On the existence of monotone selections

For a correspondence from a partially ordered set to a lattice, three sets of sufficient conditions for the existence of a monotone selection are obtained. (1) The correspondence is weakly ascending while every value satisfies a completeness condition, e.g., is chain-complete. (2) The correspondence is ascending while the target is a sublattice of the Cartesian product of a finite number of chains. (3) Both source and target are chains while the correspondence is generated by the maximization of a strongly acyclic interval order with the single crossing property. The theorems give new sufficient conditions for the existence of (epsilon) Nash equilibria