# Re: Binary Relations, draft 1

• To: Taco Hoekwater <taco.hoekwater@wkap.nl>
• Subject: Re: Binary Relations, draft 1
• From: Hans Aberg <haberg@matematik.su.se>
• Date: Mon, 16 Nov 1998 19:09:19 +0100
• Cc: math-font-discuss@cogs.susx.ac.uk
• Content-Length: 4620

At 17:36 +0100 1998/11/16, Taco Hoekwater wrote:
> HA> However, your curled 037/038 look very good. I have
> HA> noted that if one writes a paper with several different relations,
>then it
> HA> is difficult to find good variations. So from that perspective, I think a
> HA> closed variation of \succ and prec should be added.
>
>OK, that will look a little different (prolonged, of course).
>
>We are left with the question whether the equilateral triangles are
>relations or operations, with a "probably not" as temporary answer.

No, I think that equilateral triangles can be binary operators, like the
TeX \triangleleft and \triangleright. These should then be small in size,
similar to that of \circ, \bullet and those.

If you design these binary operations and want to figure out to make it
look good, I would suggest you to look at simple expressions such as (in
TeX) $a+b$, $d\cdot$, $f\circ g$, $h\bullet k$, and then try to copy image
of proportions that this gives in your mind.

In addition to the outline binary operations, just as \circ has a filled-in
version \bullet, one might think of filled in versions of \triangleleft,
\triangleright, and \diamond.

I have also found that both outline and filled similarly sized and
positioned small squares are excellent to use as binary operations.

>Building on that, we would like to have a variation on the triangles as
>binary relations, and these probably should look like closed succ and
>prec. (with the advantage that these can never be confused with the
>equilateral triangle operators). Am I correct?

Yes, that is my suggestion.

> HA> This is not a negated subset, but a strict subset, so therefore it should
> HA> be in this font. (One could also in principle negate the strict subset
> HA> symbol.)
>
>I see. But I am running out of room in this font, so some stuff *has*
>to be moved into the negations font, which is half-empty. The idea
>behind these two fonts is that I can put the negated versions of
>various relations in the same slot in that other font. But not all
>relations can be (or hardly ever are) negated, so there is quite a lot
>of room left. It makes sense to add the overflow characters into these
>slots.

Well, I just point out the logic: It is better to have the symbol somewhere
in Unicode than not at all clearly.

>Both of the pairs are completely identical in rendering at the
>moment. The symbols have descriptions to go with them, and these
>descriptions are:
>
>mostpos (5) : most positive [inverted lazy S]
>ac (224):     most positive
>
>this hints that both are the same character, and one should be
>removed.
>
>congruence (159): congruence sign (lazy S)
>race (222):       reverse most positive, line below
>
>this is less clear. is the "reverse most positive" something different
>from "congruence sign"? If so, there may have to be a difference in
>the rendering too, to avoid confusion.

I think I recall I have seen conguence somewhere, but not "reverse most
positive": From the mathematical point of view, these are most surely
different semantic concepts. But the symbol can still be the same: The
Unicode de facto breaks down for math, and one must try doing the best one
can of the situation, given the rules.

If the symbols are so close that you can not distinguish them when
setting them side by side (considering the input from others), then I would
suggest you to remove duplicates.

> HA> These are clearly a "times sign" with extra strokes on (because that
>is how
> HA> they arise mathematically, as a Cartesian product with some additional
> HA> properties). The bowtie looks different though, being prolonged, so
> HA> therfore I think it is different from the "doubly closed times sign", and
> HA> that the bowtie should only be a relation, not an operator.
>
>This is clear enough for bowtie and the closed times. But there are
>characters 235 and 237 (left filled times and right filled times), and
>these are flagged as relations. If I understand you right, they should
>probably be operators, and removed from this font.

To me, 234-238 looks as relations the way they have been designed. I have
never seen anything like that in use, so I cannot tell more. But if these
should be called "left filled times" and "right filled times", then they
should be binary operations; the design should be more square. I would then
call 234-238 "left filled bowtie", and so forth.

Hans Aberg
* Email: Hans Aberg <mailto:haberg@member.ams.org>