Strategic complementarity and substitutability without transitive indifference

We study what useful implications strategic complementarity or substitutability may have when the indifference relation(s) need not be transitive. Two results are obtained about the existence of a monotone selection from the best response correspondence when both strategies and parameters form chains. Two more results are obtained about the existence of a Nash equilibrium in games with strategic complementarities where strategy sets are chains, but monotone selections from the best response correspondences need not exist.