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Re: Math Arrows and Harpoons

At 17:42 +0100 1998/11/11, Taco Hoekwater wrote:
> HA> In general, all symmetry cases for binary relations and arrows
> HA> and such should be added, becuase at some point one may need it
> HA> in mathematics: It proves to be difficiult to find such symbols
> HA> which are good and readable, so these symbols will surely show up
> HA> in new contexts.
>
>I would prefer something a little bit more specific. There are so many
>possible symmetry cases that we might end up with a full font that
>contains only harpoons...
>
>Lots of the "missing cases" are already in the STIX tables for
>precisely this reason, but there is just not enough room to capture
>all possible cases.

I think this is the problem with treating such symbols as characters and
putting them into a character font: One should have some more general
underlying mechanism describing them.

  In the absence of this, perhaps there should be two fonts, one for arrows
and symbols which are complete, and one for arrows described as components
(head, tail, stem).

> I added 129 (leftzigzagarrow), 171 (shortleftarrowrightarrow), 224
>(leftdottedarrow).

  For 129 and 224, this seems reasonable: If I were to use the right
pointing version, I would probably want to use the left pointing version.
The same applies to 171, but I do not know what it is used for.

>134--137 are arrows pointing to a "filled square". I interpreted this
>using the accompagnying .gif files as "square" meaning "diamond".

  This looks nice, even though I have no idea what the original intended
use. It is always nice to be able to create new types of arrows, so with a
font that could design arrows in their components, I could think of
variations with both filled and unfilled diamonds, squares, circles.

>The sidebearings for the fish tails and the arrow tails are unclear to me.
>Are these extensions to other characters or are they stand-alone
>relations?

  If you mean characters such as 88 and 116, I feel pretty sure they are
exclusively used as a components of a larger character and never standalone.

  I do not immediately recall that I have seen the fish tail 88 in print,
though; but it seems me that it should be a double hook (the latter which
sometimes is used to denote a monomorphism). (For example, 80 has a single
hook.)

>Are chars 193--198 correct interpretations X "over" Y

  I am not sure what you mean here: interpretation relative to what? If you
simply mean if an arrow stem drawn above another arrow stem, this looks OK
to me. There is the variation, often used when drawing by hand, that one of
the stems get a small bend, but I think that is rare in print. It would
look like
                   |
     ---->   and   C    (Illustration of two arrows, both having a
                   |     continuous, single line, uninterrupted stem.)
                   v
superimposed over each other (with the bend "C" over the first arrows stem).

>Greg Kuperberg wrote:
>
>>I can never decide what kind of harpoon, arrow, or pigtail is most
>>appropriate to denote a group action G -> X.
>
>It would be great if someone could help him (and me).

This is his problem: I think there are many ways to avoid a special arrow,
one may write G -> End(X), or G -> Aut(X), or G x X -> X, or something,
with a plain vanilla arrow.

But it is always nice to have a good set of arrows to choose from: Perhaps
if the arrows are built up by its components, this might be possible.

>There are some other "symmetry cases" that I did not do, like the missing
>"similar above left arrow". Should these be added?

  I wish such cases could be treated by some more general mechanism, and
not just a font: For example, I would use \cong to denote an isomorphism,
whereas, = could be used to denote that a canonical injection is an
identity. Then, in a category theory diagram, these could appear on arrows
in any direction; if the arrow points up/down, then sometimes this symbol
be on the right, sometimes on the left.

  In general, I do not think of an arrow mathematically as a single symbol,
but as a compound symbol with different components: head, tail, stem, and
one (or perhaps two) stem symbol. This is often how these arrows are used:
altering a component adds something to the mathematical semantics.

  Then, when attempting to force this fact of mathematical usage into the
idea of a character font, where each symbol has its own number, one ends up
in a straightjacket. No matter how well the design is carried out arrows
are fit into a character font, some people will end up not finding their
intended use of an arrow in that font.

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