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Added as examples: $0$, $\mathbb{Z}/2\mathbb{Z}$ and $\mathbb{B}=\{0,1\}$ with $1+1=0$. Proved that they are exactly the boolean rigs of cardinal less or equal than $2$.
I donâ€™t know if boolean rigs in the sense of this entry are always commutative. In The variety of Boolean semirings, they show that they are commutative assuming that $1+x+x=1$. If some noncommutative boolean rig exists, it must be of cardinal $\ge 3$, not be a ring (because boolean rings are commutative) and not verify this equation.
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