Natural order of an idempotent semiring

Statement

Let $S$ be an idempotent semiring. Its natural order is

a\le_S b\quad\Longleftrightarrow\quad a+b=b.

This relation is a partial order, $a+b$ is the join $a\vee b$ , and both addition and multiplication are monotone in each variable.

Why it is an order

Idempotence gives reflexivity, commutativity gives antisymmetry, and associativity gives transitivity. Distributivity proves monotonicity of multiplication: if $a+b=b$ , then $ca+cb=c(a+b)=cb$ , so $ca\le_S cb$ , and similarly on the right. The additive identity $0$ is the least element.

Tropical order warning

For the max-plus tropical semifield, $\le_S$ is the usual order: $\max(a,b)=b$ exactly when $a\le b$ . For the min-plus presentation,

a\le_S b\quad\Longleftrightarrow\quad \min(a,b)=b,

so the natural order is the reverse of the usual numerical order. Switching between max-plus and min-plus therefore reverses the order even though the two presentations are isomorphic after negating finite elements.

References

Grigori L. Litvinov, Viktor P. Maslov, and Grigori B. Shpiz, “Idempotent (Asymptotic) Mathematics and the Representation Theory,” 2002. arXiv:math/0206025. Relevant: canonical order and idempotent addition.