$\boldsymbol{T}$-algebra homomorphisms between rational function semifields of tropical curves

Abstract: We prove that an injective $\boldsymbol{T}$-algebra homomorphism between the rational function semifields of two tropical curves induces a surjective morphism between those tropical curves, where $\boldsymbol{T}$ is the tropical semifield $(\boldsymbol{R} \cup { -\infty }, \operatorname{max}, +)$.