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January 1994
- 1 participants
- 1 discussions
Date: Fri, 14 Jan 1994 09:05:25 +0100
From: Axel Poigne <poigne(a)gmd.de>
I yesterday attended a talk about fuzzy logic (I know this to be
``degoutant'', but...) where a ``Lukaciewicz norm'' was discussed as a
t-norm.
If I recollect correctly, a t-norm is a has a binary operator _\wedge_
which is associative, commutative, and monotonic, the latter being a
mystery, the order being due to the real interval [0,1]. Moreover it seems
that a negation \neg is assumed to exist, since an operator \vee is
defined by de Morgan law. Quite clearly, min determines a norm as well as
the multiplication. It seems to be an assumption that negation is always
\neg a = 1-a.
Now the Lukaciewicz norm is of the form a \wedge b = min{a + b, 1}. As
consequence, a \vee b = max{a + b - 1, 0}. This norm satisfies a \wegde
\neg a = 0 and a \vee \neg a = 1, but \vee and \wedge are not
distributive, which is true for the other norms. (I hope this to be a
correct recollection of what I heard)
Trying to make head and tail of this, I wonder whether one really should
say that one has a lower semi-lattice for the order, or even an Heyting
algebra (in fact the \sqcap and \bigsqcup is about in all the arguments),
and just add a binary monotonic, etc operator _ \otimes _ (replacing the
\wedge in the t-norm). This structure rather looks like a quantale (units
are available as well). I have no idea how negation fits the picture, but
it reminds me of classical linear logic.
Does this ring a bell ? I am just puzzled, having no idea about fuzzy
logic, and little knowlege about linear logic. I know that Michael Barr
has written a paper on Fuzzy sets as toposes but he uses only geometric
logic, meaning a Heyting algebra.
Axel
A related question : these people seem to use \bigsqcup in general to
compute suprema. It appears to be more consistent to use \bigoplus on
occasions. How would this be defined in linear logic ? (Sorry, my linear
logic is very poor)
>From cat-dist(a)mta.ca Ukn Jan 14 13:18:16 1994
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Date: Fri, 14 Jan 1994 12:56:25 +0400 (GMT+4:00)
From: categories <cat-dist(a)mta.ca>
Subject: I like my coffee crisp
To: categories <categories(a)mta.ca>
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Date: Fri, 14 Jan 94 9:21:59 EST
From: Al Vilcius <vilcius(a)mathstat.yorku.ca>
This is an appeal for help to my friends and acquaintances on
CATEGORIES:
I have been enlisted to give a talk on "Fuzzy vs. Probability
Theory" to an audience in finance whose backgrounds include
applied mathematics, physics, engineering, and finance. The
audience is not familiar with categories, no less toposes, which
means that subtleties such as adding fuzzy equality to yield
variable sets (sheaves) would be lost. Nevertheless, they are
intrigued by the "fuzzy stuff" that is currently popular.
My predicament is then to choose between:
(1) torturing my conscience by giving an insubstantial and
superficial talk on memberships vs distribution functions;
(2) torturing the underlying mathematics into layman's prose.
I am hoping that some of the learned readers of CATEGORIES may
have already performed torture # 2 in a humane fashion (either in
public or in private) and have some material and/or suggestions
on how best to commit this heinous act.
My preferred approach would be a la M. Barr via variable sets and
sheaves, combined with the description of fuzzy and probabilistic
algebraic theories given by E. Manes. I am already aware of many
other fine (and some not so fine) works on fuzzy sets and fuzzy
logic, however, I don't know how to make these understandable to
non-categorists. I may well have to resort to torture # 1, but
would like to avoid doing so if possible.
All comments and suggestions, either privately to me at
vilcius(a)clid.yorku.ca or publicly on CATEGORIES, on how I could
have my "coffee crisp" would be most welcome.
Thank you ............................... Al Vilcius, Toronto
/\ /
/ \ /
/--->\ /
/ \/
>From cat-dist(a)mta.ca Ukn Jan 14 22:00:51 1994
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Date: Fri, 14 Jan 1994 21:53:59 +0400 (GMT+4:00)
From: categories <cat-dist(a)mta.ca>
Subject: simple characterization of weak cartesian closedness
To: categories <categories(a)mta.ca>
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Date: Fri, 14 Jan 94 18:55:12 +0100
From: Thomas Streicher <streiche(a)informatik.uni-muenchen.de>
I wonder whether the following trivial observation is generally known
:
a category C with finite products is WEAKLY CARTESIAN CLOSED iff
for all objects A , B in C the functor C( _ x A , B) is a
retract of a representable functor.
(The embedding part of the retraction gives a choice of functional
abstraction which is stable under substitution) and the projection
part gives evaluation).
Especially this entails that if C has splitting of idempotents then
the notions of cartesian closedness and weakly cartesian closedness
are equivalent.
I don't think that the remark above is a deep insight !!
BUT usually people refer to the quite heavy machinery of Hayashi's
semifunctors when they speak about the categorical semantics of typed
lambda calculus without eta-rule.
Thomas Streicher
>From cat-dist(a)mta.ca Ukn Jan 14 22:13:22 1994
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Date: Fri, 14 Jan 1994 22:04:18 +0400 (GMT+4:00)
From: categories <cat-dist(a)mta.ca>
Subject: cantor-bernstein
To: categories <categories(a)mta.ca>
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Date: Fri, 14 Jan 94 15:44:32 EST
From: Peter Freyd <pjf(a)saul.cis.upenn.edu>
Pierre Ageron asked (way back on 1 Dec):
The statement and the AC-free proof of Cantor-Schroeder-Bernstein's theorem
have obviously some categorical content. Has this been already investigated ?
The property, when it holds, is an important property on the category. It is,
however, a rare property.
There are, as usual, different ways to interpret the property in general
categories. I would opt for the following:
CANTOR-SCHROEDER-BERNSTEIN PROPERTY: If two objects be retracts of each other
they are necessarily isomorphic.
