Fixed Points and Solvability of Systems of Fuzzy Relation Equations

摘要

The problem of solvability of a system of fuzzy relation equations with sup — *-composition is considered in a finite semilinear space over a residuated lattice. In this setting the problem of solvability given above is similar to the problem of solvability of a system of linear equations in the form Ax = b. We put emphasis on a right-hand side vector b and consider the problem of solvability as a problem of characterization of all vectors b for which the original system of (fuzzy relation) equations is solvable. We prove that a system of equations with sup —*-composition is solvable if and only if b is a fixed point of the shrivel operator (introduced in this paper). Moreover, a set of all fixed points is a semi-linear subspace of an original space. Some other results are presented as well.