nLab protoring

Definition

A protoring is a commutative monoid $(M, 0, +)$ with a strict total order $\lt$ such that

• $a \lt b$ if and only if $a + c \lt b + c$.

• The commutative subsemigroup of strictly positive elements $M_+ \coloneqq \{x \in M \vert 0 \lt x\}$ form a semiring $(M, +, 1, \cdot)$.

• $a \lt b$ if and only if $a \cdot c \lt b \cdot c$ and $c \cdot a \lt c \cdot b$.

Examples

• Every ordered integral domain is a protoring.

See also

• ordered group

• ordered ring

• ordered integral domain

• archimedean protoring

• difference protoring

 References

• Davorin Lešnik, Synthetic Topology and Constructive Metric Spaces, (arxiv:2104.10399)