I trust that CSB holds for any boolean topos. (Anyone want to confirm?)
Kaplansky in his booklet on infinite abelian groups pointed out that CSB holds
in the category of countable torsion abelian groups (as a consequence of the
Ulm invariants). He raised it as one of three test problems for advances in
the theory of abelian groups. Does CSB continue to hold, for example, if
countablility is dropped? (Kaplansky did not, of course, talk about retracts.
He talked about two groups appearing as direct summands of each other.)
Someone found a counterexample in the latter 50's. (Anybody know who?)
If _A_ and _B_ are categories, _A_ a retract of _B_, it is routine that
a counterexample for CSB in _A_ is transported to a countexample in _B_.
_Abelian_Groups_ is a retract of _Topological_Spaces_ (via Moore spaces and
homology) hence there are a pair of spaces which appear as retracts of each
other but are different enough to have different homology groups. That fact
became better known in the late 50's than the fact about abelian groups. (And
in the late 50's it was damned difficult to explain why it should be viewed
as a trivial corollary.)
There's a stronger property: if two objects be retracts of each other the
retraction maps are isomorphisms.
The two most immediate examples are the categories of finite sets and of
finite dimensional vectors spaces. But note that any category that is
locally finite (i.e. all hom-sets are finite)--or any linear category that
is locally finite dimensional--immediately inherits the property. By moving
to a 2-category setting one may state the obvious general theorem of
which these are special cases.
I am not sure if any of this should be viewed as having "categorical content."
best thoughts,
peter
>From cat-dist(a)mta.ca Ukn Jan 14 22:19:20 1994
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Date: Fri, 14 Jan 1994 22:12:16 +0400 (GMT+4:00)
From: categories <cat-dist(a)mta.ca>
Subject: syntactic criterion for join?
To: categories <categories(a)mta.ca>
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Date: Fri, 14 Jan 1994 16:47:01 -0500
From: David Espinosa <espinosa(a)cs.columbia.edu>
Does anyone know a syntactic criterion for the existence of a natural
transformation join : TTA -> TA for a given endofunction T built from
+, *, -> ?
There is a well-known (correct me if I'm wrong) syntactic criterion
for covariance which determines whether T can be extended to an
endofunctor. Can this criterion be extended to the existence of join?
Also, does anyone know a reference for the covariance criterion?
David
>From cat-dist(a)mta.ca Ukn Jan 14 22:23:22 1994
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Date: Fri, 14 Jan 1994 22:15:51 +0400 (GMT+4:00)
From: categories <cat-dist(a)mta.ca>
Subject: Re: Fuzzy +
To: categories <categories(a)mta.ca>
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Date: Fri, 14 Jan 94 13:18:44 EST
From: Michael Barr <barr(a)triples.Math.McGill.CA>
One thing to say is that I did not write a paper on fuzzy sets as toposes,
but someone named Eytan did. I wrote a paper on fuzzy sets as non-toposes
and it differs from Eytan's in being correct. On the other hand, fuzzy
sets are a quasi topos, which means they do have a first order logic.
That said, it has to admitted that the first order logic is probably not
what they really had in mind as fuzzy logic and what they did have in
mind (using operators like truncated sum and negations like - minus is
closer to linear logic than to classical, even intuitionistic classical,
logic. I once started to write a paper on this, but have not completed
it it; maybe one day I will. And, BTW, Andy Pitts, unbeknownst to me,
also once wrote a paper on fuzzy sets as a non-topos. His is also correct.
Michael
>From cat-dist(a)mta.ca Ukn Jan 14 22:27:53 1994
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Date: Fri, 14 Jan 1994 22:19:28 +0400 (GMT+4:00)
From: categories <cat-dist(a)mta.ca>
Subject: Re: I like my coffee crisp
To: categories <categories(a)mta.ca>
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Date: Fri, 14 Jan 94 12:28:42 PST
From: "Michael J. Healy (206) 865-3123" <mjhealy(a)espresso.rt.cs.boeing.com>
> Date: Fri, 14 Jan 94 9:21:59 EST
> From: Al Vilcius <vilcius(a)mathstat.yorku.ca>
>
> This is an appeal for help to my friends and acquaintances on
> CATEGORIES:
>
> I have been enlisted to give a talk on "Fuzzy vs. Probability
> Theory" to an audience in finance whose backgrounds include
> applied mathematics, physics, engineering, and finance. The
> audience is not familiar with categories, no less toposes, which
> means that subtleties such as adding fuzzy equality to yield
> variable sets (sheaves) would be lost. Nevertheless, they are
> intrigued by the "fuzzy stuff" that is currently popular.
>
> My predicament is then to choose between:
>
> (1) torturing my conscience by giving an insubstantial and
> superficial talk on memberships vs distribution functions;
>
> (2) torturing the underlying mathematics into layman's prose.
>
> I am hoping that some of the learned readers of CATEGORIES may
> have already performed torture # 2 in a humane fashion (either in
> public or in private) and have some material and/or suggestions
> on how best to commit this heinous act.
>
> My preferred approach would be a la M. Barr via variable sets and
> sheaves, combined with the description of fuzzy and probabilistic
> algebraic theories given by E. Manes. I am already aware of many
> other fine (and some not so fine) works on fuzzy sets and fuzzy
> logic, however, I don't know how to make these understandable to
> non-categorists. I may well have to resort to torture # 1, but
> would like to avoid doing so if possible.
>
> All comments and suggestions, either privately to me at
> vilcius(a)clid.yorku.ca or publicly on CATEGORIES, on how I could
> have my "coffee crisp" would be most welcome.
>
> Thank you ............................... Al Vilcius, Toronto
>
> /\ /
> / \ /
> /--->\ /
> / \/
>
>
I have a related predicament. I'm an industrial mathematician with a
need to learn what I can as soon as possible about a mathematical
background for fuzzy logic. I am also furiously learning what I can
about category theory and logic in connection with some work in formal
methods for software engineering and machine learning (neural networks).
