Finite Subdirect Products Research Articles

On the modularity of the lattice of Baer-σ-local formations

Throughout this paper, all groups are finite. A group class closed under taking homomorphic images and finite subdirect products is called a formation. The symbol σ denotes some partition of the set of all primes. V. G. Safonov, I. N. Safonova, A. N. Skiba (Commun. Algebra. 2020. Vol. 48, № 9. P. 4002–4012) defined a generalized formation σ-function. Any function f of the form f : σ È {Ø} → {formations of groups}, where f(Ø) ≠ ∅, is called a generalized formation σ-function. Generally local formations or so-called Baer-σ-local formations are defined by means of generalized formation σ-functions. The set of all such formations partially ordered by set inclusion is a lattice. In this paper it is proved that the lattice of all Baerσ-local formations is algebraic and modular.

Laws of the lattice of all $${\mathfrak {X}}$$-local formations of finite groups

Let $${\mathfrak {X}}$$ be a class of simple groups with a completeness property $$\pi ({\mathfrak {X}}) = \mathrm {char} \, {\mathfrak {X}}$$ . A formation is a class of finite groups closed under taking homomorphic images and finite subdirect products. Forster introduced the concept of $${\mathfrak {X}}$$ -local formation in order to obtain a common extension of well-known Gaschutz–Lubeseder–Schmid, and Baer theorems (Publ Mat Univ Autonoma Barcelona 29(2–3):39–76, 1985). In the present paper, it is shown that any law of the lattice of all formations is true in the lattice of all $${\mathfrak {X}}$$ -local formations.

On Categorical Equivalence Between Formations of Monounary Algebras

A formation is a class of algebras that is closed under homomorphic images and finite subdirect products. Every formation can be considered as a category. We prove that two formations of monounary algebras with finitely many cycles are equivalent as categories if and only if they coincide.

Ultraproducts preserve finite subdirect reducibility

An algebraic structure A is said to be finitely subdirectly reducible if A is not finitely subdirectly irreducible. We show that for any signature providing only finitely many relation symbols, the class of finitely subdirectly reducible algebraic structures is closed with respect to the formation of ultraproducts. We provide some corollaries and examples for axiomatizable classes that are closed with respect to the formation of finite subdirect products, in particular, for varieties and quasivarieties.

On lattices of formations of monounary algebras with finitely many cycles

A formation is a class of algebras that is closed under homomorphic images and finite subdirect products. As already known the lattice of all formations of finite monounary algebras is distributive. We show that for the lattice of formations of at most countable monounary algebras this is not true. Also we find a necessary and sufficient condition for a lattice of at most countable monounary algebras with finitely many cycles to be distributive.

Formations of Monoids, Congruences, and Formal Languages

The main goal in this paper is to use a dual equivalence in automata theory started in [RBBCL13] and developed in [BBCLR14] to prove a general version of the Eilenberg-type theorem presented in [BBPSE12]. Our principal results confirm the existence of a bijective correspondence between formations of (non-necessarily finite) monoids, that is, classes of monoids closed under taking epimorphic images and finite subdirect products, with formations of languages, which are classes of (non-necessarily regular) formal languages closed under coequational properties. Applications to non-r-disjunctive languages are given.

О наследственности формаций унаров

Volgograd State Socio-Pedagogical University, Russia, 400066, Volgograd, Lenin Ave., 27, rasal@fizmat.vspu.ruA class of algebraic systems is said to be a formation if it is closed under homomorphic images and finite subdirect products. It hasbeen proven that any formation of at most countable monounary algebras is a hereditary formation.Key words: unar, formation, hereditary formation.

Formations of finite monounary algebras

A class of finite monounary algebras is said to be a formation if it is closed under homomorphic images and finite subdirect products. Let us denote by \({\mathbb{F}}\) the collection of all formations of finite monounary algebras. Then \({\mathbb{F}}\) can be partially ordered by the class-theoretical inclusion and it forms a complete lattice. The main aim of this paper is to show that \({\mathbb{F}}\) is isomorphic to the lattice of all hereditary subsets of a certain poset.

Formations of lattice ordered groups and of GMV-algebras

Abstract A class of lattice ordered groups is called a formation if it is closed with respect to homomorphic images and finite subdirect products. Analogously we define the formation of GMV-algebras. Let us denote by ℱ1 and ℱ2 the collection of all formations of lattice ordered groups or of GMV-algebras, respectively. Both ℱ1 and ℱ2 are partially ordered by the class-theoretical inclusion. We prove that ℱ1 satisfies the infinite distributivity law and that ℱ2 is isomorphic to a principal ideal of ℱ1.

