Arrows progress continued

[Brian Rice]

Hello again,
This is a continuation of my previous post.

Incidentally, the citation of _Lattices and Order_ should be:

Introduction to Lattices and Order. B.A. Davey and H.A. Priestley. 
(C) Cambridge University Press, 1990.

which I highly recommend for an interesting viewpoint on orders, 
algebras (especially boolean algebra), symmetry, and other abstract 
concepts in formal terms. I've had this book for a while.

To continue from before...

Before I finish, I should return to a notion I touched on at the 
beginning of this essay. The notion of pairings of arrows is 
interesting, and even more interesting in light of the discussion of 
frames. For example, suppose I consider for any frame a graph that 
represents the set of all pairs of arrows over that frame. That graph 
would have to represent the *values* of all the arrows! This can be 
seen because every arrow is a kind of pairing, though not necessarily 
unique in value (e.g. (1,2) should mean different things in different 
contexts without the context having to consider the identity.).

To which I add:

Of course every arrow obviously has some notion of value, even those 
incompletely-specified arrows; so this value graph represents a 
reflection on the content of the particular frame. This notion is not 
really satisfactory as is, since viewing the system at the meta-level 
allows one to see that the relationship between the reflective aspect 
and the base-level is not clearly uniquely specified. In other words, 
there could be multiple notions of value applied to the same domain.
One way around this is to force one per domain and allow for multiple 
value-interpretations by designing a graph which implements an 
equivalency relation over the domain's arrows: basically an explicit 
mapping from every arrow to some element of a graph which effectively 
behaves as the set of possible values.

--------------------

I have more to say of course. The following is almost a random note, 
but it specifically answers an aspect of the arrow notion as a whole.

There is an interesting relationship between information as arrows 
and as bits. Consider that arrows encode two ordered selections; one 
could consider them to be similar to Lisp CONS cells (list cells). As 
such, one can map its accessors CAR and CDR to 0 and 1 respectively 
as bits. Coupling the idea of concatenation of accessors to the 
sequential receipt of information bits, one obtains an encoding:
1101101 -> cdr cdr car cdr cdr car cdr

Which means that results like Chaitin's algorithmic information 
postulates relate to arrow in a direct way.

More to come...
~