The spectra of reducible matrices over complete commutative idempotent semifields and their spectral lattices

Abstract

Previous work has shown a relation between L-valued extensions of Formal Concept Analysis and the spectra of some matrices related to L-valued contexts. To clarify this relation, we investigated elsewhere the nature of the spectra of irreducible matrices over idempotent semifields in the framework of dioids, naturally ordered semirings, that encompass several of those extensions. This initial work already showed many differences with respect to their counterparts over incomplete idempotent semifields, in what concerns the definition of the spectrum and the eigenvectors. Considering special sets of eigenvectors also brought out complete lattices in the picture and we argue that such structure may be more important than standard eigenspace structure for matrices over completed idempotent semifields. In this paper, we complete that investigation in the sense that we consider the spectra of reducible matrices over completed idempotent semifields and dioids, giving, as a result, a constructive solution to the all-eigenvectors problem in this setting. This solution shows that the relation of complete lattices to eigenspaces is even tighter than suspected.

Keywords:

• Dioids
• complete idempotent semifields
• all-eigenvectors problem
• spectral order lattices
• eigenlattices

Acknowledgements

We would like to thank all reviewers for their extensive help in improving this paper.

Notes

No potential conflict of interest was reported by the authors.

1 Right and left eigenlattices will only be distinguished by indexing them with the standard notation for a right and left eigenvalue, respectively.

2 Essentially, this introductory material follows Davey and Priestley (2002).

3 Recall that these actually depicts the irreflexive, transitive reduction of the orders they represent.

4 Recall that it can be represented as a Hasse diagram by means of its transitive–reflexive reduction.

Additional information

Funding

FJVA was partially supported by EU FP7 project LiMoSINe [contract 288024] for this research. CPM has been partially supported by the Spanish Government-Ministry of Economy and Competitiveness project TEC2014-61729-EXP and TEC2014-53390-P.