Construction of a complementary quasiorder

For a monounary algebra $\mathcal{A}=(A,f)$ we study the lattice $\operatorname{Quord}\mathcal{A}$ of all quasiorders of $\mathcal{A}$, i.e., of all reflexive and transitive relations compatible with $f$. Monounary algebras $(A, f)$ whose lattices of quasiorders are complemented were characterized in 2011 as follows: ($*$) $f(x)$ is a cyclic element for all $x \in A$, and all cycles have the same square-free number $n$ of elements. Sufficiency of the condition ($*$) was proved by means of transfinite induction. Now we will describe a construction of a complement to a given quasiorder of $(A, f)$ satisfying ($*$).