Tropical semimodules of dimension two

The tropical arithmetic operations on $\mathbb R$ are defined as $a\oplus b=\min\{a,b\}$ and $a\otimes b=a+b$. In the paper, the concept of a semimodule is discussed, which is rather ill-behaved in tropical mathematics. The semimodules $S\subset\mathbb R^n$ having topological dimension two are studied and it is shown that any such $S$ has a finite weak dimension not exceeding $n$. For a fixed $k$, a polynomial time algorithm is constructed that decides whether $S$ is contained in some tropical semimodule of weak dimension $k$ or not. This result provides a solution of a problem that has been open for eight years.