So I really need to find an appropriate, no-nonsense (i.e., mathematical)
formalism that meets all these requirements; given that, I can afford to
invest considerable effort learning it. My current choice is to study
categorical or category-related theories, and am currently reading up
on Steven Vickers' work on topological systems as well as Goguen and
Burstalls' work on institutions. If anybody has information that might
help, or could elaborate on your reply to Al Vilcius so that a categorical
novice might understand as well, I would be most grateful. I did study
topology and algebra in grad school many years ago.
Thank you,
Mike Healy
mjhealy(a)atc.boeing.com
>From cat-dist(a)mta.ca Ukn Jan 16 12:52:30 1994
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Date: Sun, 16 Jan 1994 12:37:25 +0400 (GMT+4:00)
From: categories <cat-dist(a)mta.ca>
Subject: Acyclic models
To: categories <categories(a)mta.ca>
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Date: Sat, 15 Jan 94 11:43:33 EST
From: Michael Barr <barr(a)triples.Math.McGill.CA>
There have been two apparently quite different categorical versions of
acyclic models. The first, as found for example in Barr-Beck, COCA,
1966, says that if K = {Kn}, augmented over K(-1) and similarly L were
chain complex functors and G is a cotriple such that (Kn)epsilon: (Kn)G
--> Kn has a natural splitting when n >= 0 and if the complex LG -->
L(-1)G --> 0 has a natural contracting homotopy, then any natural
transformation K(-1) --> L(-1) can be extended to a unique up to
homotopy map K --> L. In many cases the required naturality is too hard
to verify (or false) and so a second form of the theorem is used. Here
we simply suppose that the complex (Kn)G* --> Kn --> 0 is acyclic (an
easy consequence of the splitting above), that LG --> L(-1)G --> 0 is
acyclic and that K(-1) is isomorphic to L(-1) and conclude that H(K) is
isomorphic to H(L). (Kn)G* stands for the standard powers-of-G
resolution coming from eps. This version is easy to apply, but suffers
from three defects. First, it works only in the case of isomorphism,
not arbitrary maps. Second, it does not in itself give naturality,
although that could probably be remedied by using a category of
relations. Third, and probably most important, it gives no uniqueness.
This means, for example, that although you can use it (in conjunction
with an argument involving simplicial subdivision) to show that singular
and simplicial homology of triangulated spaces are isomorphic, you
cannot show this way that the isomorphism is induced by the inclusion of
the simplicial chains into the singular ones.
I have recently discovered a version of acyclic models that repairs all
three defects. Moreover, it gives a single proof of both forms as well
as third form involving what I will call weak homotopy equivalence.
(This is not a Quillen model category in general, although there would
appear to be considerable overlap.) Let C be the category of
chain complexes of functors from some category X to an abelian category
A. Say that an arrow K --> L in C is a weak homotopy equivalence if for
each object x of X, Kx --> Lx is a homotopy equivalence (has a homotopy
inverse and homotopies, etc., but not assumed natural). Let Sigma stand
for one of the classes:
(a) homotopy equivalences
(b) weak homotopy equivalences
(c) homology isomorphisms
and let D denote the category of fractions gotten from C by inverting
Sigma. Let (G,eps) be a pair consisting of an endofunctor on X and a
natural transformation G --> Id. Say that the augmented object K -->
K(-1) --> 0 of C is Sigma-trivial if the 0 endomorphism is in Sigma.
Say that the object K of C is G presentable (w.r. to Sigma) if for each
n >= 0, the chain complex (Kn)G* --> Kn --> 0 is Sigma-trivial and K is
G acyclic (w.r. to Sigma) if KG --> K(-1)G --> 0 is Sigma-trivial. Then
Theorem: If K is G presentable and L is G acyclic, both w.r. to Sigma,
then any natural transformation K(-1) --> L(-1) can be extended in D to
an arrow, unique in D, K --> L.
In case (a), this is the theorem of Barr-Beck, 1966 and in case (c),
this repairs the three defects cited above, while case (b) appears to be
genuinely new.
The proof is embarrassingly easy. Consider the diagram
(alpha K)G* K(eps*)
K(-1)G* <----------- KG* ---------> K
|
|
|
|
v (alpha L)G* L(eps*)
L(-1)G* <----------- LG* ---------> L
alpha K and alpha L are the augmentation arrows. The G-presentability
implies that K(eps*) is in Sigma and the G-acyclicity that (alpha L)G*
is. When these are inverted, we get the desired map K --> L as the
composite. A paper on the subject will be posted in the usual ftp
location within a week or two.
>From cat-dist(a)mta.ca Ukn Jan 16 12:53:10 1994
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Date: Sun, 16 Jan 1994 12:44:32 +0400 (GMT+4:00)
From: categories <cat-dist(a)mta.ca>
Subject: Re: cantor-bernstein
To: categories <categories(a)mta.ca>
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Date: Sat, 15 Jan 94 11:36:22 EST
From: Michael Barr <barr(a)triples.Math.McGill.CA>
One additional example and you don't even need retracts. In the category
of finitely generated modules over a commutative ring, all epis are
isos. As a result, if you have epis in both directions, they are
isos. So the dual category category is S-B. This is fairly
easy if the ring has ACC, but there is a trick that works for any
ring to reduce it to that case. Since f.d. vector spaces are self-dual,
this example encompasses that one.
Michael
>From cat-dist(a)mta.ca Ukn Jan 17 09:36:47 1994
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Date: Mon, 17 Jan 1994 09:26:04 +0400 (GMT+4:00)
From: categories <cat-dist(a)mta.ca>
Subject: Re: simple characterization of weak cartesian closedness
To: categories <categories(a)mta.ca>
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Date: Mon, 17 Jan 94 9:44:41 MET
From: Simone Martini <martini(a)di.unipi.it>
Thomas Streicher asks
>> whether the following trivial observation is generally known:
>>
>>a category C with finite products is WEAKLY CARTESIAN CLOSED iff
>>for all objects A , B in C the functor C( _ x A , B) is a
>>retract of a representable functor.
