Zamulin's note

[EricWagner@aol.com]

>From my point of view (as a mathematician) a preordered set
consists of a set $P$ together with a relation $\leq$ on $P$
(i.e., a subset of $P\times P$) that is reflexive and
transitive).  (One should probably be even more precise and
include the source and target in the definition of a relation).
Using the same symbol for a preordered set its underlying set is
a common abuse of notation.  Using the word "preorder" as
denoting both the preordered set and its underlying relation is
another common abuse of notation (or language).  I don't think that
either abuse would bother most mathematicians but they may well
be more important from a computer science point of view where the
notation may be processed by various programmed tools.*

I would say that a subsorted signature \Sigma = (S, TF, PF, P, \leq) 
where (S, TF, PF, P) is a many-sorted signature and
(S,\leq) is a preordered set.	(Actually, I also have some
disagreement with this version of a many-sorted signature -- see
my upcoming talk in WADT99.)

Regards,  Eric Wagner

* For some time I have be saying that one difference between
mathematics (as practiced by mathematicians) and computer science
(as practiced by computer scientists) is that In mathematics the
concepts are very precise but the notation is sloppy, while in
computer science the concepts are sloppy but the notation is very
precise.
This, of course, is just a reflection of mathematics being
largely conceptual (so one can deduce what the sloppy notation
means) while computer science deals with computers and programs
where correct syntax is generally crucial but meaning is
frequently determined by programs that are not completely
understood.
As I see it, we are trying to make both the concepts and the
notation precise.

E.

[[[ AT:

I think this goes in line with Sasha's proposal --- so, no change of
decision here, we adopt the notation without any subscript.

And we all are looking forward to Eric's talk in Bonas!

Andrzej Tarlecki
]]]