Distributivity inequalities of monotonic operations

The aim of this work is to complement recent results about distributive binary operations. Our investigation was motivated by non-distributive logical connectives and their generalizations in fuzzy set theory. In spite of the title we neither solve the concerned functional inequalities nor look for characterization of the solutions of such inequalities. We only lay bare conditions for their fulfilment. For monotonic operations with neutral or zero element we present the sufficient conditions to obtain the subdistributivity or superdistributivity in the case of the lack of distributivity for the considered pairs of operations. In particular we describe families of increasing binary operations subdistributive or superdistributive with respect to idempotent uninorms and nullnorms.