Idempotent semifields and lattice-ordered groups

Statement

The category of commutative idempotent semifields with unit-preserving semiring homomorphisms is equivalent to the category of lattice-ordered abelian groups with $\ell$ -group homomorphisms.

The functor from semifields sends $S$ to its multiplicative group $S^\times=S\setminus\{0_S\}$ , equipped with the natural order of the idempotent addition. The inverse construction adjoins a new bottom element $\bot$ to an $\ell$ -group $\Gamma$ , uses the group law as multiplication, and defines

x\oplus y=x\vee y,\qquad 0_S=\bot.

Why the nonzero elements form a lattice

If $a,b\in S^\times$ , their join in the natural order is $a+b$ , which is again nonzero because it lies above $a$ . Since multiplication by any nonzero element is an order automorphism, meets can be recovered from joins: in multiplicative notation,

a\wedge b=\bigl(a^{-1}\vee b^{-1}\bigr)^{-1}.

Thus $S^\times$ is an abelian $\ell$ -group, not merely an ordered group.

Constructing the semifield

Let $\Gamma$ be an abelian $\ell$ -group, written multiplicatively with identity $1$ . Extend multiplication to $S=\Gamma\sqcup\{\bot\}$ by making $\bot$ absorbing, and set addition equal to join after declaring $\bot\leq\gamma$ for every $\gamma\in\Gamma$ . Translation-invariance of the lattice order gives distributivity:

a(b\vee c)=ab\vee ac.

Every element of $\Gamma$ remains multiplicatively invertible, so $S$ is an idempotent semifield.

Morphisms and examples

A semiring homomorphism restricts to a group homomorphism preserving joins; an $\ell$ -group homomorphism extends uniquely by sending $\bot$ to $\bot$ . These assignments are mutually inverse on morphisms as well as on objects.

The trivial $\ell$ -group corresponds to the Boolean semifield. The totally ordered additive group $\mathbb R$ corresponds to the max-plus tropical semifield. Partially ordered $\ell$ -groups give idempotent semifields whose natural orders are not total.

Convention warning

Here “characteristic one” means additively idempotent. The adjoined $\bot=0_S$ is the semiring's additive identity and is not the identity element of the group. Omitting it produces a parasemifield, not the unital semiring used elsewhere in this corpus.

References

Guillaume Tahar, “Ordered algebraic structures and classification of semifields,” 2017. arXiv:1709.06923. Relevant: the equivalence between characteristic-one semifields and lattice-ordered groups.
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 semifields.