Product of formations of algebraic systems

Abstract

In [i], Mal'tsev has introduced the definition of the product of classes of algebraic systems, having a considerable importance in the theory of varieties.But the application of this definition to the theory of formations is limited (for example, it is easy to give an example when the product, in the sense of [I], of two formations of groups is not a formation).The principal reason why the concept of product from [1] cannot be used to full extent in the theory of formations consists in the fact that a formation, unlike a variety, is not always a hereditary class.In the present paper we give a modification of the Mal'tsev multiplication [1] using the concept of a replica.In the case when one considers the formation of groups, our definition coincides with the well-known concept of the product of the formations of groups [2]. In the present paper we also introduce the concept of the formation of algebraic systems and we investigate the cases when the product of formations is a formation. We make use of the terminology in [3]. All the considered algebraic systems have the same signature. We recall that a collection ~ of algebraic systems is said to be a class if with each of its system ~ contains also all of its isomorphs; the systems from ~ are called ~-systems. We note that [3] contains another definition of the concept of replica; the difference consists in the fact that there, for ~ , one takes a homomorphism rather than an epimorphism, which requires the necessity of considering hereditary classes.If the system ~ has an ~-replica, then this replica is uniquely defined within an isomorphism (see [3]).Now we give our fundamental Definition I. Let ~ be some nonempty class of algebraic systems; let ~ and ~ be some of its subclasses. We denote by ~ the collection of all those ~-systems which satisfy the following condition: there exists an ~-morphism o~:~ ~i ' whose kernel congruence 8 is such that all the cosets =~ (~) that are ~-subsystems of the *Dedicated to the birthday of A. I. Mal'tsev.

Formations of Monoids, Congruences, and Formal Languages

“…Subdirect products allow us to introduce the notion of formation of monoids, which is a particular case of the most general notion of formation of algebraic structures, introduced and studied by Shemetkov and Skiba in [26].…”

“…Formations of finite groups are important for a better understanding of the structure of finite groups, and the more general notion of formation of algebraic structures, introduced and studied by Shemetkov and Skiba in [26], plays a central role in universal algebra. Therefore it seems quite natural to seek an Eilenberg type theorem establishing a connection between formations of finite monoids and formations of regular languages, which are classes of regular languages closed under Boolean operations and derivatives with a weaker property on the closure under inverse monoid morphism.…”

On the abnormal structure of finite groups

“…This class was studied in [5]. The reader is also referred to [1,2,3,8,10,15,18,19] for other interesting related results.…”

On two questions of L. A. Shemetkov concerning hypercyclically embedded subgroups of finite groups

“…The symbol Að p À 1Þ denotes the formation of all abelian groups of exponent dividing p À 1; see [22].…”