19 April 2014 ~ 0 Comments

Magmas and Triangles

This week I worked on some kind of magmas. Magmas are the most simple algebraic structures. They are sets with an operation (for example addition) but this operation has no special property except closure. Closure is the property that the addition of 2 elements of a set S gives an element of the set S.

It’s rather easy to create a magma since the only thing you have to do is to choose a set (for example the numbers from 0 to 9) and to create an addition table where the result of the addition of 2 elements is randomized between the elements of the set (for example 1+1=3). Using this method the probability to create a magma is neally 100% when the set S contains a sufficient number of elements.

 

What magmas did I worked with ?

I just used the following operation. Let S be the set of natural integers

if a > 0 then a*b = b + (a-1) f(b)

if a = 0 then a*0 = 0

where f(b) can be choosen at will.

if for each b in intergers, f(b) = b then we have the multiplication everyone studied at primary school. if f(b) <> b for some b then we start to have special properties.

This operation is a kind of generalization of the well known multiplication.

 

Why is it useful to work with magmas ?

In my case I use magmas to work on some triangles in groups. I discovered some times ago that there were triangles in groups. I conjectured some ideas and then worked on demonstrations of these conjectures. For one of my first conjecture (in the case of cyclic groups), I used presentation of groups. For the group of quaternions, I used presentations too.

But to understand why there were some triangles in a group, I have to understand which special property of a group creates triangles. The good way to do this is to use algebraic structure with no special property and to add little by little some new properties. This is what I’m working on.

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