The logical environment of this research is the Intuitionistic Logic and principled-philosophical orientation of the Bishop's Constructive Mathematics. In this paper, basing our consideration on the sets with the apartness relation, we analyze the lattices of all co-filters of an ordered semigroup under a co-quasiorder as a continuation of our article \cite{Rom18}. We prove a number of results related to co-filters in a semigroup with apartness and the lattice of all co-filters of such semigroups.