Semilinear space; Residuated lattice; Solvability of a system of equations; Contraction operator; Dilation operator; Fixed point

The notion of a semilinear space over residuated lattice is introduced. Two problems of solvability of systems of linear-like equations with sup-* or inf--> compositions are considered in a finite semilinear space. We prove that each system of equations is solvable if and only if its right-hand side is a fixed point of the respective contraction or dilation operator. Sets of fixed points are characterized as subsemimodules over respective reducts of the residuated l-monoid.