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Re: Binary Relations, draft 1

>>>>> "HA" == Hans Aberg <haberg@matematik.su.se> writes:

 >> Currently, my "normal" triangles are actually closed versions of
 >> "less" and "greater". So, I would then like to use the real
 >> equilaterals as variations.
 HA> So then these should merely be renamed.

 HA> The reasoning is like this: I am not sure that there should be equilateral
 HA> triangles used as binary relations; perhaps as operators, but not as
 HA> relations.

 HA> AS relations they shoudl be prolonged. They then need only look good, not
 HA> be say a closed <. 

Just happened to be convenient. 

 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.
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?

 HA> On 175/176, I would want to have variations with the = negated ("not
 HA> equal") under the \subset: There is some confusion in math as to which
 HA> symbol is the subset symbol (180 or 182), so if one thinks of \subset
 >> as an
 HA> implication and not a <, this is nice to have for clarity.
 >> All negations are to appear in the next font.

 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

 >> >> Is "mostpos" (5) the same shape as "ac" (224)?
 >> >> Is "congruence" (159) the same shape as "race" (222)?
 HA> For these, one can always think of the curl starting up/down, as in
 HA> 222/224. But I do not know if that is how people are using them.
 >> Your explanation is unclear to me. My english is not all that good.

 HA> I am also trying to learn English. :-)

 >> If
 >> they *are* different, then it might be nice to design them to be
 >> different as well. But which one of the two should be changed and how?

 HA> I only mean that a curl can always look as
 HA>      _                _
 HA>     | |_|    or    |_| |
 HA> starting up or down.

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

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.

 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.

Greetings, Taco