О насыщенных формациях конечных унаров

Abstract

A class of algebraic systems which is closed under homomorphic images and finite subdirect products is called a formation. Formations was widely used in group theory. Particularly, the saturated formations of groups is one of the most studied formations. A formation of finite groups is said to be a saturated formation if G/Φ(G) ∈ F implies G ∈ F for an arbitrary finite group G and it’s Frattini subgroup Φ(G). A generalization of these definitions is as follows. A congruence θ on the algebraic system A is called a Frattini congruence if the union of all θ-classes generated by the elements of B differs from A for each proper subsystem B of the algebraic system A. A class X is saturated in the class Y, if A ∈ Y and A/θ ∈ X for some Frattini congruence θ on A implies A ∈ X. We consider finite formations of monounary algebras in this paper. An element a of a monounary algebra ⟨A, f⟩ is cyclic if f n (a) = a for some positive integer n. A monounary algebra is cyclic if all of it’s elements are cyclic. First we give a condition for a congruence of finite monounary algebra to be a Frattini congruence. Then we prove that the empty formation, the formation of all finite cyclic monounary algebras and the formation of all finite monounary algebras are the only saturated formations in the class of all finite monounary algebras.