Heyting field

A Heyting field is one of the inequivalent ways in constructive mathematics to capture the classical notion of a field. It is essentially a field with an apartness relation.

Definition
A commutative ring is a Heyting field if it is a field in the sense that and if it is moreover local: Not only does the non-invertible $$0$$ not equal the invertible $$1$$, but the following disjunctions are granted more generally
 * $$\neg(0=1)$$
 * Each non-invertible element is zero
 * Either $$a$$ or $$1-a$$ is invertible for every $$a$$

The third axiom may also be formulated as the statement that the algebraic "$$+$$" transfers invertibility to one of its inputs: If $$a+b$$ is invertible, then either $$a$$ or $$b$$ is as well.

Relation to classical logic
The structure defined without the third axiom may be called a weak Heyting field. Every such structure with decidable equality being a Heyting field is equivalent to excluded middle. Indeed, classically all fields are already local.

Discussion
An apartness relation is defined by writing $$a \# b$$ if $$a-b$$ is invertible. This relation is often now written as $$a\neq b$$ with the warning that it is not equivalent to $$\neg(a=b)$$.

The assumption $$\neg(a=0)$$ is then generally not sufficient to construct the inverse of $$a$$. However, $$a \# 0$$ is sufficient.

Example
The prototypical Heyting field is the real numbers.