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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 ¬ {\displaystyle } , either a {\displaystyle a} or 1 − a {\displaystyle 1-a} is invertible for every a {\displaystyle 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 {\displaystyle a} # b {\displaystyle b} if a − b {\displaystyle a-b} is invertible. This relation is often now written as a {\displaystyle a} ≠ b {\displaystyle b} with the warning that it is not equivalent to ¬ {\displaystyle }. For example, the assumption ¬ {\displaystyle } is not generally sufficient to construct the inverse of a {\displaystyle a} , but a {\displaystyle a} ≠ 0 {\displaystyle 0} is.
The prototypical Heyting field is the real numbers.