
Date: Mon, 2 Jul 90 10:23:54 -0700
From: plotkin@src.dec.com (Gordon Plotkin)

For an example recent use of ordered algebras see the book by Matthew
Hennessy,1988, Algebraic Theory of Processes. As references he gives
Guessarian, 1981, Algebraic Semantics, LNCS Vol99 , the ADJ Paper, Initial
Algebra Semantics and Continuous Algebras, JACM, Vol24, no1, pp 102-124
and recommends Nivat and Reynolds,1885, Algebraic methods in Semantics; it has
useful papers and further references in it. I am sure there is lots, lots
more, hardly a new thought!

 Gordon Plotkin

Date:        Wed, 04 Jul 90 17:10:55 EDT
From:        Michael Barr <INHB@MUSICB.MCGILL.CA>

Dear Bob:

Here is some more on Vaughn's problem.  First off, I checked out the
reference to the Bloom and Meseguer papers and they both concerned
theories for which the operations were order-preserving (or even
omega-sup preserving) and this is neither what Vaughn was proposing nor
what I, in the last note, suggested.  Thus neither one would allow
boolean algebras to be models and neither is really relevant.

The other thing I checked out is Linton's paper on functorial semantics
in SLNM vol. 80.  This is the first paper in volume, with a title
something like, ``An outline of functorial semantics''.  His theory
really does implement my suggestion of not having the operations
preserve the order at all.  He doesn't look at the HSP question.  This
note will consist of a brief sketch of Linton's paper, followed by some
ideas of my own on the HSP problem.

Let B be a category.  By a B-based theory, I will mean a category T,
plus a functor F:B* --> T (B* is the opposite of B) that is an
isomorphism on objects.  Think of T as having all the arrows of B (Manes
would call them the crisp maps), together with others (that Manes calls
fuzzy maps).  This is not literally true because F needn't be faithful,
but it is a good way to think of this.  A model of this theory is a
functor M:T --> Sets with the property that MF is representable.  In
fact, it might be better to describe a model as an M, together with an
object b of B such that MFa == B(a,b).  (Use == for isomorphic).  The
reason the second way is better is that it avoids having to make
choices.

If you work out what this all means, it turns out that a model of the
theory is an object b of B together with a family of maps Mf:B(a1,b) -->
B(a2,b) correspnding to each morphism f:Fa_1 --> Fa_2 of T.  Recall that
every object of T is of the form Fa for a unique object a of B.  This is
subject to the conditions that if f=Fg:Fa1 --> Fa2, then Mf is the map
B(g,b) induced by g and that M(f.g)=Mf.Mg.

Where do these theories come from?  Well one way is as follows.  Let U:A
--> B be a functor.  Let T have objects Fa in one-one correspondence
with the objects of B.  Define T(a1,a2) = NT(B(a1,U-),B(a2,U-)).  This
might be a proper class and certain care has to be taken in that case,
but basically that is how such a category arises.  If U has a left
adjoint L, then we will get T(a1,a2)=B(ULa2,ULa1), so the question of
size cannot arise then.  In that case, T is simply the Kleisli category
of the triple.

Things start to get complicated (not to mention ugly) when you start to
look at HSP theorems.  If all you want to do is add equations, there is
no problem.  Here's why.

An equation takes the form f=g for f,g:n --> m in T.  Although quotient
categories are generally quite nasty, they aren't if you never identify
objects and we don't.  It is easy to see that if M' is a submodel of M
and M |- f=g, then so does M'.  Products are even easier and it follows
from standard category theory that we have an epi-reflective
subcategory.  For H, we take the set of epimorphisms that are split
epimorphisms as posets.  Assumng that alpha:M -->> M' is such an
epimorphism, represented by the split epimorhism b -->> b', and that M
satisfies the equation, the fact that M' does follows from the square

                 B(n,b) ---->> B(n,b')
                  !  !          !  !
                Mf!  !Mg     M'f!  !M'g
                  !  !          !  !
                  v  v          v  v
                 B(m,b) ---->> B(m,b')

especially the fact that the upper arrow is epic.  For the converse, the
crucial is that every object is the target of a poset-split epi from a
free algebra and hence if an HSP subcategory contains every free
algebra, it contains every algebra.  If it doesn't, look at the
reflections of those free algebras and there you will find the
equations.

