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. A commutative ring is a Heyting field if ¬ $(0=1)$ , either $a$ or $1-a$ is invertible for every $a$ , and each noninvertible element is zero. The first two conditions say that the ring is local; the first and third conditions say that it is a field in the classical sense.

The apartness relation is defined by writing $a$ # $b$ if $a-b$ is invertible. This relation is often now written as $a$ ≠ $b$ with the warning that it is not equivalent to ¬ $(a=b)$ . For example, the assumption ¬ $(a=0)$ is not generally sufficient to construct the inverse of $a$ , but $a$ ≠ $0$ is.