Fuzzy Relation Equations in Semilinear Spaces

We introduce a notion of an idempotent semilinear space and consider two systems of linear-like equations. These systems are equivalent to systems of fuzzy relation equations with sup-*and \(\inf\)-→ compositions. We show that the theory of Galois connections can be successfully used in characterizing whether these systems are solvable and, if so, finding their solutions sets. Moreover, because the two types of systems of linear-like equations are dual according to this theory, it is sufficient to investigate only one system.