Construction of a complementary quasiorder

Abstract

For a monounary algebra A = (A, f) we study the lattice QuordA of all quasiorders of 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 ∊ 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 (*).