Subeigenvectors and supereigenvectors of fuzzy matrices

A fuzzy algebra is a triple (B,⊕,⊗), where (B,≤) is a nonempty, bounded, linearly ordered set and a⊕b=max⁡{a,b}, a⊗b=min⁡{a,b} for a,b∈B. A vector x is said to be a λ-eigenvector of a square matrix A if A⊗x=λ⊗x for some λ∈B. The aim of the paper is to solve subeigenproblem and supereigenproblem for some λ∈B, that is to find a solution x of A⊗x≤λ⊗x and A⊗x≥λ⊗x, x is called subeigenvector and supereigenvector, respectively. The problems are related to and motivated by the similar problems of tropical linear algebra. In this paper the properties of subeigenvectors and supereigenvectors are described and the values λ associated with subeigenvectors and supereigenvectors are characterized. As a consequence, efficient algorithms for checking all equivalent conditions are introduced.