The spectra of irreducible matrices over completed idempotent semifields

Motivated by some spectral results in the characterization of concept lattices we investigate the spectra of reducible matrices over complete idempotent semifields in the framework of naturally-ordered semirings, or dioids. We find non-null eigenvectors for every non-null element in the semifield and conclude that the notion of spectrum has to be refined to encompass that of the incomplete semifield case so as to include only those eigenvalues with eigenvectors that have finite coordinates. Considering special sets of eigenvectors brings out finite complete lattices in the picture and we contend that such structure may be more important than standard eigenspaces for matrices over completed idempotent semifields.