I cannot say about "generally known", but..
this property it is quoted as one of the elementary characterizations of wCCC
in a paper of mine (Categorical Models for non-extensional lambda-calculi,
Mathematical Structures in Computer Science (1992), vol 2, pag 327--357).
The paper, which has a definite didactic pace, discusses also the
case where there is only an epy natural transformation from C(_,A=>B) to
C( _ x A , B), which gives models of typed, non extensional, Combinatory
Logic; and it gives conditions on the existence of models of
type-free lambda-calculus as reflexive objects in wCCCs.
Simone Martini
Universit\`a di Pisa,
Dipartimento di Informatica.
>From cat-dist(a)mta.ca Ukn Jan 17 09:40:11 1994
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Date: Mon, 17 Jan 1994 09:29:33 +0400 (GMT+4:00)
From: categories <cat-dist(a)mta.ca>
Subject: Re: simple characterization of weak cartesian closedness
To: categories <categories(a)mta.ca>
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Date: Mon, 17 Jan 1994 11:37:33 +0100 (MET)
From: Raymond Hoofman <raymond(a)fwi.uva.nl>
Quoting Thomas Streicher,
> I wonder whether the following trivial observation is generally known
> :
>
> a category C with finite products is WEAKLY CARTESIAN CLOSED iff
> for all objects A , B in C the functor C( _ x A , B) is a
> retract of a representable functor.
Yes, this is the "degenerate" case of a semi-adjunction between a functor G
and a semi-functor F: the semi-isomorphism
D(F(-), ...) \cong_{s} C(-, G(...))
becomes a retraction (see [1], also [2]).
> I don't think that the remark above is a deep insight !!
> BUT usually people refer to the quite heavy machinery of Hayashi's
> semifunctors when they speak about the categorical semantics of typed
> lambda calculus without eta-rule.
However, if the products of your typed lambda calculus also do not satisfy
the eta-rule, the semi-isomorphism above does not degenerate, and it is
less obvious how to give a simple characterization without semi-functors (apart
from saying that the Karoubi envelope of the category is Cartesian closed).
[1] The theory of Semifunctors, R. Hoofman, MSCS 3
[2] Collapsing Graph models by preorders, R. Hoofman & H. Schellinx, LNCS 530
With kind regards,
Raymond Hoofman.
>From cat-dist(a)mta.ca Ukn Jan 17 16:49:30 1994
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Date: Mon, 17 Jan 1994 16:18:00 +0400 (GMT+4:00)
From: categories <cat-dist(a)mta.ca>
Subject: Fuzzy references
To: categories <categories(a)mta.ca>
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Date: Mon, 17 Jan 1994 12:57:49 -0400
From: Mike Wendt <wendt(a)cs.dal.ca>
==========================================================================
Hi Al:
I'm not sure if this is what you want but here are a couple of references
I've noticed recently (my search for categorical-measure-theory-type stuff):
Bandemer, H., Nather, W, "FUZZA DATA ANALYSIS," Theory and Decision Library,
Kluwer Academic Press, Series B, Vol. 20 (Norwell, Mass., 1992).
Rodabaugh, S., Klement, E., Hoehle, U. (eds.), "APPLICATIONS OF CATEGORY
THEORY TO FUZZY SUBSETS," Kluwer Academic Press, Series B, Vol. 14
(Norwell, Mass., 1992).
I'm sorry, I can't give you a review of these books yet. I have only peeked
in the first one. It seems interesting enough and is at an introductory
level.
Regards,
-Mike Wendt
==========================================================================
>From cat-dist(a)mta.ca Ukn Jan 17 16:49:47 1994
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From: categories <cat-dist(a)mta.ca>
Subject: Re: Fuzzy +
To: categories <categories(a)mta.ca>
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Date: Mon, 17 Jan 94 15:36:35 +0100
From: Pierre Ageron <ageron(a)univ-caen.fr>
In my thesis "Structure des logiques et logique des structures", I tried (very
shortly) to understand what the algebraic and categorical counterparts of
fuzzy logic are.
There are different proposals in the literature, the reason is that there
are plenty of interesting operations on the interval [0,1] and that it is very
difficult to tell which ones are relevant for fuzzy logic. The most general
axiomatization was given by Rene Guitart in his 1979 thesis (or a 1982
paper in the Cahiers). He considered complete ordered abelian monoids (in
their 1990 book, Barr and Wells restricted to complete Heyting algebras).
I observed that every complete ordered abelian monoid has a canonical
Lafont algebra structure: this means that (this) fuzzy logic is the extension
of intuitionistic linear logic with infinitary versions of the additive
connectives "plus" and "with".
Guitart defined the notion of "algebraic universe": essentially a category
equipped with a monad P looking like the monad of subsets on Ens (I mean
Set !). This notion subsumes the notion of elementary topos and allows
to give higher order semantics for logics other than intuitionistic logic.
In the case of fuzzy logic, the point is that every complete ordered abelian
monoid defines such a structure on Ens. The Kleisli category of P is the
category of fuzzy relations. All this is explained in my thesis using
the notations of linear logic.
All that framework gives Tarskian semantics for (propositional or
higher order) fuzzy logic. It is not clear whether there are Heytingian
semantics for fuzzy logic, i.e. a proof theory. The difficulty is that
every small complete category is a poset (but this result by Freyd uses
AC, so hope remains...)
