the
> > identity.  (This isn't a particularly insightful statement,
> > but it does justify using the term "identity" to describe A
> > falseness.)
>
> I'm confused by this. I wasn't so much worried about inverses as
> closure. What's the monoid operation you're assuming? I don't think the
> usual cross product works, does it? For a random example, consider FCH
> group T, which contains, among others, FCHs 24536 and 43526. Permuting
> 24536 by 24536 results in 25346, which is in T. But permuting 43526 by
> 24536 results in 45236, which is in F. So the cross product T x T can't
> be any one of the FCH groups, can it? Is there some other operation of
> one FCH group on another you are considering that is closed?

That's why I said "combinations of falseness 'groups'".  It
is these that are closed.  So T x T == (T union F) (and
probably some others thrown in for good measure).  As I said
before, this isn't a particularly insightful observation.
However it does provide a mechanism for asserting that A is
the identity as A x G = G for all G, whether G is a single
falseness group or a combination of them.  As there's no
idea of an inverse, multiplying by more and more falseness
"groups" eventually leads to the union of all falseness
"groups", and  at that point, you can multiply by anything
with no result.  This is why it is a monoid rather than a
group.

Richard