Warning: what follows is ugly and inelegant.  Read on at your own risk.

Now really, it seems highly unlikely that you aren't going to want to
impose inequalities as well as equalities.  (Time out to complain
about ``inequation''.  If a neologism is genuinely more informative than
the old word, fine.  ``Inequality'' is criticized as not being
antonomous to ``equality''.  A reasonable criticism.  In what earthly
way is ``inequation'' an improvement?  Down with it!)  Now I have
dropped the order relation from the operations and now I will need to
reintroduce them for adding inequalities.  Yes, but that seems to be
what is required.  Anyone who can find an elegant way of doing this,
hats off.

Structure is given by a map T(n,m) --> Hom(B(n,b),B(m,b)), the latter
being the set of functions in sets.  B(n,b) and B(m,b) are taken to be
the discrete sets of maps so that the operations don't preserve order.
Now I am going to `remember' that b is ordered and that b(m,b) and in
turn Hom(B(n,b),B(m,b)) inherit order relations from that of b.  Then I
am going to suppose that T is a posets-category so that T(n,m) is
ordered and suppose that the above arrow preserves order.  This doesn't
make the operations order preserving, but rather that if f < g in the
order, then the interpretation of f is < the interpretation of g.  Not
that f and g individually preserve order.  You could transpose the above
map and find a different way of putting it, but it looks just as ugly.
If T arises from an underlying posets functor, you will see that T does
naturally become a posets-category.  Now we can talk about M |- f<g and
such like.  And we can look at the full subcategory of M that satisfy
that condition.  It is no trouble to show that it is closed under
products, strict subobjects and posets split epimorphisms and, while I
haven't actually checked the details, I believe the converse is true as
well.  The same argument with free algebras ought to work.  Of course,
now the free algebra on some poset might be abstractly isomorphic to its
previous value, but have a richer order relation.  That leads to a new
inequality.

For both equalities and inequalities, the antecedants will be Horn
clauses in the predicates of equalities and inequalites.  This
additional expressive power is what you get by using operations based on
arbitrary posets, rather then sets.

For the purposes of computer science, one probably ought to restrict to
the case of an aleph_0, or at most aleph_1 accessible theory.  This
basically means that need look only at finite (or finite and countable)
ary operations.

Regards, Mike

Date: 11-JUL-1990 16:14:54.31
From: PJOHNSON@Wesleyan.bitnet

Dear Mike,  Why the fuss about allowing operations that don't preserve
the order.  Surely no such operations could arise from algebras for
triples over POS.  When handed a category B that has a cartesian closed
structure, and (in your notation) a B-theory F, then it is a reasonable
condition on a functor M that the composite MF is representable, in the
sense of B.  To be more precise, M is a model if MF is an exponential
functor, and all that makes no reference to any ambient set theory.  Lacking
this definition, I don't see how the equivalence of theories and triples
could possibly hold (my doubts in the second line).  What am I missing here?
Sincerely, Paul

From:       Paul Taylor <pt@doc.ic.ac.uk>
Date:       Thu, 12 Jul 90 11:37:27 BST

A new version of my commutative diagrams package is now available.  Users
of the version released last September will know that it uses a matrix
syntax for the objects and morphisms, and is able to stretch horizontal
arrows to meet the objects at the endpoints. The new version is also able
to stretch verticals, and the diagonals have been re-worked.

Those who have Mike Barr's package will recognise the following diagram
from his manual, or from the errata to Toposes, TRiples and Theories
which he circulated on 25 April.  BITNET users suffering gross corruption
may like to know that "\rArr?\mu" has a caret (superscript) symbol in
the middle, whilst "\dArr^?\sigma TT'?" (following 4) has a tilde.
$$\CellSize=4em
\diagram
TT   & \rArr^\mu       & T \\
     & \SE_{TT\eta'}   & 1      & \SE^{T\eta}     \\
\dArr^{T\eta'T} & 2    & TTT' &\pile{
                                 \rArr^{\mu T'}\\
                                 \rArr_{T\sigma}} & TT'   \\
     &                 & \dArr^{T\eta'TT' }& 3    & \dArr^{T\eta'T'} &
                                                       \llap{6\quad}\SE^{id} \\
TT'T &\rArr^{TT'T\eta'}& TT'TT' &\rArr^{TT'\sigma}& TT'T' & \rArr^{T\mu'}&TT'\\
\dArr^{\sigma T} & 4   & \dArr~{\sigma TT'}& 5    & \dArr~{\sigma T'}& 7 &
                                                                \dArr_\sigma \\
T'T & \rArr_{T'T\eta'} & T'TT' & \rArr_{T'\sigma} & T'T'  & \rArr_{\mu'} & T'\\
\enddiagram$$

