Various notions of "temperament extension"

There are several various related notions of "temperament extension" that
are possible to define on the set of all subgroup temperaments.
Furthermore, for every concept of "extension," there is a dual concept of
"restriction." Each of these lead to slightly different mathematical
structures.

Of the major ones that make any sort of sense, the following
characteristics are universal, and in a sense the "weakest" sensible
criteria shared by all (assume A and B are arbitrary subgroup temperaments):

If A is a "restriction of" B, meaning that B is an "extension of" A, then
1) Subgroup inclusion: dom(A) <= dom(B).
2) Kernel inclusion: ker(A) <= ker(B).
3) Weak rank preservation: rank(A) <= rank(B).

Nobody could ever possibly come up with a notion of anything we call a
subgroup "restriction" or "extension" that doesn't support at least the
preceding three properties. However, different notions of extension can
also have additional properties. Some examples, some of which imply one
another:

4) Strong rank preservation: rank(A) = rank(B). Under this criterion, 2.3.7
49/48 64/63 isn't a "restriction" of 7-limit blacksmith, as the rank has
gone down. Blacksmith simply has no restriction to the 2.3.7 subgroup (or,
if you prefer a pathological supercontorted restriction with a single
unmapped generator).

5) Generator preservation: in plain English, the generators need to not
split in half or anything when you extend A to B. In mathematical terms, as
a multilinear function, the restriction of B to A's subgroup must equal A
exactly (note that this doesn't imply #4). Under this criterion, 2.3.5
meantone isn't a "restriction" of 2.3.5.7.11 mohajira. Mohajira simply has
no restriction to the 2.3.5 subgroup (or, if you prefer, a contorted
restriction with sqrt(3/2) as a generator).

6) "Strict" kernel inclusion: ker(A) < ker(B). Under this criterion, 2.9.5
meantone isn't a restriction" of 2.3.5 meantone.

It's important, when considering what's best above, to also discuss the
implications on our notion of "extension" as well.

For instance, say that you don't buy into #4, because for instance, you
think the restriction of Blacksmith to the 2.3.7 subgroup really should be
the rank-1 2.3.7 49/48 & 64/63 temperament. This, in turn, implies that
Blacksmith is an extension of that temperament. Even more interestingly,
this would imply that 2.3.5.7 81/80 is an extension of 2.3.5 meantone, for
instance. This is a lot different from what we usually consider a
temperament "extension," but it's interesting.

Another example: say you don't want to buy into #6, because you want 2.9.5
81/80 to be defined as the restriction of 2.3.5 81/80 to the 2.9.5
subgroup. That, then, implies that 2.3.5 81/80 is an extension of 2.9.5
81/80. Also very different, but also interesting.

This leads to a lot of different variations on the theme of temperament
extension. However, it seems that there are three such variations that
stick out as being more common than the rest:

1) KEENAN'S EXTENSION: supports all 6 properties. Ranks and generators have
to be preserved, and this implies strict kernel extensions. This is the
logical outcome of Keenan's rules on subgroup restrictions, flipped around.

2) GENE'S EXTENSION: supports the first four properties. Rank has to be
preserved, but generators notably do NOT have to be preserved: mohajira is
an extension of meantone. This is taken from Gene's rules on "temperament
families," which are defined on the wiki and on Carl's mirror of his site,
but just generalized straightforwardly to any chain of subgroups
totally-ordered by subgroup inclusion, not just full-limits.

3) MIKE'S EXTENSION: supports properties 1-3 and 5. Like Keenan's
extension, but where rank doesn't have to be preserved; the restriction of
2.3.5.7 81/80 to the 2.3.5 subgroup is 2.3.5 81/80, and hence, the former
rank-3 temperament is just an extension of the latter.

This brings us back to our notion of subgroup temperament families, which
is going to quickly turn into a study in posets. When defining the partial
order, which sort of extension is the better way to go? I'll discuss this
more in the next post, but I believe that for the specific aim of
organizing subgroup temperaments into families, Keenan's will turn out to
be the most useful, for the same reason that his subgroup restriction rules
are also useful. I do think that both my and Gene's notions of extension
can also be quite useful, of course, in other situations.

Mike