Abstract. Let (A, f ) be a monounary algebra. We describe all monounary algebras (A, g) having the same set of quasiorders, Quord(A, f ) = Quord(A, g). It is proved that if Quord(A, f ) does not coincide with the set of all reflexive and transitive relations on the set A and (A, f ) contains no cycle with more than two elements, then f is uniquely determined by means of Quord(A, f ). In the opposite case, Quord(A, f ) = Quord(A, g) if and only if Con(A, f ) = Con(A, g). Further, we show that, except the case when Quord(A, f ) coincides with the set of all reflexive and transitive relations, if the monounary algebras (A, f ) and (A, g) have the same quasiorders, then they have the same retracts. Next we characterize monounary algebras which are determined by their sets of retracts and connected monounary algebras which are determined by their sets of quasiorders.
IntroductionA quasiorder of an algebra is a binary relation on its support, which is reflexive, transitive and compatible with all fundamental operations of the algebra.In many papers quasiorders of algebras are studied. The system of all quasiorders of an algebra is a complete algebraic lattice with respect to inclusion. Also, by [2], [12], every algebraic lattice is isomorphic to the quasiorder lattice of a suitable algebra.Let us notice that quasiorders of an algebra A can be considered as a common generalization of partial orders which are compatible with all operations of A and its congruences.We will deal with monounary algebras. The first goal of this paper is, for a given monounary algebra (A, f ), a characterization of all monounary algebras (A, g) such that the algebras (A, f ) and (A, g) have the same sets of quasiorders. This problem is in a close connection with the papers [3]-[6], 2000 Mathematics Subject Classification: 08A60.