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Re: 0^0=1
Posted:
May 3, 2012 2:48 AM
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Virgil writes:
> In article
> <8c8fc0a4-4e89-4c18-9df6-f2e79e2feb6e@z17g2000yqf.googlegroups.com>,
> 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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