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Re: Binary Relations, draft 1

At 22:56 +0100 1998/11/15, Taco Hoekwater wrote:
> HA> The symbols
> HA>     037 037 22B2 vartriangleleft
> HA>     038 038 22B3 vartriangleright
> HA> should definitely be triangles (straight corners). I think these
>together with
> HA>     039 039 22B4 trianglelefteq
> HA>     040 040 22B5 trianglerighteq
> HA> should be designed as the AMS-fonts
> HA>     \vartriangleleft
> HA>     \vartriangleright
> HA>     \trianglelefteq
> HA>     \trianglerighteq
> HA> that is, somewhat prolonged (not exactly an equilateral triangle). But I
> HA> see that you have 146/147: Perhaps 037/038 are closed variations of TeX
> HA> \succ and \prec; it is always nice having extra relations to choose from.
>
>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.
>
>If the "var" versions have to be triangles, then they will be
>triangles. But see below for a remark on triangles as operators.
>
>If you really like having closed versions of succ and prec, that is
>fine, then I will add those. On the other hand, if the wish to have
>them is only an academic desire to have as many symbols as possible,
>then I object. (If you want weird symbols as "on-the-fly" relations,
>you might as well use a dingbat font).

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

AS relations they shoudl be prolonged. They then need only look good, not
be say a closed <. However, your curled 037/038 look very good. I have
noted that if one writes a paper with several different relations, then it
is difficult to find good variations. So from that perspective, I think a
closed variation of \succ and prec should be added.

> 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.

This is not a negated subset, but a strict subset, so therefore it should
be in this font. (One could also in principle negate the strict subset
symbol.)
> >> "closure" (233) and "asymp" (9) use the symbols from "frown" (54) and
> >> "smile" (55). Is that correct?
>
> HA> You seem to have made the "frown" (54) and "smile" (55) thicker to the
> HA> middle, as of a mouth. I think that this might be incorrect; TeX and
> HA> AMS-fonts seem to not have done that. (Most these relations seem to be
> HA> designed as sans-serif, of equal thickness.) Otherwise I think that
>you are
> HA> not forced to use the "frown" and the "smile" for the other two; just
>make
> HA> them look good.
>
>Amsfonts are (implicit apology to the ams people) *ugly*. Functional,
>but very *ugly*. I will do not consider these fonts to be usable as a
>reference representation of anything. But you might still be right, of
>course. Will check some other fonts.

The only thing that matters is that it looks good in print: When drawing by
hand, I do not thionk that one does anything particular about this.

> >> 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.

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?

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

> >> The naming scheme hints that "bowtie" is a doubly closed
>multiplication sign.
>
> HA> TeX has a binary relation called \bowtie; it looks like bowtie and
>not as a
> HA> double close x (the bowtie is more prolonged). But I do not recall any
> HA> usage.
>
>The table also has operators "left closed times sign" and "right
>closed times sign", that really look like times with a bar on one
>side. For the filled version, these are marked as "relations" instead
>of operators, and those turned out as char 235 and 237.

For the "left closed times sign" |x and "right closed times sign" x| it is
clear that these are operators, often used to denote a trivial extension in
group theory, homological algebra and such theories. For completeness, one
might add an operator "doubly closed times sign" |x|, even though I do not
recall any use of it.

These are clearly a "times sign" with extra strokes on (because that is how
they arise mathematically, as a Cartesian product with some additional
properties). The bowtie looks different though, being prolonged, so
therfore I think it is different from the "doubly closed times sign", and
that the bowtie should only be a relation, not an operator.

> HA> In general, in the few cases where a similar symbol exist both as a
>binary
> HA> relation and as a binary operator (as in the case of AMS-font
> HA> \vartriangleleft and TeX \triangleleft), the relations seems to be
> HA> prolonged relative to the operator.
>
>I also have triangle as a binop (which is equilateral, indeed). The
>current variations are all flagged as relations.

What I mean here is that these symbol variations, binary operator/relation,
are both semantically and graphically different, and when used side by side
a mathematician would recognize the difference: The relation will be
somewhat prolonged relative the operator. That is what the empiricism
seemed to be to me.

  Hans Aberg
                  * Email: Hans Aberg <mailto:haberg@member.ams.org>
                  * Home Page: <http://www.matematik.su.se/~haberg/>
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