Definition are at the very first page of this room (should be just 2 clicks away)
Next:
Recall that in all rings, semirings, near rings and near semirings, (which include familar things like the reals, the complex etc.) they all have the property:
Let the aforementioned ring type structures be $R$ $$\forall n \in R, 0r=0$$
This means, in all these ordinary systems, we cannot have a multiplicative inverse of zero e.g. $q0=1$ as a contradiction will result
In order to allow that, we must relax this theorem, commonly known as the property of zero.
That is, in the algebra that you will be seeing here, 0 is not a multiplicative absorber. However it can still be an identity
To begin let us revise all known ways to proof the above theorem:
Let's take an example:
$0=0*1=0*(1+0)=0+0*0=0*0$
so as you can see, $0^2=0$. In order to allow $0^2\neq 0$, using the above diagram, we need to discard axioms from fields such that the proof cannot be completed regardless of how you do it
One possible set of axiom to discard will be: right multiplicative identity and right distributive law
(or R_dist., Rid* in the diagram)
You do something similar in order to prevent $0n=0=n0$
That's how it initially started when I tried to introduce division by zero into the structure. Later on, as shown in the previous discussions above, I have proved a couple of theorems that characterise these structures
These algebras are very different from the usual algebras we came across because they are very unatural as tobias puts in. We literally need to break the usual rules to make them exists. It seems, however that the interesting ones tend to be not associative, however
I suspect the above proof diagrams can be formalised as something in category theory, but I am not very good at that right now to use them yet
The whole project started simply because I want to see whether one can really not able to divide by zero in all conceivable mathematial structures
thus nobody had any idea if these unatural objects will ever be useful. The machinations in investigating them, hwoever seemed to help on investigating rock paper scissors commutative magmas
