Increasing Selections from Increasing Multifunctions

We study when the existence of an increasing selection can be derived from the monotonicity of a multifunction w.r.t. an extension of the basic order from points to subsets. The most interesting results emerge when the target is a lattice and monotonicity is interpreted in one of the ways suggested by A.F. Veinott, Jr. An increasing selection exists if (1) the multifunction is weakly ascending while every value satisfies a completeness condition, e.g., is chain-complete, or (2) the multifunction is ascending while the target is a sublattice of the Cartesian product of a finite number of chains.