I find this structure to be useful in my work, but I cannot find any name for it. I think it might be a "commutative division semiring" but I'm not certain because I cannot find any literature on it.
Basically, I'm looking for a name of some set $S$ where $S$ forms a commutative monoid under addition, the nonzero elements of $S$ an abelian group under multiplication, and the distributive law holds. An example is the set $\{False,True\}$ with logical OR as addition and logical AND as multiplication.
Best Answer
I think you are looking for semifields in this sense:
However, I think the passage I copied contains an error when it says all elements. If you look at the references you'll find that they do exclude $0$.
You'll also find, in Golan's book at least, the name "division semiring" used if commutativity of multiplication is required.
The two most useful resources on semirings that I ever found were these:
and
I have never read this but it looks like something to consult: