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Secret
Secret
12:01 PM
Properties of distributive algebras If an algebra is a ringnoid for all elements, (i.e. is distributive) then multiplication is a homomorphism of addition.
Proof: Let the ringnoid be $S$. Given that $\forall a,b,c, a(b+c)=ab+ac$ then by the definition of homomorphism, multiplication by $a$ is a homomorphism
 
 
3 hours later…
Secret
Secret
2:39 PM
Theorem 9 also implies if $x1=1$ and $1$ is a left identity, then $xy=y$ for all $y$ and hence turning $x$ into an identity of the same type as $1$
 
Secret
Secret
3:18 PM
Theorem $\Omega_1$: Division by zero no-go theorem (Associativity): Finite associative division by zero is not interesting
Proof: Suppose a associative division by zero algebra $S$ has a pair of inverses $q0=1$ or $0q=1$. Then the left semigroup action of $0$ and $q$ combined in an appropriate way given any element $x\ in S$ must be equivalent to the action of the multiplicative identity $1$ (may be one sided and may be more than one). That is, consider the following associative laws:
$$0qx=1x$$ or $$0qx=x1$$ or $$q0x=1x$$ or $$q0x=x1$$
Then the left semigroup action due to multiplication of 0 followed by q (or vise versa depending on what type of zero inverses and multiplicative identity is present) must be in…
 

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