Radicals of 0-regular algebras

Summary We consider a generalisation of the Kurosh--Amitsur radical theory for rings (and more generally multi-operator groups) which applies to 0-regular varieties in which all operations preserve 0. We obtain results for subvarieties, quasivarieties and element-wise equationally defined classes. A number of examples of radical and semisimple classes in particular varieties are given, including hoops, loops and similar structures. In the first section, we introduce 0-normal varieties (0-regular varieties in which all operations preserve 0), and show that a key isomorphism theorem holds in a 0-normal variety if it is subtractive, a property more general than congruence permutability. We then define our notion of a radical class in the second section. A number of basic results and characterisations of radical and semisimple classes are then obtained, largely based on the more general categorical framework of L. M\'arki, R. Mlitz and R. Wiegandt as in [13]. We consider the subtractive case separately. In the third section, we obtain results concerning subvarieties and quasivarieties based on the results of the previous section, and also generalise to subtractive varieties some results for multi-operator group radicals defined by simple equational rules. Several examples of radical and semisimple classes are given for a range of fairly natural 0-normal varieties of algebras, most of which are subtractive.     

Radical constructions in the presence of two universal classes

Summary The constructions for the lower radical class, the upper radical class, and the smallest semisimple class were introduced in 1953 by KUROSH. Generalizations of the lower radical class and smallest semisimple class constructions were given in 1970 by LEAVITT. Here these constructions are investigated in the presence of two universal classes. The result of KUROSH for the lower radical construction is generalized and similar results are obtained for the other constructions. Entrata in Redazione il 12 ottobre 1970.     

Strict radicals and endomorphism rings

We consider the determination of ring radicals by classes of modules as first discussed by Andrunakievich and Ryabukhin, but in cases where the modules concerned are defined by additive properties. Such a radical is the upper radical defined by the class of subrings of a class of endomorphism rings of abelian groups and is therefore strict. Not every strict radical is of this type, and while the A-radicals are of this type, there are others, including some special radicals. These investigations bring radical theory into contact with (at least) two questions from other parts of algebra. Which rings are endomorphism rings? For a given ring R, which abelian groups are non-trivial R-modules?     

On the convergence properties of basic series representing special monogenic functions

In this paper, it is shown that certain classes of special monogenic functions cannot be represented by the basic series in the whole space. New definitions for the order of basis of special monogenic polynomials are given together with theorems on representation of classes of special monogenic functions in certain balls and at a point. Received: 8 January 2002     

Special radicals of near-rings and Γ-near-rings

It was previously shown that every special radical classR of rings induces a special radical class R of -rings. Amongst the special radical classes of near-rings, there are some, called the -special radical classes, which induce, special radical classes of -near-rings by the same procedure as used in the ring case. The -special radicals of near-rings possess very strong hereditary properties. In particular, this leads to some new results for the equiprime andI ₃ radicals.     

Invariant affinor metric structures on Lie groups

We introduce the class of special metric structures on Lie groups which are connected with the radical of a fixed 1-form on a Lie group. These structures are called affinor metric structures. We introduce and study some special classes of invariant affinor metric structures and generalize many results of the theory of contact metric structures on Lie groups.     

A Representation of Virial Coefficients That Avoids the Asymptotic Catastrophe

We investigate the notion of an asymptotic catastrophe in representations of Mayer coefficients. The manifestations of the catastrophe and its formal definition are given. The significance of the definition introduced for an asymptotic catastrophe is clarified. Virial-coefficient representations that are free of the asymptotic catastrophe phenomenon are given. Sets of labeled graphs (blocks) nonseparable in the Harary sense are expanded into classes labeled by cycle ensembles satisfying specific conditions, and the representations are based on these expansions. These cycle ensembles are called frame cycle ensembles. The same classes can be labeled by special blocks, which are called frames. The frames are brought into one-to-one correspondence with the frame cycle ensembles. In the block classification, frames play a role similar to the role of trees in the tree classification of connected labeled graphs. A tree classification of frame cycle ensembles is introduced. We prove that the described virial-coefficient representations are free of the asymptotic catastrophe phenomenon.     

Radical and Torsion Theory for Acts

After recapitulating the rudiments of the Kurosh-Amitsur radical theory of S-acts, hereditary radicals are discussed. The hereditary Hoehnke radical assignments r which designate a Rees congruence r(A) to each S-act A, are the hereditary Kurosh-Amitsur radical assignments. Then the corresponding radical as well as semisimple classes are characterized. Equivalence classes of injective S-acts determine hereditary torsion assignments t, these are just the hereditary Hoehnke radicals, but the congruence t(A) of A need not be a Rees congruence. Torsion and torsionfree classes are characterized; several hereditary torsion assignments may determine the same torsion class which is always a radical class closed under taking subacts. Examples show that a hereditary torsion assignment need not be a hereditary radical.     

