nLab ordered Kock field

Idea

A definition of ordered field in constructive mathematics which uses denial inequality. It is primarily used in synthetic differential geometry.

Definition

An ordered Kock field is a Kock field $R$ with a strict weak order $\lt$ which is not necessarily connected:

1. for all $a \in R$, $\neg(a \lt a)$

2. for all $a \in R$ and $b \in R$, if $a \lt b$, then $\neg(b \lt a)$

3. for all $a \in R$, $b \in R$, and $c \in R$, if $a \lt c$, then $a \lt b$ or $b \lt c$

4. $0 \lt 1$

5. for all $a \in R$ and $b \in R$, if $0 \lt a$ and $0 \lt b$, then $0 \lt a + b$

6. for all $a \in R$ and $b \in R$, if $0 \lt a$ and $0 \lt b$, then $0 \lt a \cdot b$

7. for all $a \in R$, $a \neq 0$ if and only if $a \lt 0$ or $0 \lt a$

Properties

The order in an ordered Kock field is a strong linear affirmative antithesis partial order.

 See also

• Kock field

• ordered local ring

• Archimedean ordered Kock field?

References

• David Jaz Myers, Orbifolds as microlinear types in synthetic differential cohesive homotopy type theory (arXiv:2205.15887)