Paper copies of the manual for my package will be available at Como, and
I shall dispatch electronic copies of the package & manual just before
I depart for Italy myself (to allow for last minute bug-fixes while I
print my own papers!).  Known past users should get it automatically,
but to be sure, please email me, including the word "diagrams" in your
subject line.

Paul Taylor, Dept. of Computing, Imperial College, London SW7 2BZ, UK
email   pt@doc.ic.ac.uk   pt%ic.doc@ukacrl   or   ...!mcvax!ukc!icdoc!pt
phone +44 71 589 5111 x 5057 fax +44 71 581 8024

*************** PS LONDON PHONE NUMBERS HAVE CHANGED *******************

From:       Mr David Murphy <dvjm@cs.glasgow.ac.uk>
Date:       Wed, 18 Jul 90 16:32:25 BST


I am becoming interested in sheaves valued over sets with order; poset
valued sheaves and frame (or locale) valued sheaves in particular.
Is there much work on the properties of categories of these sheaves ?
Can I expect any kind of logic in these categories, or does the logical
structure only appear in the set valued case ? I'm sorry this question
is a little vague, but if I knew the question I really wanted to ask,
I'd understand my problem much better ...

Thanks.

David Murphy
dvjm%uk.ac.glasgow.cs@nsfnet-relay
dvjm%uk.ac.glasgow.cs@ukc.uucp
dvjm%uk.ac.glasgow.cs@UKACRL

Date:        Wed, 18 Jul 90 20:22:03 EDT
From:        Michael Barr <INHB@MUSICB.MCGILL.CA>

Dear Bob:

I now realize that I was overlooking something important in my previous
memos on the subject of theories based on categories other then sets.  I
am convinced that the questions raised are important, but they are not
as easy as I thought.  I also remain convinced that operations should
not be assumed to be morphisms in the underlying category.  Just as
operations in a theory are not normally morphisms of the theory.

What I forgot is that arrows have codomains as well as domains.  Thus a
morphism n to m in a category is not---except in a metaphorical sense---
simply an n-fold of m-ary operations.  So suppose 2 and 3 represent the
2 and 3 element chains, resp.  An arrow 2 --> 3 does not simply
represent a pair of 3-ary operations, but rather a pair of 3-ary
operations of which the first is less than or equal to the second.  This
means that such inequalities can be built directly into the theory.  It
also means that the simple-minded HSP theorem I stated before is wrong.
(It appears to be correct for the special kind of theories in which
every arrow n-->m factors through the set of components of n, a very
special case in which there are no non-trivial inequalities between
distinct operations.)

Now one the most interesting examples is going to be the theory whose
models is the set of omega-complete posets; that is posets in which
every increasing chain has a sup.  A lot can be learned by looking at
that example in some detail.

Begin with an operation theta:omega+1-->omega.  A model of such an
operation will assign to each increasing sequence x={x_0<x_1<...} a
sequence {theta_0(x)<theta_1(x)<...<theta_omega(x)}.  The first thing
that has to be done is to put in the equation theta_i(x)=x_i, which is
unproblematic and proceeds in the same way as in sets-based theories.  A
model for the resultant theory assigns to each increasing sequence an
upper bound for that sequence.  It needn't be the least upper bound,
even if the sequence is eventually constant or even constant (although
the latter could be added as an equation).  What is clearly needed is
to add the condition that if x_i<y for all i, the theta_omega(x)<y.

Now if we think of this in terms of subcategories, it is clear that the
full subcategory of all models that satisfies is HSP, meaning quite
precisely that it is closed under strong subobjects, arbitrary products
and surjective epimorphisms.  Not ALL epimorphisms, even regular ones;
here is an example.  Let B consist of countably pairs of elements
u_i<v_i, not otherwise comparable.  Let theta_omega be the sup
operation, in this case pretty trivial.  Let C be the quotient gotten by
identifying v_i=u_{i+1}.  C now has a plethora of countable increasing
sequences, all of which have to be provided with values of theta_omega.
Since this has to be done freely, the value of theta_omega on each
sequence (save only for the few that come from sequences in B) must be
different from those of all the others, even though all the proper
increasing sequence would have the same sup.  So this regular image
doesn't lie in the subcategory (which means the inclusion doesn't
preserve coequalizers, why should it?, and the category doesn't have
effective equivalence relations).

