Endomorphisms of semimodules over semirings with an idempotent operation

For an arbitrary endomorphism $A$ of the free semimodule $K^n$ over an Abelian semiring $K$ with operations $\oplus$ and $\odot$ it is shown under the assumption that $\oplus$ is idempotent (and under certain other restrictions on $K$) that there exists a nontrivial “spectrum”, i.e., there exist a $\lambda\in K$ and a nontrivial subsemimodule $J$ such that $Af=\lambda\odot f$ for any $f\in J$. The same result is also obtained for endomorphism analogues of integral operators (in the sense of the theory of idempotent integration). In terms of this spectrum investigations are made of the asymptotic behavior of endomorphisms under iteration and of convergence of the “Neumann series” appearing in the solution of the equations $y=Ay\oplus f$. The simplest examples are connected with the semiring $\{K=R\cup \{-\infty\},\ \oplus=\max,\ \odot=+\}$ and arise, for example, in dynamic programming problems.