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lojban and group theory
For reference, I'm using the text *Topics in Algebra* by i. n. herstein,
Wiley pub. as my reference for group, ring, and field theory.
On Thursday, January 24, 2002, at 04:51 pm, John Cowan wrote:
Brook Conner wrote:
(mathematically speaking, IEEE floating point numbers are none of
these,
are not even a group (I don't think NaN has an additive inverse, and
Inf
has very peculiar behavior)).
The additive inverse of NaN is NaN, I think, depending on your
definition of inverse: NaN + NaN is not 0, but 0 - NaN is definitely
NaN.
A group is a set G and an operation (usually denoted +) such that:
1. For a,b in G, a + b is in G. IEEE FP does that.
2. For a, b, c in G, a + (b + c) = (a + b) + c). This is associativity.
IEEE FP does this.
3. There exists an element 0 s. t. a + 0 = 0 + a = a. You can prove that
0 is unique, though it isn't part of the definition. If a is NaN, IEEE
does that.
4. For any a in G, there exists one element b s. t. a + b = 0. NaN + NaN
is NaN, and NaN - NaN is NaN. There is nothing you can add to NaN and
get 0.
Ergo, strictly speaking, IEEE FP numbers do not form a group in the
algebraic sense. They have lots of other useful properties, like
commutativity (a+b = b+a, something not necessarily true for a group).
So when looking at the semantics of lojban in a programmatic sense, we
need to be careful about exactly what we say those semantics are. And as
all this IEEE bumpf shows, to some extent, the right
formality/expressiveness/utility balance is domain-specific.
Mathematicians would not be satisfied if mekso *had* to be IEEE floats
or even doubles, though your average Perl programmer would probably be
just fine with that.
As for not being a group, if they weren't a group over the defined
IEEE operations, that would mean that something not an IEEE-float
was being delivered, which is self-contradictory, since every
bit combination has an IEEE meaning.
No, the fact that every bit combination means something just means the
operation is closed. That's part 1 of the definition of a group.
Another problem of semantics is choosing the subset of the vocabulary
and making sure the user knows what it is. Are you going for an
imperative model? Lots of "ko" running around.
"ko" can be defaulted, though, if that's a convention established
between speaker and listener.
Ah, missed that. Thanks.
Brook