/sci/ - Science & Math

>>9041713
If we have no distributivity at all we can in fact write down many structures that work easily.
Take the cyclic group [math]C_2[/math] twice and let [math]K=\{0,1,a\}[/math]. Then [math]a=a^{-1}=-a[/math],
[math]0*a=0*0 = 0*1 =0[/math] and [math]1+a=1+1 = 1+0 = 1[/math].
This is essentially consistent, but boring, because you can't relate + and * in any way. This makes the notion of a field extremely unique, in a way. Two groups in which one has an extra element can produce a consistent algebra. Having two "interlocking" groups only yields trivial algebras. The next question would be whether you can make consistent algebras with more than 2 groups/binary operations that have more than one unique element in one set. However, that is far from my field of expertise and the notation becomes tedious.