So provisionally at least, it seems that that ought to be taken as the
definition of HSP.  It is not hard to show that a full HSP subcategory
of a category of models that contains all the free algebras is the whole
thing.  This is clearly a necessary condition.  If such a subcategory
doesn't contain the free algebras, then any algebra it doesn't contain
gives rise to an equation.  For example, in the case at hand, the free
algebra generated by a countable chain is uncountable, since it contains
one element corresponding to each possible subchain (including ones that
are repetitious, even constant) and once the condition of being a sup is
imposed, then the value of theta_omega at each eventually constant
sequence is set equal to the element in question and the sups of all the
not-eventually-constant sequences are identified.

So far so good.  It looks like one could hope to say that corresponding
to an HSP category, you get a quotient theory by imposing either
equations or inequalities.  The problem is that this doesn't quite lead
to a quotient theory.  Here's why.  A theory is a category with a
contravariant functor from the base that is an isomorphism on objects.
This means that a quotient theory wouldn't identify any objects.  But a
quotient category that doesn't identify objects is more-or-less like a
quotient among semigroups; in particular it is surjective.  But I will
show now that HSP categories don't lead to surjectives at the object
level.

Let F--|U be the free and underlying functor between B and the category
E of models.  Let E_0 be an HSP subcategory and let I:E_0-->E be the
inclusion and L its left adjoint. Then LF--|UI is the adjoint pair
between E_0 and B.  Let K and K_0 be the Kleisli categories for the
triples.  These are dual to the theory categories.  The functor K-->K_0
is induced by the morphism of triples.  K(m,n)=B(m,UFn) and K_0(m,n)=
B(m,UILFn) and the induced map is surjective iff the map UFn-->UILFn
is split epi, which isn't in the example above.

I believe that the map from E-->ILE is surjective in this example, but
it is not always the case.  To see this consider the theory that has
just one 2-ary operation.  In other words it has an operation, call it
sigma that takes pairs x<y to sigma(x,y).  Let us assume the equation
sigma(x,x)=x.  Then the free algebra generated by x<y has just 3
elements x,y and sigma(x,y).  Now add the inequality y<sigma(x,y).  now
the free algebra generated by x<y is infinite.  It contains not only the
three elements above, but also sigma(y,sigma(x,y)) and so on.  On the
other hand, it does determine an HSP subcategory.

It is clear that an equational theory cannot be just a quotient theory
in a simple-minded sense; yet it does seem to be the insertion of
(in)equalities in the right places.  So the question that seems to be
open is to see in what sense the insertion of these inequalities
corresponds to equations.

For the converse, going from quotient theories to subcategories,
analogous questions arise. If we suppose we have a quotient theory, then
the algebras give a subcategory closed not only under strong subobjects,
but under all of them.  So here too we see that the traditional idea of
equational subcategory is not the right one.  So what is the right one?

Another question that is unrelated (as far as I can see) is what is a
sketch in this context?  When I said take an operation from omega+1 -->
omega, it was clear that I had in mind some idea of sketching theories,
not the theory itself.  Although Peter Johnstone described the use of
sketches as retrograde in a review of TTT (analogous to going back from
groups to generators and relations), the fact is that a sketch is
frequently finite and a the theory isn't.  In fact, many interesting
sketches are quite manageably finite and this is important in computing.
I think of the difference between sketches and theories as analogous to
that between lazy evaluation and (whatever is opposite to lazy, perhaps
beaverish).

Regards, Michael

P.S. All those who object to non-categorical material on the net, please
stop reading here.  This does not bear on category theory at all (except
in a very remote sense of having to do with software verification), but
I think it deserves as wide circulation as possible.  I got it from my
son who found it on a Microsoft BB.  I cannot vouch for its accuracy,
but it is consistent with whatever information has leaked about the HT
fiasco.
Subject: Hubble Telescope
Date: Thu Jul 05 10:47:24 1990

Here's something...(and how about that quote in the first
signature!). I don't know what axaf is.

