The Lattice of Compatible Quasiorders of Acyclic Monounary Algebras

Acyclic monounary algebras are characterized by the property that any compatible partial order r can be extended to a compatible linear order. In the case of rooted monounary algebras A = (A, f ) we characterize the intersection of compatible linear extensions of r by several equivalent conditions and generalize these results to compatible quasiorders of A. We show that the lattice QuordA of compatible quasiorders is a disjoint union of semi-intervals whose maximal elements equal the intersection of their compatible quasilinear extensions. We also investigate algebraic properties of the lattices QuordA and ConA.