I was thinking about some similarities in the definitions of group and field, and if it would be possible to generalize in some sense, like follows.(adsbygoogle = window.adsbygoogle || []).push({});

A field is basically a set F, such that (F,+) is a commutative group with identity 0, and (F-{0}, .) is a commutative group with identity 1, and . distributes over +, that is a.(b+c) = a.b + a.c

If I call n the number of operations in the algebraic structure, then n=2 for fields, if I set n=1 there is left only the first operation + (and no distributive law) so I get the definition of a commutative group.

Is it possible to generalize and get some consistent definition of an "n-field"? That is, for example, a 3-field has 3 operations (+, . ,*) with 3 distinct identities (0,1,e) such that (F-{0,1},*) is a commutative group with identity e and

1) a.(b+c) = a.b + a.c (. distributes over +)

2) a*(b.c) = a*b . a*c (* distributes over .)

I can't find any inconsistency, but neither can I construct an example of such a 3-field.

If it could be possible, one in principle could prove things about n-fields, and then letting n=2 one would get a proof about fields, and letting n=1 one would get a proof about commutative groups (that's the goal I had in mind).

Can anyone construct an example of a 3-field, or show is has no sense at all?

Thanks.

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# Groups, Fields, and beyond

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