Subdirect products of an idempotent semiring and a b-lattice of skew-rings

Bandelt and Petrich [Subdirect products of rings and distributive lattices, Proc. Edinburgh Math. Soc. (2) 25(2) (1982) 155–171] characterized a class of additive inverse semirings which are subdirect products of a distributive lattice and a ring. The aim of this paper is to characterize a class of additively regular semirings which are subdirect products of an idempotent semiring and a [Formula: see text]-lattice of skew-rings.

On maximal subformations of n-multiple Ω-foliated formations of finite groups

Only finite groups are considered in the article. Among the classes of groups the central place is occupied by classes closed regarding homomorphic images and subdirect products which are called formations. We study Ω-foliateded formations constructed by V. A. Vedernikov in 1999 where Ω is a nonempty subclass of the class I of all simple groups. Ω-Foliated formations are defined by two functions — an Ω-satellite f : Ω ∪ {Ω 0} → {formations} and a direction ϕ : I → {nonempty Fitting formations}. The conception of multiple locality introduced by A. N. Skiba in 1987 for formations and further developed for many other classes of groups, as applied to Ω-foliated formations is as follows: every formation is considered to be 0-multiple Ω-foliated with a direction ϕ; an Ω-foliated formation with a direction ϕ is called an n-multiple Ω-foliated formation where n is a positive integer if it has such an Ω-satellite all nonempty values of which are (n − 1)-multiple Ω-foliated formations with the direction ϕ. The aim of this work is to study the properties of maximal n-multiple Ω-foliated subformations of a given n-multiple Ω-foliated formation. We use classical methods of the theory of groups, of the theory of classes of groups, as well as methods of the general theory of lattices. In the paper we have established the existence of maximal n-multiple Ω-foliated subformations for the formations with certain properties, we have obtained the characterization of the formation ΦnΩϕ (F) which is the intersection of all maximal n-multiple Ω-foliated subformations of the formation F, and we have revealed the relation between a maximal inner Ω-satellite of 1-multiple Ω-foliated formation and a maximal inner Ω-satellite of its maximal 1-multiple Ω-foliated subformation. The results will be useful in studying the inner structure of formations of finite groups, in particular, in studying the maximal chains of subformations and in establishing the lattice properties of formations.

Abelian endoregular modules

In this paper, we introduce the notion of abelian endoregular modules as those modules whose endomorphism rings are abelian von Neumann regular. We characterize an abelian endoregular module [Formula: see text] in terms of its [Formula: see text]-generated submodules. We prove that if [Formula: see text] is an abelian endoregular module then so is every [Formula: see text]-generated submodule of [Formula: see text]. Also, the case when the (quasi-)injective hull of a module as well as the case when a direct sum of modules is abelian endoregular are presented. At the end, we study abelian endoregular modules as subdirect products of simple modules.

Structure of Strict Abundant Semigroups

We introduce a new class of semigroups called strict abundant semigroups, which are concordant semigroups and subdirect products of completely [Formula: see text]-simple abundant semigroups and completely 0-[Formula: see text]-simple primitive abundant semigroups. A general construction and a tree structure of such semigroups are established. Consequently, the corresponding structure theorems for strict regular semigroups given by Auinger in 1992 and by Grillet in 1995 are generalized and extended. Finally, an example of strict abundant semigroups is also given.

Subgroups of 3-Factor Direct Products

Extending Goursat’s Lemma we investigate the structure of subdirect products of 3-factor direct products. We construct several examples and then provide a structure theorem showing that every such group is essentially obtained by a combination of the examples. The central observation in this structure theorem is that the dependencies among the group elements in the subdirect product that involve all three factors are of Abelian nature. In the spirit of Goursat’s Lemma, for two special cases, we derive correspondence theorems between data obtained from the subgroup lattices of the three factors (as well as isomorphisms between arising factor groups) and the subdirect products. Using our results we derive an explicit formula to count the number of subdirect products of the direct product of three symmetric groups.

ON THE NUMBER OF SUBSEMIGROUPS OF DIRECT PRODUCTS INVOLVING THE FREE MONOGENIC SEMIGROUP

The direct product $\mathbb{N}\times \mathbb{N}$ of two free monogenic semigroups contains uncountably many pairwise nonisomorphic subdirect products. Furthermore, the following hold for $\mathbb{N}\times S$, where $S$ is a finite semigroup. It contains only countably many pairwise nonisomorphic subsemigroups if and only if $S$ is a union of groups. And it contains only countably many pairwise nonisomorphic subdirect products if and only if every element of $S$ has a relative left or right identity element.