this post was submitted on 12 Mar 2025
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There's not much coherent algebraic structure left with these "definitions." If Ωx=ΩΩ=Ω then there is no multiplicative identity, hence no such thing as a multiplicative inverse.
No; 1 is the multiplicative identity.
1Ω=Ω, and for all x in C 1x=x. Thus, 1 fulfills the definition of an identity.
1 = Ω0 = Ω(Ω + Ω) = ΩΩ + ΩΩ = Ω + Ω = 0
so distributivity is out or else 1 = 0
Correct; multiplying by Ω doesn't distribute over addition.
Distributivity is a requirement for non associative algebras. So whatever structure is left is not one of those