Pierre AGERON
>From cat-dist(a)mta.ca Ukn Jan 17 16:50:40 1994
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id AA24033; Mon, 17 Jan 1994 16:50:39 +0400
Date: Mon, 17 Jan 1994 16:26:39 +0400 (GMT+4:00)
From: categories <cat-dist(a)mta.ca>
Subject: Re: cantor-bernstein
To: categories <categories(a)mta.ca>
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Date: Mon, 17 Jan 94 10:13:02 EST
From: Peter Freyd <pjf(a)saul.cis.upenn.edu>
Mike Barr writes:
One additional example and you don't even need retracts. In the category
of finitely generated modules over a commutative ring, all epis are
isos. As a result, if you have epis in both directions, they are
isos. So the dual category category is S-B. This is fairly
easy if the ring has ACC, but there is a trick that works for any
ring to reduce it to that case.
Wonderful thought: all epis are isos. Anyway, I see a proof that any
epi endo on a finitely presented module over a commutative ring is
iso, but finitely generated?
There's a metaprinciple that says that a result like this should
generalize from commutative to PI rings (that is, rings that satisfy
some non-trivial Polynomial Identity). Can anyone confirm? A corollary
would be that in any additive category if two objects each appear as
retracts of the other, and if the ring of endomorphisms of one of them
is a PI ring then the retractions are isos.
best thoughts,
peter
>From cat-dist(a)mta.ca Ukn Jan 17 22:08:34 1994
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Date: Mon, 17 Jan 1994 21:46:07 +0400 (GMT+4:00)
From: categories <cat-dist(a)mta.ca>
Subject: mathematics made hard
To: categories <categories(a)mta.ca>
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Date: Mon, 17 Jan 94 17:49:08 EST
From: Peter Freyd <pjf(a)saul.cis.upenn.edu>
This is about the opposite of category theory. I'm going to give
a soft proof of something and ask how to get the hard proof. All
of this because of Mike Barr's note about modules of commutative rings.
Let M be a finitely presented module over a commutative ring R
and f:M -> M an epimorphic endomorphism. We will show that
f is necessarily an isomorphism. First specialize to the case that
R is Noetherian. The kernels of the powers of f form an ascending
chain of submodules of M, hence must stabalize. That is, there
k+1 k k+1
is a natural number k such that Ker(f )= Ker(f ). Since f is
epi, it is a cokernel for its kernel and there must exist g:M -> M
k+1 k
such that f g = f . (I'm composing maps in the diagramatic order.)
Using for the second time that f is epi we may cancel to obtain
fg = 1. Since fgf = f1 we cancel once more (using that f is epi
for the third time) to obtain gf = 1.
Now, let r and n be natural numbers and
r n
R -> R -> M -> O
an exact sequence. There must be an rxn matrix K, an rxr matrix
A' an nxn matrix A, another nxn matrix B, and an nxr matrix
C such that
KA = A'K
BA + CK = I. (K describes the map
r n n
from R to R that defines M, A describes the endomorphism on R
r
that "lifts" f, A' describes the endomorphsim on R . Since f is
n+r n
epi the map R -> R obtained by stacking A and K is also
n n+r
epi, hence it has a left-inverse (B,C):R -> R .)
Specialize to the case that R is the the "generic ring",
that is the ring generated by the 2nn+2nr+rr entries of K,A,A',B,C
with nr+nn equations. We may infer that there is an
rxr matrix X and an nxr matrix Y such that
KB = XK
AB + YK = I.
The entries of X and Y are necessarily given by polynomials in
the generating "variables" and the last two matrix equations must
result in rr+nr equations that are direct consequences of the
nr+nn defining equations. Hence the original theorem works for any
finitely presented module over any commutative ring.
Now for the hard part: what are these polynomials? In the case n = 1
its easy (and reveals quickly the need for commutativity).
Try it for n=2, r=1. Given:
ac+be = ga
ad+bf = gb
hc+ie+la = 1
hd+if+lb = 0
jc+ke+ma = 0
jd+kf+mb = 1
find, for a start, a polynomial on these variables, x, such that
ah+bj = xa
ai+bk = xb.
>From cat-dist(a)mta.ca Ukn Jan 17 22:18:57 1994
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Date: Mon, 17 Jan 1994 22:05:50 +0400 (GMT+4:00)
From: categories <cat-dist(a)mta.ca>
Subject: Re: cantor-bernstein
To: categories <categories(a)mta.ca>
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Date: Mon, 17 Jan 94 20:15:09 EST
From: Michael Barr <barr(a)triples.Math.McGill.CA>
> Mike Barr writes:
>
> One additional example and you don't even need retracts. In the category
> of finitely generated modules over a commutative ring, all epis are
> isos. As a result, if you have epis in both directions, they are
> isos. So the dual category is S-B. This is fairly
> easy if the ring has ACC, but there is a trick that works for any
> ring to reduce it to that case.
>
> Wonderful thought: all epis are isos. Anyway, I see a proof that any
> epi endo on a finitely presented module over a commutative ring is
> iso, but finitely generated?
>
> There's a metaprinciple that says that a result like this should
> generalize from commutative to PI rings (that is, rings that satisfy
> some non-trivial Polynomial Identity). Can anyone confirm? A corollary
> would be that in any additive category if two objects each appear as
> retracts of the other, and if the ring of endomorphisms of one of them
> is a PI ring then the retractions are isos.
>
> best thoughts,
> peter
>
>
I will try to recall the argument (on-line).
Given an epi-endomorphism f, look at the ascending chain ker(f),
ker(f^2), ker(f^3),.... In the noetherian case, this stabilizes so
that ker(f^n) = ker(f^{n+1}). Assume thatn is as small as possible,
so that ker(f^{n-1}) < ker(f^n). Choose an element x in the ker of
f^n, not in the lesser one. x = f(y) for some y, since f is onto.
