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Re: Binary Relations, draft 1

At 17:36 +0100 1998/11/16, Taco Hoekwater wrote:
> HA> I think that the blackboard bold characters should be added to
>Unicode, for
> HA> the reason that they are used in mathematics as symbols with semantic
> HA> meaning, and not as a special style font. For example, it is pretty much
> HA> standard to use blackboard bold R, Z denoting the sets of reals and
> HA> integers, and it is becoming to be more common to use say blackboard bold
> HA> small letters to denote certain fixed objects (such as blackboard bold i,
> HA> j, k to denote the imaginary units in quaternions).
>
>They are all included in the submission tables. But they are separate
>from the geometrical symbols (at least in the source) in the sense
>that they are very much tied in with the text font. Using CM
>Blackboard next to Times is ugly.

This does not matter much, because when letters are used in math mode, they
are typeset with extra space around, so that they stand out singly. So
strictly speaking, the submission should have a special set of regular
letters, plus the undotted i,j, for mathematical use to indicate individual
mathematical variables -- that is, if one should keep the idea "one
character = one meaning".

On the contrary, abbreviated names, such as "sin", "cos", "Hom", "Ext", and
so on, are typeset as regular text words (more densely, using ligatures),
with a style (upright/slanted, sans serif/serif, bold/non-bold) added.

>The included alphabets in the submission are "BlackBoard", "Script"
>and "Fraktur", btw. I will probably design (or steal/borrow) a
>"Calligraphic" as well. They are the very last thing on my list,
>because there are no problems with the meanings (A blackboard R is a
>blackboard R), so most of the time will be spent in the real design
>(which will be rather tricky), instead of on discussions. We need the
>discussions for almost anything else, so it was safer to do those
>first (more time == more discussion == less chances of mistakes).

The question on what different math styles to use was discussed in the
LaTeX3 project group. I was able to identify more styles that
mathematicians want to use than is currently available in TeX/AMS-fonts:

  First there is a regular font like times with serifs, coming in styles
upright/italics, bold/non-bold. The there is a sans serif font, like
helvetica, also coming in styles upright/slanted, bold/non-bold. (A total
of eight variations.)

  Then there are the "BlackBoard Bold" and the "Fraktur", which only comes
in one variation each, as alternatives do not look good. (A total of two
variations.)

  Then, for the script fonts, there appears two variations, as you say: The
AMS-fonts "American style" Calligraphic, which is not so scripty, and the
"European style" Script, which is a great deal more scripty. I think that
the latter one is in reality a variation of the handwritten script used in
some European countries. Then these two type of script fonts should exist
both as non-bold (plain) and bold. (A total of four variations.)

  So, summing it up, there should be a total of fourteen of these
_letters_. As for the numbers and some common symbols, I do not know: These
might be needed in bold/non-bold variations for some purposes, but I do not
think that the other styles are used in mathematics. Some may want to use
Blackboard Bold numbers, I think, but I doubt it is used for the other
fonts.

  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/>