What is the real name of the following definition?

**Definition.** A **semifield** is a commutative ring R with an operation (-)^{-1} \colon R \to R such that for all x \in R, x x^{-1} x = x and x^{-1} x x^{-1} = x^{-1}.

This is nice because semifields are models of a purely algebraic theory, so they should have a good category unlike fields.

But also, note that any field can be given a semifield structure where 0^{-1} = 0!

Can you think of any examples of semifields which are not fields? I suppose maybe the obvious answer probably “the free semifield on one generator.” But are there more interesting answers?

Well these seem very close to what are called meadows, though meadows have the additional condition that (x^{-1})^{-1} = x. See this thread by Sridhar Ramesh in response to my question “are there any nice categories of fields?” A nice highlight for me: you can think of meadows as fields valued over arbitrary boolean algebras rather than the usual two-valued algebra.

1 Like

The element x^{-1} x is idempotent, and is a multiple of x, for all x. I guess you need to find rings where such elements exist.

By the way, do you require that (xy)^{-1} = x^{-1} y^{-1} ?

1 Like

Not necessarily. I’m not super invested in semifields, I just came up with the idea and thought it was cool .