- adam

|The originator of this message is Dimitri Mihalas.
|
|For those who want the condensed version, the software that
|Perkin-Elmer used to polish the mirror was faulty.
|
|Robert W. Spiker, UVa Dept. of Astronomy
|-------------------------------+ It is truly written that a man has five
|rws3n@astsun.astro.virginia.edu| times as many fingers as ears, but only
|  or @bessel.acc.virginia.edu  | twice as many ears as noses.
|
|
|Message follows:
|----------------------------------------------------------------------
|in case you have not heard: from a reliable inside source i found out
|that the problem with ST is that the SOFTWARE driving the polisher was
|defective. the corrections for spherical aberration were put in with the
|wrong sign. consequently the mirror is not corrected for sph. abb., but
|has an added dose of it.
|
|the error was not detected during testing because no test with collimated
|light was ever done. (editorial remark: unthinkable!) apparently this was
|a $30M economy measure in the face of the Challenger accident. likewise
|none of the optics were ever tested in vacuum. the primary was and is
|"perfect" relative to the specified curve; but alas the specification
|was wrong. sigh.
|
|>from my amateur astronomer days (does that include 1990?) i recall that
|spherical aberration is EASY to detect with the foucault test, which is
|done with a pinhole, not collimated light. it is hard to believe that
|ANYONE could have made such a blunder..
|
|the only reason that people know this much is that the same software
|was used for AXAF. the errors there were so huge as to be immediately
|noticeable, and when the software was corrected, the mirror was "perfect".
|i don't know whether the information from axaf was available prior to
|the launch of ST, but it seems that it had to be. in which case one
|wonders why PE didn't issue a "hold everything!".
|
|the future: no chance of bringing the whole telescope down for a refit.
|best plan is to design compensating optics into the lightpath for future
|instruments: relatively easy to do. but that will still take 3-5 years.
|
|i suppose it's "win a few, lose a few..." but i personally think that
|nasa, the government, and the people should stick it into PE and TURN
|it hard until they agree to refund the cost of the mistake and of the repairs.
|i'm sick of seeing defense and defense-related contractors get away
|with bloody murder and just get fatter and fatter on the profits.
|
|back to theory
|dimitri
|---------------------------------------------------------------------------

From:        INHB000 <INHB@MUSICB.MCGILL.CA>

Dear Bob:

This is, we all hope, the last communication on the subject of HSP
theorems.  I will probably be giving a talk on it in Montreal and will
try to have a paper ready for distribution then.  I will cast this in
terms of posets, although it really all goes through for an arbitary
locally presentable category and acessible theory.

Let E/M be a factorization system on Pos such that maps in E are epi and
in M are mono.  At first I thought there were only the two extremal such
factorizations, but now I think there are others.  In any case, let it
be chosen and fixed.  Suppose now that T is an accessible triple on Pos.
The Kleisli category K is then accessible, as is the free functor F:Pos
-->K.  It is folklore that the category of T algebras is equivalent to
the category that has as objects pairs (G,b) where b is a poset and M:K*
(that is the opposite of K) to Set such that GF=Hom(-,b). In fact, you
don't have to include b as part of the structure, but it is convenient.
Morphisms are natural transformations of functors (which induce unique
morphisms between the representing objects.

Let us say that (G,b) is an M/U-subobject of (G',b') if b-->b' is in M.
By an HSP subcategory of the category C of T-algebras, I will mean a
subcategory closed under U-split epis, M/U-subobjects and products.  One
way of getting an HSP subcategory is the following.  By a Horn, I mean a
diagram in K of the form
                    Ff
                 Fn----->Fm
                  \
                   \
                    \
                   g \
                      \
                       \
                        v
                         Fk

such that f is in E.  An algebra (M,b) satisfies this Horn if there is a
factorization as indicated:
                     MFf
                MFn<-----Fm
                  ^       ^
                   \      |
                    \     |
                  Mg \    |
                      \   |
                       \  |
                        \ |
                         Fk

The full subcategory of objects that satisfy a Horn or any class thereof
is an HSP subcategory of the category of algebras.  It would be nice if
the converese were true, but it isn't.  What happens is that you can
iterate this construction and get a new HSP subcategory.  In the right
kind of example, the itereated construction is not describable as the
objects that satisfy a Horn.  I will give some examples later.

