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Topic: 0^0=1
Replies: 145   Last Post: May 7, 2012 8:08 AM

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 Jussi Piitulainen Posts: 289 Registered: 12/12/04
Re: 0^0=1
Posted: May 3, 2012 2:48 AM

Virgil writes:

> In article
> jbriggs444 wrote:
>

> > Addition and multiplication are both group operations over the
> > natural numbers.

>
> Multiplication is NOT a *group* operation over the natural numbers,
> nor over any set of two or more numbers of which one is zero.

Correct. It's a monoid operation. Inverses were not used in the good
analogy where the following defines the powers x^n, n = 0, 1, 2, ...,
for an associative binary operation @ with identity e.

x^0 = e
×^n = x^(n - 1) @ x for n = 1, 2, 3, ...

For the addition of numbers, this defines x*n; for the multiplication
of numbers, or square matrices, the powers x^n; for the concatenation
of strings, repetition; for the idempotent operations - union,
intersection in a powerset; gcd, lcm; min, max in the presence of a
smallest and a largest element - it's not so interesting but it's
valid; it is good for the composition of mappings S -> S and of binary
relations in S; when there is an e, x^0 = e is generally rock solid.

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