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fishfry science forum Guru Wannabe
Joined: 29 Apr 2005
Posts: 299

Posted: Mon Jul 17, 2006 2:58 am Post subject:
Re: Operator Overloading (Sets and Numbers)



In article <1153081607.125821.107190@35g2000cwc.googlegroups.com>,
"Doug Goncz" <DGoncz@aol.com> wrote:
Before going on, some context.
Computer programming is not math and vice versa. That said, there is
definitely some operator overloading going on in math, though they don't
call it that. But it's not complicated.
Here's an example. When one is constructing the naturals, integers,
rationals, and reals out of the empty set and the standard set of
axioms, one typically does gets to a certain stage ... say we've
constructed the integers. Then we invoke a construction known as the
field of quotients of an integral domain, to get the rationals.
Now the rationals contain a subring that is isomorphic to the integers.
So the math books all say, "Let's identify the subring of the rationals
that's isomorphic to the integers, with the integers themselves."
So now, when I say 1 + 1 = 2, is that '+' the addition operator of the
integers? Or of the rationals?
That's operator overloading as used in math. It's an essentially trivial
technicality to deal with the fact that the addition operator on the
integers is not the same mathematical object as the addition operator on
the rationals, which is different again from the addition operator on
the reals. But we use the same symbol, '+'. 

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DGoncz@aol.com science forum Guru Wannabe
Joined: 25 Oct 2005
Posts: 122

Posted: Sun Jul 16, 2006 8:26 pm Post subject:
Operator Overloading (Sets and Numbers)



Hello, sci.math readers.
A search in Google Groups records of sci.math for "operator
overloading" by relevance returns no thread with this Subject.
http://groups.google.com/group/sci.math/search?q=%22operator+overloading%22&start=0&
I have a personal slant to operator overloading and I want to see how
my ideas fit into what is in common use.
The Fundamental Theorem of Arithmetic is that every number has a unique
prime factorization, and every prime factorization evaluates to a
unique number, as far as I understand it. Also, it is implied that the
infinite set of primes is the only subset of Z+ having this quality of
relating numbers in Z+ to unique, representative sets.
In an algebra where X represents the unique multiset prime
factorization (UMPF) of x, and Y the UMPF of y, and upf is a function
returning the UMPF of an expression, we know:
X /\ X = X
X \/ X = X
upf(gcd(x,y)) = X /\ Y
upf(lcm(x,y)) = X \/ Y
x /\ y is undefined
x \/ y is undefined
upf(x+y) could be anything but maybe gcd(x+1, y+1) divides it.
upf(xy) could be anything but we know gcd(x1, y1) divides xy, from
Kyle, so maybe it could be factored easily.
upf(x*y) = X + Y
upf(x/y) = X  Y
X*Y is undefined
X/Y is undefined
upf(x^y) = X * y (multipying each multiplicity in X by y)
upf(x^(1/y) = X / y (dividing each multiplicity in X by y, which can
produce a real...)
log base x of y = the multiplicity of x in Y (I think only if x is a
prime...)
X*Y might be something
X/Y might be something
Do you see the pattern? Addition, multiplication, and exponentiation,
and their inverses, in Z (in Z+?) have (some) corresponding operations
one level *down* in precedence among the UMPFs..
Is there an algebra overloading these operators so that;
x /\ y = (x,y) = X /\ Y
x \/ y = [x,y] = X \/ Y
x + y = X + Y
x  y = X  Y
x * y = X * Y
x / y = X / Y and
x ^ y = X ^ Y
or something similar, or is it important to keep the line drawn between
sets and numbers, and keep this downstepping relationship?
If it's important to keep the existing relationship, can you fill in
the blanks for set operators corresponding to arithmetic operators?
Can we say P moves all operators down one step, in the way it acts to
generate factorings? If so, we have a set that is an operator, acting
on numbers, sets, and operators, and I have *never* heard of *that*!
Can we overload the "operator", P?
Doug Goncz
Replikon Research
Seven Corners, VA 220440394 

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