0 = f^n(x) = f^{n+1}(y), so that 0 = f^{n}(y) = f^{n-1}(x), a
contradiction. That takes care of the noetherian case and doesn't
even use commutativity, it would seem. For the general case,
suppose R is the ring, M the module, f: M --> M the endomorphism and
x an element with f(x) = 0. Now pick a set of generators for M, say
y_1,...,y_n. What you have to do is to find a suitable finite
subset of R, with just the right elements in it to express all the
f(y_i), x and at least one preimage of each y_i as linear
combinations of the y_i using coefficients from that subset. Now
let S be the subring of R generated by that finite set of elements
and N be the least S-submodule of M containing all the y_i. If I
have left anything required out of S, add that too. Anyway, S is
noetherian (this does use commutativity, I believe) and f induces a
counter-example on N.
I believe this argument is due to one of the Rutgers people like
Faith or Osofsky, but I am far from certain of that. It will be
true for PI rings if affine PI rings have acc on left ideals. For
commutative rings it is essentially the Hilbert basis theorem.
Michael
>From cat-dist(a)mta.ca Ukn Jan 19 22:39:16 1994
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Date: Wed, 19 Jan 1994 22:19:51 +0400 (GMT+4:00)
From: categories <cat-dist(a)mta.ca>
Subject: RE Fuzzy +
To: categories <categories(a)mta.ca>
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Date: Wed, 19 Jan 1994 11:45:07 -0600
From: Lawerce Neff Stout <lnstout(a)uxh.cso.uiuc.edu>
I've done quite a lot of work on categories of fuzzy sets. The main paper
is in the volume edited by H\"ohle and Rodabaugh referred to earlier.
Barr is correst that fuzzy sets form a quasitopos, but the logic of that
quasitopos is that of the underlying set category, hence not interesting
as a place to do fuzzy mathematics. The fuzzy connectives come from a
second monoidal closed structure obtainable from, for example, the t-norms
usually referred to in the fuzzy literature. This gives a very satisfactory
logic if one uses what I called unballanced subobjects (the map involved is
both monic and epic). There is a weak representor for these subobjects
(representation is not unique though there is an ordering on maps which
allows a canonical choice of representative to be made) allowing an internal
representation of a large fragment of higher order fuzzy logic.
I have a more recent paper (to appear in the proceedings of the 1992 Linz
seminar, being published by Kluwer sometime later this year) in which I look at categories of fuzzy sets with values in a Quantale or Projectale. That paper is
available from me by e-mail (I don't have ftp facilities available). It
includes a characterization of categories of fuzzy sets in terms of the
representability of the logic and the property of being topological over
Sets.
Larry Stout
>From cat-dist(a)mta.ca Ukn Jan 21 09:51:34 1994
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Date: Fri, 21 Jan 1994 09:30:23 +0400 (GMT+4:00)
From: categories <cat-dist(a)mta.ca>
Subject: categorical treatment of F_omega?
To: categories <categories(a)mta.ca>
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Date: Thu, 20 Jan 1994 17:24:00 -0500
From: David Espinosa <espinosa(a)cs.columbia.edu>
1. Could someone send me a good reference for a categorical treatment
of the Girard / Reynolds F_2 polymorphic type system? That is,
polymorphic functions as (some form of) natural transformations?
2. More importantly, has there been a categorical treatment of
Girard's F_omega type system?
David
>From cat-dist(a)mta.ca Ukn Jan 22 13:58:28 1994
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Date: Sat, 22 Jan 1994 13:49:21 +0400 (GMT+4:00)
From: categories <cat-dist(a)mta.ca>
Subject: Terminology question
To: categories <categories(a)mta.ca>
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Date: Sat, 22 Jan 1994 2:44:17 -0500 (EST)
From: D_FELDMAN(a)UNHH.UNH.EDU
Is there a standard terminology for the following sort of gadget or something
very similar?
Define a {\em ??????} to be a pair $(S,F)$ consisting of a category $S$ and
a functor $F$ from ${\bf S}$ to ${\bf Bij}$, the category of finite
sets and bijections, satisfying\\
(i) If $F(s_1)=t_1$ and $\tau:t_1\rightarrow t_2$, then there exists a
unique object $s_2\in {\bf S}$ and a unique ${\bf S}$-morphism
$\sigma:s_1\rightarrow s_2$ such that $F(\sigma)$=$\tau$.\\
(ii) For $t \in {\bf Bij}$, $F^{-1}(t)$ is an (effectively
computable) finite set.
David Feldman
>From cat-dist(a)mta.ca Ukn Jan 24 13:43:31 1994
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Date: Mon, 24 Jan 1994 13:15:21 +0400 (GMT+4:00)
From: categories <cat-dist(a)mta.ca>
Subject: Re: Terminology question
To: categories <categories(a)mta.ca>
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Date: Mon, 24 Jan 1994 16:13:07 +0000 (GMT)
From: Edmund Robinson <edmundr(a)cogs.susx.ac.uk>
>
> Date: Sat, 22 Jan 1994 2:44:17 -0500 (EST)
> From: D_FELDMAN(a)UNHH.UNH.EDU
>
> Is there a standard terminology for the following sort of gadget or something
> very similar?
>
> Define a {\em ??????} to be a pair $(S,F)$ consisting of a category $S$ and
> a functor $F$ from ${\bf S}$ to ${\bf Bij}$, the category of finite
> sets and bijections, satisfying\\
> (i) If $F(s_1)=t_1$ and $\tau:t_1\rightarrow t_2$, then there exists a
> unique object $s_2\in {\bf S}$ and a unique ${\bf S}$-morphism
> $\sigma:s_1\rightarrow s_2$ such that $F(\sigma)$=$\tau$.\\
> (ii) For $t \in {\bf Bij}$, $F^{-1}(t)$ is an (effectively
> computable) finite set.
>
> David Feldman
>
>
I think this would traditionally be described as a "finite discrete
opfibration (over Bij)". The functors corresponding to condition (i)
are discrete opfibrations, and the finite comes from condition (ii).
Neither of these uses any special property of Bij (such as the fact
that it is a groupoid). It might be more modern to use "cofibration"
instead of "opfibration". See Barr & Wells "Toposes, Triples and
Theories" p231 ex [OPF] for more conventional definitions, and perhaps
Benabou "Fibred categories and the foundations of naive category
theory" (J. Symbolic Logic (50) No. 1, 1985, 10-37) for more of an
indication of why these sorts of structures are so common.