On QB ∞-Rings

In this article, we introduce a new class of rings, the QB ∞-rings. We establish various properties of this concept. These show that, in several respects, QB ∞-rings behave like QB-rings. We prove that the notion of QB ∞-rings is a Morita invariant property of rings and every finite subdirect product of QB ∞-rings is a QB ∞-ring. We also exhibit examples to point out that the class of QB ∞-rings is much larger than the class of QB-rings.

Exchange Rings Satisfying the n-Stable Range Condition, II

This is a continuation of the paper [14]. It is shown that any finite subdirect product of exchange rings satisfying the n-stable range condition is still an exchange ring satisfying the n-stable range condition. Furthermore, we give necessary and sufficient conditions on matrices over an exchange ring R, under which R satisfies the n-stable range condition. This generalizes the corresponding results for unit-regular rings and the stable range one condition.

Quasi-Identities of Congruence-Distributive Quasivarieties of Algebras

LetR be an arbitrary quasivariety of algebras and A an algebra not necessarily inR. A congruence θ ∈ ConA is an R-congruence if A/θ ∈ R. The set ConRA of all R-congruences of A makes an algebraic lattice under inclusion. A quasivariety R of algebras is congruence-distributive if the lattice of Rcongruences is distributive for every algebra A. In the case when R is a variety this definition coincides with that of a congruence-distributive variety. An algebra A in a quasivariety R is called finitely subdirectly R-indecomposable if, for every representation of A in the form of a finite subdirect product ∏ {Ai ∈ R : i < n}, one of the projections πi : A → Ai is an isomorphism. Note that if R is congruence-distributive then every finitely subdirectly R-indecomposable algebra is (absolutely) finitely subdirectly indecomposable (see [1, 2]). It is well known [3] that every congruence-distributive variety of algebras of finite signature with the finitely axiomatizable class of finitely subdirectly indecomposable algebras has a finite basis of identities. It was asked in some articles (see [1, 4, 5]) whether any congruence-distributive quasivariety R of algebras with the finitely axiomatizable class of finitely subdirectly R-indecomposable algebras has a finite basis of quasi-identities. A positive answer to this question is known in two particular cases: for finitely generated quasivarieties [1] and for quasivarieties with equationally definable intersections of principal congruences [5]. We say that a lattice L has at most countable width if every antichain in L is at most countable. A lattice L satisfies the minimality condition if every chain in L has a least element. A lattice with the maximality condition is defined by reversal. We say that a class R of algebras is finitely axiomatizable relative to a class K if there exists a finite set Ψ of propositions such that R = K ∩ Mod(Ψ). In this article we prove the following two theorems.

Weakly Special Classes of Semiprime Rings

This paper investigates closure properties possessed by certain classes of finite subdirect products of prime rings. If ℳ is a special class of prime rings then the class ℳ∞ of all finite subdirect products of rings in ℳ is shown to be weakly special. A ring S is said to be a right tight extension [resp. tight extension] of a subring R if every nonzero right ideal [resp. right ideal and left ideal] of S meets R nontrivially. Every hereditary class of semiprime rings closed under tight extensions is weakly special. Each of the following conditions imposed on a semiprime ring yields a hereditary class closed under right tight extensions: ACC on right annihilators; finite right Goldie dimension; right Goldie. The class of all finite subdirect products of uniformly strongly prime rings is shown to be closed under tight extensions, answering a published question.

TIGHT EXTENSIONS AND Λ-CLASSES

Certain classes of associative rings are characterized as finite subdirect products of specialized prime rings. These characterizations depend on results about Λ-classes and tight extensions, concepts that were fist introduced by Olson, Le Row and Hey man.

PI-rings and semiprime rings having faithful modules with krull dimension

It is proved that if a PI-ring R has a faithful left R-module M with Krull dimension, then its prime radical rad(R) is nilpotent. If, moreover, the R-module M and the left idealR(rad(R)) are finitely generated, then R has a left Krull dimension equal to the Krull dimension of M. It turns out that a semiprime ring, which has a faithful (left or right) module with Krull dimension, is a finite subdirect product of prime rings. Moreover, first, a right Artinian ring R such that rad(R)2=0 has a faithful Artinian cyclic left module, and second, a finitely generated semiprime PI-algebra over a field has a faithful Artinian module. We give examples showing that the restrictions imposed are essential, as well as an example of a finitely generated prime PI-algebra over a field, which is not Noetherian and has a Krull dimension.

Finite subdirect products of rings and morita duality

Finite subdirect products of rings and morita duality

Strongly semiprime rings

For a ring with 1, we show that every proper kernel functor generates a proper torsion radical if and only if the ring is a finite subdirect product of strongly prime (also called ATF) rings. This is equivalent to every essential right ideal containing a finite set whose right annihilator is zero. We use this characterization to quickly prove a number of properties of rings satisfying this condition, and apply the results to the problem: when is every kernel functor a torsion radical.