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Topic: Quasi-fields
Replies: 38   Last Post: Jul 24, 2011 7:21 PM

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 William Elliot Posts: 1,948 Registered: 5/30/08
Re: Quasi-fields
Posted: Jul 17, 2011 9:06 AM

On Sun, 17 Jul 2011, Frederick Williams wrote:

> In Dauben's biography of Abraham Robinson (page 103 of the first
> printing) a quasi-field is said to be a commutative field with
> distribution
>
> a(b + c) = ab + ac (1)
>
> replaced by
>
> a(b_1 + b_2 + ... + b_n) = ab_1 + ab_2 + ... + ab_n. (2)
>
> But, if in (2) we put b_3 = ... = b_n = 0, and apply the rules
>
> x + 0 = x and x0 = x
>

Why does x0 = x?

> we get (1), don't we?

You can't use x0 = 0 until you prove it for quasi-fields.

> So a quasi-field is a field.

No, unless n = 2. What you get is a0 = na0; 0 = (n-1)a0;
a(b + c) = a(b + c + (n-2)0) = ab + ac + (n-2)a0 = ab + ac - a0.

a0 = ab + ac - a(b + c)

> Dauben writes that quasi-fields have the property
> that the zero element for addition posses an inverse.
>

If for all a, a0 = 0, then it's a field.
If some a with a0 /= 0, then 1 = (a0)^-1 a0 = 0^-1 a^-1 a0 = 0^-1 0.

1*0 = 0; 2*0 = (1 + 1)0 = 1*0 + 1*0 - 1*0 = 0.

> I'm intrigued!

What's an example of a quasi-field?

> I know nothing about these things beyond what Dauben
> writes. Googling 'quasi-field' finds
> http://en.wikipedia.org/wiki/Quasifield, but that is something else.
> Dauben refers to:
>
> Robinson, 'On a certain variation of the distribution law for
> commutative algebraic field' Proceedings of the Royal Society of
> Edinburgh, 61, 1941, pp 93-101.
>
> to which I have no access.
>
> --
> When a true genius appears in the world, you may know him by
> this sign, that the dunces are all in confederacy against him.
> Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting
>