The theorem is that every HSP subcategory is given as the intersection
of a possibly transfinite chain of HSP category each derived from the
preceding ones either as an intersection at limit ordinals or by
satisfying a class of Horns at non-limit ordinals.  And, of course,
conversely.

I still don't know how to characterize a single step HSP subcategory.
That is probably still interesting.  Nonetheless, this seems to be the
best one cne can do in general.  I also have a condition sufficient that
a sequence of Horn constructions is a single one.

A couple of examples.  Consider a theory with one 2-ary operation we
will call sigma such that sigma(x,x)=x.  The free algebra generated by
2, which we will represent as x<y has just three elements x,y and
sigma(x,y).  Now form the Horn
                    Ff
              F(1+1)---->F2
                  \
                   \
                    \
                   g \
                      \
                       \
                        v
                         F2
where f:1+1-->2 is the inclusion.  Here we are using the epi/regular
mono factorization system.  g takes the first generator of F(1+1)
(which, as it happens is 1+1) to y and the second one to sigma(x,y).  An
algebras this Horn iff it satisfies
             x<y ==> y<sigma(x,y)
In the full subcategory defined by this Horn, the free algebra generated
by 2 is now infinite, since it now includes elements like
sigma(x,sigma(x,y)) and sigma(y,sigma(x,y)).  We could now take a Horn
that forced these last two elements to be equal.  (The top of the Horn
is F(1+1)-->F1).  The solutions of this Horn are still an HSP
subcategory of the category of algebras.  But it cannot be given by one
step because the terms that were set equal weren't even elements in the
first free algebras.

Here is an example in which we use the regular epi/mono factorization
system.  Here we can use only equations, since the top of the Horns must
be coequalizers, which means we are putting in equations.  It might seem
that in this case, Horns can be composed, but isn't so.  Consider two
operations phi and psi of type 1-->2.  This means essentially that we
have operations phi_0<phi_1 and psi_0<psi_1.  Suppose in addition we
have a 3-ary operation tau such that tau(x,x,y)=x and tau(x,y,y)=y.
Then the free algebra generated by a single x has infinitely many
elements like phi_0(x)<phi_1(x) and psi_0(phi_1(x))<psi_1(phi_1(x)), but
no terms built from tau because there are no non-trivial 3 chains.  Now
add the equation phi_1=psi_0 and all of a suddent there are 3 chains
such as phi_0<phi_1=psi_0<psi_1.  Now we could add the equation
x=tau(phi_0(x),phi_1(x),psi_1(x)).  This equation cannot be stated in
the first algebra because the necessary elements weren't there.

Here is an interesting example.  Begin with the theory of one omega-ary
operation I will call lim.  So to each sequence x_0<x_1<... there is an
element lim x_i.  Add the Horn that embodies the inequalities x_j<lim
x_i for each j.  This is done by a Horn whose top is F(1+1)-->F2 as with
the case of a binary op above.  The next Horn looks like
                    Ff
              F(1+1)---->F2
                  \
                   \
                    \
                   g \
                      \
                       \
                        v
                      F(omega+1)
The map f is as before.  For g, take the map that takes the first
element to lim x_i and the other to x_{omega}.  The Horn forces the
condition lim x_i<x_{omega} and an algebra satisfies it, in addition to
the ones above iff lim x_i is the least upper bound.  The algebras are
the omega-CPOs.

There is nothing to prevent you from sticking to discrete theories, in
which all the ops have discrete domain and domain.  We can also make the
ops be increasing in some variables and decreasing by looking at Horns
of the form
                    Ff
              F(1+1)---->F2
                  \
                   \
                    \
                   g \
                      \
                       \
                        v
                      F(n2+m2)
Here by n2, I mean the sum of n copies of 2 and similarly with m2.  Thus
n2+m2 is the sum of n+m arrows, say x_i<y_i for i=1,...,n+m.  Given an
operation sigma on n+m discrete elements, let g take the first element
of 1+1 to sigma(x_1,...,x_n,y_{n+1},...,y_{n+m}) and the second element
to sigma(y_1,...,y_n,x_{n+1},...,x_{n+m}).  An algebra satisfies this
Horn iff it is increasing in the first n variables and decreasing in the
last m. Thus this analysis actually can be specialized to the question
as first posed.

Michael