Another way of looking at the structure would be to turn it around and
say that you have a functor G: Bij -> FiniteSet given on objects by
G(t) = F^{-1}(t).
best wishes,
Edmund Robinson
>From cat-dist(a)mta.ca Ukn Jan 25 07:26:29 1994
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Date: Tue, 25 Jan 1994 07:21:14 +0400 (GMT+4:00)
From: categories <cat-dist(a)mta.ca>
Subject: Re: Terminology question
To: categories <categories(a)mta.ca>
Cc: cdl2 -- France Dacar <France.Dacar(a)ijs.si>,
Robert Dawson <rdawson(a)husky1.stmarys.ca>,
"Oege de.Moor" <Oege.de.Moor(a)prg.ox.ac.uk>,
"Valeria de.Paiva" <vcvp(a)cl.cam.ac.uk>,
"Ruy de.Queiroz" <rjq(a)doc.ic.ac.uk>,
"Fer-Jan De.Vries" <ferjan(a)cwi.nl>,
Kyung-Goo Doh <kg-doh(a)u-aizu.ac.jp>,
James Dolan <jdolan(a)ucrmath.ucr.edu>,
Xiaomin Dong <xdong(a)clid.yorku.ca>,
Winfried Drecmann <mas031(a)bangor.ac.uk>,
Dominic Duggan <dduggan(a)watmsg.waterloo.edu>,
Gerald Dunn <gdunn(a)nmsu.edu>, Hans Dybkjaer <dybkjaer(a)ruc.dk>,
Abbas Edalat <ae(a)doc.ic.ac.uk>,
David Espinosa <dae(a)martigny.ai.mit.edu>,
Michel Eytan <eytan(a)dpt-info.u-strasbg.fr>,
Joe Fasel <jhf(a)c3.lanl.gov>, David Feldman <d_feldman(a)unhh.unh.edu>,
Zbigniew Fiedorowicz <zigf(a)mps.ohio-state.edu>,
Juarez Muylaert Filho <jamf(a)doc.ic.ac.uk>,
Stacy Finkelstein <stacy(a)saul.cis.upenn.edu>,
Kathleen Fisher <kfisher(a)cs.stanford.edu>,
Maria Frade <mjf(a)di.uminho.pt>, Peter Freyd <pjf(a)saul.cis.upenn.edu>,
Tom Fukushima <fukushim(a)cpsc.ucalgary.ca>,
Jonathan Funk <jfunk(a)morgan.ucs.mun.ca>,
Fabio Gadducci <gadducci(a)di.unipi.it>,
Vijay Gehlot <gehlot(a)saul.cis.upenn.edu>,
Wolfgang Gehrke <wgehrke(a)risc.uni-linz.ac.at>,
Silvio Ghilardi <ghilardi(a)vmimat.mat.unimi.it>,
Paul Glenn <glenn(a)cua.edu>, Joseph Goguen <Joseph.Goguen(a)prg.ox.ac.uk>,
Marek Golasinski <mg001(a)vm.cc.uni.torun.pl>,
Al Goodloe <agoodloe(a)mason1.gmu.edu>,
Bob Gordon <gordon(a)euclid.math.temple.edu>,
Francoise Grandjean <grandjean(a)agel.ucl.ac.be>,
John Gray <gray(a)math.uiuc.edu>, Luzius Grunenfelder <luzius(a)cs.dal.ca>,
Stefano Guerrini <guerrini(a)di.unipi.it>,
Alessio Guglielmi <guglielm(a)di.unipi.it>,
James Harland <jah(a)cs.mu.oz.au>, Robert Harper <rwh(a)cs.cmu.edu>,
Magne Haveraaen <magne(a)eik.ii.uib.no>,
"Michael J. Healy" <mjhealy(a)espresso.rt.cs.boeing.com>,
Michel Hebert <mhebert(a)egaucacs.bitnet>,
Murray Heggie <heggie(a)cad.uccb.ns.ca>,
Luis Javier Hernandez <zl(a)cc.unizar.es>, Walt Hill <whill(a)netcom.com>,
SATO Hiroyuki <schuko(a)sun4.cc.kyushu-u.ac.jp>,
Bernard Hodgson <bhodgson(a)mat.ulaval.ca>
Message-Id: <Pine.3.05.9401250714.C24955-b100000(a)nimble.mta.ca>
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Date: Tue, 25 Jan 1994 10:18:25 +0000
From: Steven Vickers <sjv(a)doc.ic.ac.uk>
Do others suffer the same heartsink as I do when confronted with a posting
like this?
>Define a {\em ??????} to be a pair $(S,F)$ consisting of a category $S$ and
>a functor $F$ from ${\bf S}$ to ${\bf Bij}$, the category of finite
>sets and bijections, satisfying\\
> (i) If $F(s_1)=t_1$ and $\tau:t_1\rightarrow t_2$, then there exists a
>unique object $s_2\in {\bf S}$ and a unique ${\bf S}$-morphism
>$\sigma:s_1\rightarrow s_2$ such that $F(\sigma)$=$\tau$.\\
> (ii) For $t \in {\bf Bij}$, $F^{-1}(t)$ is an (effectively
>computable) finite set.
For human readers (and after all, is this message _ever_ going to be
presented to a Latex interpreter?) most of the $'s and \'s here are not
only completely unnecessary, but, worse, a positive barrier to
understanding.
I would expect - but this is something that can be put to the test - that
even people completely familiar with Latex would find it easier to read the
following version. It certainly involves less typing.
>Define a ?????? to be a pair (S,F) consisting of a category S and
>a functor F from S to Bij, the category of finite
>sets and bijections, satisfying -
> (i) If F(s_1)=t_1 and tau:t_1 -> t_2, then there exists a
>unique object s_2 in S and a unique S-morphism
>sigma: s_1 -> s_2 such that F(sigma)=tau.
