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Inquisitively,

Edwin

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- Thread starter Edwin
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Inquisitively,

Edwin

- #2

Hurkyl

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Hint: ( what is the zero element? ) (highlight to see)

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mathwonk

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hence it cannot be field unless n is a power of some prime number.

it also cannot have a unit for multiplication, since that could only be the constant function 1, which being constant, is not an automorphism.

it is probably not closed under either addition or multiplication but i have not cgecked it.

try some simple examples, like a quadratic extension. if the group consistssay of id and conjugation, try adding z and zbar.

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Thanks for your help guys!

Best Regards,

Edwin

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Inquisitively,

Edwin

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mathwonk

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what a great questiuon! i hVE NO IDEA!

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mathwonk

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of course for endomorphisms we hVE ddition, multiplication and compositiopn, thTS THREE OPERtions.

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Hurkyl

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Why stop at binary operations, or associative ones? You might want to look into universal algebra. Or maybe model theory.Why did mathematicians not develop more abstract theory for two, three, or n associative binary operations on a set?

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I have at home a book called 'A Modern View of Geometry' (can't recall the author's name just now) in which the properties of projective spaces are gradually developed through a series of finite point models of the axioms. Hand in hand with this, the author sets up an algebraic structure, which functions as a sort of primitive Cartesian geometry. As more axioms are added, the algebra becomes richer and the proto-projective space takes on more of the characteristics we expect from a recognizable space.

The reason I mention this is that the fundamental algebraic object he uses is a TRINARY ring. This is the only place I have seen a trinary ring actually in use as an analytical tool, instead of being the object of the analysis itself.

As a side note, I noticed a very strange coincidence: the 7 point projective space mentioned in the above reference has the same structure as the multiplication table for the octonians, in the sense that each projective line contains exactly 3 points and there is an isomorphic mapping of the unit basis of the octonians to points in the projective space, such that the product of any two elements of the octonian basis (except 1) is precisely the third point on the line containing the two elements that were multiplied.

I don't know what to make of this, but I thought it was an interesting connection between octonians and projective geometry.

The reason I mention this is that the fundamental algebraic object he uses is a TRINARY ring. This is the only place I have seen a trinary ring actually in use as an analytical tool, instead of being the object of the analysis itself.

As a side note, I noticed a very strange coincidence: the 7 point projective space mentioned in the above reference has the same structure as the multiplication table for the octonians, in the sense that each projective line contains exactly 3 points and there is an isomorphic mapping of the unit basis of the octonians to points in the projective space, such that the product of any two elements of the octonian basis (except 1) is precisely the third point on the line containing the two elements that were multiplied.

I don't know what to make of this, but I thought it was an interesting connection between octonians and projective geometry.

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