# Name for a field without the requirement of having an additive inverse?

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.

I think you are looking for semifields in this sense:

In ring theory, combinatorics, functional analysis, and theoretical computer science (MSC 16Y60), a semifield is a semiring (S,+,·) in which all elements have a multiplicative inverse.

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:

Golan, Jonathan S. Semirings and their Applications. Springer Science & Business Media, 2013.

and

Gondran, Michel, and Michel Minoux. Graphs, dioids and semirings: new models and algorithms. Vol. 41. Springer Science & Business Media, 2008.

I have never read this but it looks like something to consult:

Glazek, Kazimierz. A guide to the literature on semirings and their applications in mathematics and information sciences: with complete bibliography. Springer Science & Business Media, 2002.

• The Wiki article isn't perfect, but you get the drift. – rschwieb Dec 6 '18 at 14:32
• I'll take a look into Golan's book. But it would be the definition you gave for division semiring with an identity, so it is a monoid aditively, instead of just a semiring. – Garmekain Dec 6 '18 at 15:50
• @Garmekain I'm pretty sure the tendency is to define semirings on abelian monoids not just semigroups. I think the first two references above require that, anyhow. – rschwieb Dec 6 '18 at 16:15

For the example you gave, there is a name for that special semiring, and it is called the two-element boolean algebra or boolean semiring.