> (ii) For t in Bij, F^{-1}(t) is an (effectively
>computable) finite set.
(I have ignored the puzzle of whether {\bf ...} is mathematically
meaningful - in the original $S$ turns into ${\bf S}$. If it _is_
mathematically meaningful, then in Latex it should be macroized.)
Steve Vickers.
p.s. Having rephrased the question, I still don't know the answer - sorry.
>From cat-dist(a)mta.ca Ukn Jan 25 12:17:27 1994
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Date: Tue, 25 Jan 1994 11:47:26 +0400 (GMT+4:00)
From: categories <cat-dist(a)mta.ca>
Subject: Re: cantor-bernstein
To: categories <categories(a)mta.ca>
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Date: Wed, 19 Jan 1994 09:37:14 -0500
From: Stephen Chase <chase(a)math.cornell.edu>
With regard to the remarks of Barr and Freyd on surjective endomorphisms
of finitely generated modules: I haven't digested those remarks, but here
is a slick proof of the result, communicated to me years ago by Bill
Waterhouse (instead of reduction to the Noetherian case, it uses
localization):
It is enough to prove bijectivity at all localizations, so we can assume
that the commutative ring A is local with maximal ideal m. Given a
surjective endomorphism f of a finitely generated A-module M, let F be a
finitely generated free A-module mapping onto M so that the mapping
induces an isomorphism F/mF ----> M/mM. Let K = Ker(F ---> M). f then
lifts to an endomorphism g of F, which is an isomorphism because it is
so mod m. Then g(K) is contained in K, and to prove f is bijective we
need only show g'(K) is likewise contained in K (with g' the inverse of g).
But g satisfies its characteristic polynomial, which has invertible constant
term det(g) (up to sign); thus g' is a polynomial in g and so maps K into
itself.
I haven't seen this proof in the literature. However, the following
related reference might be of interest: M. Orzech, L. Ribes, "Residual
finiteness and the Hopf property in rings", J. Algebra 15 (1970), 81-88.
Sincerely,
Steve Chase
>From cat-dist(a)mta.ca Ukn Jan 25 12:22:48 1994
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Date: Tue, 25 Jan 1994 11:53:49 +0400 (GMT+4:00)
From: categories <cat-dist(a)mta.ca>
Subject: terminology
To: categories <categories(a)mta.ca>
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[Note from moderator:
1. The two posts following are being forwarded, but I don't feel that this
list is really the place for a long discussion of suitable notation for
e-mail, so I hope that any discussion will be short]
Date: Tue, 25 Jan 94 08:37:53 EST
From: Peter Freyd <pjf(a)saul.cis.upenn.edu>
I certainly agree with Steve's point. But I would go further:
Define a ?????? to be a pair (*S*, F) consisting of a category *S* and
a functor F from *S* to *Bij* , the category of finite
sets and bijections, satisfying -
(i) If F(S ) = T and f:T -> T' , then there exists a
unique object S' in *S* and a unique morphism
g: S -> S' such that F(g) = f.
(ii) For T in *Bij*, F (T) is an (effectively
computable) finite set.
But: I must confess that I also experience a little "heartsink" when I
see a list of addresses as long as that above.
best thoughts,
peter
++++++++++++++++++++++++++++++++++++++++++++
Date: Tue, 25 Jan 94 9:51:20 EST
From: Al Vilcius <vilcius(a)mathstat.yorku.ca>
Referring to the "rephrased" question of Steve Vickers:
>
> Do others suffer the same heartsink as I do when confronted with a posting
> like this?
>
> >Define a {\em ??????} to be a pair $(S,F)$ consisting of a category $S$ and
> >a functor $F$ from ${\bf S}$ to ${\bf Bij}$, the category of finite ......
Yes, I certainly do, and much prefer the "humanized" alternative:
>
> >Define a ?????? to be a pair (S,F) consisting of a category S and
> >a functor F from S to Bij, the category of finite ......
>
--
/\ / Al Vilcius, Toronto
/ \ /
/--->\ /
/ \/
>From cat-dist(a)mta.ca Ukn Jan 27 12:30:09 1994
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Date: Thu, 27 Jan 1994 11:53:21 +0400 (GMT+4:00)
From: categories <cat-dist(a)mta.ca>
Subject: RE: terminology
To: categories <categories(a)mta.ca>
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Date: Wed, 26 Jan 1994 2:35:50 -0500 (EST)
From: D_FELDMAN(a)UNHH.UNH.EDU
Thank you to all those who responded, including those who pointed
out my e-faux pas. Incidently, the complaint about S versus {\bf S}
alerted me to a typo in a paper under preparation (these should have
been the same) and so I am especially grateful for that.
David Feldman
>From cat-dist(a)mta.ca Ukn Jan 31 16:29:46 1994
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Date: Mon, 31 Jan 1994 15:59:00 +0400 (GMT+4:00)
From: categories <cat-dist(a)mta.ca>
Subject: New address of Fer-Jan de Vries
To: categories <categories(a)mta.ca>
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Date: Mon, 31 Jan 1994 20:17:52 +0100
From: F.J.de.Vries(a)cwi.nl
CWI, January 31, 1994
Dear Colleague.
The coming year I will live and work in Japan.
My addresses will be the following:
Office: from March 1st, 94, onwards
NNT,
Communication Science Laboratories
Hikaridai, Seika-cho,
Soraku-gun, Kyoto 619-02
Phone +81-7749-5-1841,
Facsimile +81-7749-5-1851
Email ferjan(a)progn.kecl.ntt.jp
Home: from Feb 1st, 94, onwards
Seresu-Gakuenmae 305
Gakuen-Naka 1-1542-190
Nara-shi, Nara 631
Phone: yet unkown...
Sayonara, Fer-Jan de Vries.
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