Boolean semifield - Knowlpedia

Example

The Boolean semifield is

\mathbb B=\{0,1\},\qquad a+b=\max\{a,b\},\qquad ab=\min\{a,b\}.

Equivalently, addition is logical OR and multiplication is logical AND. It is an idempotent semifield because $1+1=1$ and $1$ is the only nonzero element.

Order and universal behavior

The natural order is $0<1$ , and addition is join. For every nontrivial idempotent semiring $S$ , the map $\mathbb B\to S$ sending $0$ to $0$ and $1$ to $1$ is the unique unit-preserving semiring homomorphism from $\mathbb B$ .

Not the Krasner hyperfield

The Boolean semifield and the Krasner hyperfield have the same carrier, zero, one, and multiplication, but not the same addition:

1+_{\mathbb B}1=1,\qquad 1\boxplus_{\mathbb K}1=\{0,1\}.

The first is a single-valued semiring operation; the second is a hyperaddition containing the additive inverse relation.

References

Grigori L. Litvinov, Viktor P. Maslov, and Grigori B. Shpiz, “Idempotent (Asymptotic) Mathematics and the Representation Theory,” 2002. arXiv:math/0206025.
Matthew Baker and Nathan Bowler, “Matroids over hyperfields,” 2017. arXiv:1601.01204. Relevant: examples distinguishing the Krasner hyperfield from ordinary single-valued algebra.