Re: [zzdev] Subspaces & ZZ versioning

[Antti-Juhani Kaijanaho <gaia@xxxxxx>]

On 20001225T182954+0100, Benjamin Fallenstein wrote:
> Oops, just realized you've defined "subspace" to mean something else...
> funny enough, the only kind of "subspace selector" currently working is
> what your definition would call the subspace generated by a single cell
> with respect to some given dimensions. Anyway, what I meant is, as I
> said, a set of cells and a set of connections between these cells.

Actually, I now think we need to revisit those definitions.
I can see at least two good definitions for a subspace:

   1) Let Z be a ZigZag space.  Z' is a subspace of Z iff
      * it is a ZigZag space
      * its set of cells is a subset of Z's set of cells
      * its set of connections is a subset of Z's set of connections

   2) Let Z be a ZigZag space.  Z' is a subspace of Z iff
      * it is a subspace of Z (by definition 1) and
      * if d is a dimension in Z' and there is a connection along d
        in Z', then for all cells c and c' in Z', if there is a
	connection from c to c' along d in Z, then there is a connection
	from c to c' along d.

1 is your def.  2 is an extension of my earlier def
(instead of a set of dims defining the subspace, we have two disjoint
set of dims, one for defining the closure, and another for including
more connections; I've been calling these hard and soft dimensions in
my sketches).

Def 1 is nice also theoretically since it mirrors many other subspace
definitions in mathematics.  Def 2 seems more natural: you cannot arbitrarily
remove connections from a rank where the cells are in the space.

-- 
%%% Antti-Juhani Kaijanaho % gaia@xxxxxx % http://www.iki.fi/gaia/ %%%