Ordered semiring - Knowlpedia

Definition

An ordered semiring is a semiring $R$ equipped with a partial order $\leq$ such that

a\leq b\Longrightarrow a+c\leq b+c \quad\text{and}\quad ac\leq bc

for all $a,b,c\in R$ . In this knowl, semirings are commutative and unital, so the second condition also covers multiplication on the other side.

Scope

The order is part of the structure; it need not be the natural order of an idempotent semiring, and $0$ need not be the least element. Morphisms of ordered semirings preserve $0,1,+,\cdot$ and the order.

An order generated by specified inequalities is closed under reflexivity, transitivity, addition, and multiplication. This presentation viewpoint is the additive-relation layer used by ordered blueprints.

Examples

Every semiring has the discrete order, $a\leq b$ exactly when $a=b$ .
$\mathbb{R}_{\geq 0}$ with its usual operations and order is an ordered semiring.
An idempotent semiring has the canonical order $a\leq b\iff a+b=b$ , but reversing that order is also common in tropical conventions.

References

The convention follows Baker–Lorscheid, The moduli space of matroids, §2.6.