A Heyting field is one of the inequivalent ways in

constructive mathematics In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove th ...

to capture the classical notion of a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

. It is essentially a field with an

apartness relation In constructive mathematics, an apartness relation is a constructive form of inequality, and is often taken to be more basic than equality. It is often written as \# (⧣ in unicode) to distinguish from the negation of equality (the ''denial inequal ...

. A

commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...

is a Heyting field if ¬

(0=1)

, either

a

1-a

invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...

for every

a

, and each noninvertible element is zero. The first two conditions say that the ring is

local Local may refer to: Geography and transportation * Local (train), a train serving local traffic demand * Local, Missouri, a community in the United States * Local government, a form of public administration, usually the lowest tier of administrat ...

; the first and third conditions say that it is a field in the classical sense. The apartness relation is defined by writing

a

b

a-b

is invertible. This relation is often now written as 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 is. The prototypical Heyting field is the

real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...

References

* Mines, Richman, Ruitenberg. ''A Course in Constructive Algebra''. Springer, 1987. Constructivism (mathematics) {{algebra-stub