Subalgebras of algebras determined by polymorphisms of unary central relations or by polymorphisms of non-trivial equivalence relations

Abstract

The set of all clones on an arbitrary set forms a complete lattice under the set inclusion. Its co-atoms are called maximal clones. All maximal clones on a finite set are classified by I. G. Rosenberg into 6 classes by 6 types of relations. Two of them are non-trivial equivalence relations and central relations. Consider a finite algebra that all fundamental operations are polymorphisms of non-trivial equivalence relations. We describe all subalgebras of such an algebra. Moreover, all subalgebras of a finite algebra that all fundamental operations are polymorphisms of unary central relations are investigated.