03:10
@Semiclassical @Slereah Suppose by division by zero, you mean adjoining a multiplicative inverse of zero to the algebraic system. Then I proved the following:
Division by zero algebra Theorem 1: Given any algebraic structure $(S,\cdot,+)$ such that $\cdot$ left distribute over $+$ and the underlying set $S$ arbitrary, and that there exists a left or right multiplicative inverse to a left or right additive identity, the addition structure must have the following form:
(see in my main chat room in the above link for the proof)
But basically, the main thing about zero inverse that really messed up what we knew about numbers, is that multiplicative identities become idempotent under +
which one can easily show that (since the non necessary associative algebraic structure is left distributive)
for any $x$ in the algebra
In other words, natural number addition no longer makes sense in these algebraic structures because you ether get something unchanged, or you frequently have $x+y=y$
I have yet to fully investigate what happens for the multiplication structure, because that one is a lot less trivial for an underlying set of infinite cardinality
In addition, for finite structures, there are around 5 no-go theorems I proved 2 years ago which show that the addition becomes quite trivial since you have for some given $x$, $x+y=y$ for all $y$
(and hence for finite structures, I don't consider that division by zero since every element becomes a left or right absorber)
Meanwhile in the literature, there are only 2 known structures that allows some notion of division by zero:
Wheels and
Meadows.
Wheels modified the distributive law and introduce an involution operator to mean division thus result it to become an extension of rational fields
Whereas meadows deals with the issue of division by zero by introducing some sort of pseudoinverse $yxy^{-1}-(something)=x$
Both structures does not permit a zero inverse. Particular, for wheels, it has been shown in the paper that attempt to put in a zero inverse collapse the wheel into the trivial wheel $\{0\}$
Proposition 4.4 in particular for the claim I made about collapsing into the trivial wheel
Actually I might have misinterpreted proposition 4.4, it only said that if any of the new elements is the same as 0 or 1 then the wheel becomes trivial
Thus I am actually not sure what will happen if we adjoin a zero inverse into wheels
It has come to my attention that
I agree, I did make some mistakes as coming off as hostile in this post. However, it should be noted that the substantive message within this meta post still stands.
The moderators on this exchange clearly have no "chill" at all as they would automatically downvo...
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If you take free bosons they fall into th...
