Linear Projectors in the max-plus Algebra

In general semimodules, we say that the image of a linear operator B and the kernel of a linear operator C are direct factors if every equivalence class modulo C crosses the image of B at a unique point. For linear maps represented by matrices over certain idempotent semifields such as the (max,+)-semiring, we give necessary and sufficient conditions for an image and a kernel to be direct factors. We characterize the semimodules that admit a direct factor (or equivalently, the semimodules that are the image of a linear projector): their matrices have a g-inverse. We give simple effective tests for all these properties, in terms of matrix residuation.