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
[edit]A commutative ring is a Heyting field if it is a field in the sense that
- Each non-invertible element is zero
and if it is moreover local: Not only does the non-invertible not equal the invertible , but the following disjunctions are granted more generally
- Either or is invertible for every
The third axiom may also be formulated as the statement that the algebraic "" transfers invertibility to one of its inputs: If is invertible, then either or is as well.
Relation to classical logic
[edit]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
[edit]An apartness relation is defined by writing if is invertible. This relation is often now written as with the warning that it is not equivalent to .
The assumption is then generally not sufficient to construct the inverse of . However, is sufficient.
Example
[edit]The prototypical Heyting field is the real numbers.
See also
[edit]References
[edit]- Mines, Richman, Ruitenberg. A Course in Constructive Algebra. Springer, 1987.
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