Construction of a complementary quasiorder

Abstract

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 (\(*\)).