Lexico groups and related radical classes

There are defined and studied radical classes of lattice ordered groups, in some way related to the notion of lexico extensions. It is shown that such radical classes form a proper class and they fail to be torsion classes.     

Nielsen Zeta Function, 3-Manifolds, and Asymptotic Expansions in the Nielsen Theory

We prove that the Nielsen zeta function is either a rational function or a radical of a rational function for orientation-preserving homeomorphisms of closed orientable 3-dimensional manifolds which are special Haken or Seifert manifolds. In the case of a pseudo-Anosov homeomorphism of a surface, we find an asymptotics for the number of twisted conjugacy classes or for the number of Nielsen fixed-point classes with norm at most x. Bibliography: 20 titles.     

ON THE EXISTENCE AND NON-EXISTENCE OF COMPLEMENTARY RADICAL AND SEMISIMPLE CLASSES

ABSTRACT

In certain categories of mathematical structures, non-trivial complementary radical classes (torsion classes or connectednesses) can be found. The question is why this is true for some but not for all categories. The answer depends on the embedding of trivial objects into nontrivial objects and is given by our main result: Any ‘reasonable’ category has no non-trivial complementary radical and semisimple classes if and only if for every trivial object T and every non-trivial object A there is a morphism T → A. Roughly, a ‘reasonable’ category in our sense is one with at least one object into which a terminal object can be embedded and has finite products, coproducts or lexicographic products.     

Generalized Vector Equilibrium Problems with Trifunctions

In this paper, we study the existence of strong and weak solutions of the generalized vector equilibrium problems for trifunctions. Two special classes of vector-valued trifunctions are introdcued, called the classes of (SPM) and (GPM), respectively. Some existence results for strong solutions associated to functions of these classes are given.     

Generalized Vector Equilibrium Problems with Trifunctions

In this paper, we study the existence of strong and weak solutions of the generalized vector equilibrium problems for trifunctions. Two special classes of vector-valued trifunctions are introdcued, called the classes of (SPM) and (GPM), respectively. Some existence results for strong solutions associated to functions of these classes are given.     

Very large subgroups of a lattice-ordered group

This article introduces the concept of a very large subgroup in the theory of lattice-ordered groups. The existence of a minimal very large subgroup is connected to some previously known structure theory, but it is also linked to conditions not studied before. Very large subgroups are useful in studying torsion and radical classes, and among other things, extension of lattice-ordered groups using very large kernels yields an intriguing completion operation for torsion classes. In the final section there is a new contruction which produces a lattice-ordered group in which every value is essential, having no special values.     

一般的正项几何规划的一种分解方法

关于几何规划的分解方法，［１３］均给出特殊类型的几何规划的分解方法，本文则对一般的正项几何规划给出一种直接分解方法。     

SPECIAL RADICALS OF NEAR-RING MODULES

Abstract

Equiprime near-rings, which generalize the concept of prime-ness in rings, were defined by the present authors, together with S. Veldsman. This concept was shown in subsequent work to lead to a very satisfactory theory of special radicals for near-rings. In the current paper, we define equiprime N-groups for a near-ring N. It is shown that an ideal A of N is equiprime if and only if it is the annihilator of an equiprime TV-group G. Special classes of near-ring modules are defined, and a module-theoretic characterization of special radicals of near-rings is established, similar to that given by Andrunakievich and Rjabuhin for special radicals of rings.     

Radical classes of algebras with B-action

We initiate the radical theory of algebras with B-action where B is a fixed Boolean ring. We consider lattices of classes of algebras defined in terms of ideals of B. In two special cases (universal classes of -groups with B-action and idempotent algebras with B-action), these ideal-defined classes are sublattices of the lattice of radicals, and we characterise semisimplicity in such cases. Received February 2, 1998; accepted in final form June 11, 1998.     

J-半单亚直既约环类确定的上限

Ｆ．Ａ．Ｓｚａｓｚ在［１］中提出公开问题５５：设Ｋ是Ｊａｃｏｂｓｏｎ根为零的全体亚直既约环类,研究类Ｋ确定的上根．本文对此进行了研究,证明了Ｊａｃｏｂｓｏｎ根为零的全体亚直既约环类Ｋ确定的上根Ｒ是特殊根,它介于Ｊａｃｏｂｓｏｎ根与Ｂｒｏｗｎ－ＭｃＣｏｙ根之间．并给出任意结合环Ａ为Ｒ－根环的充要条件．     

Disjoint congruence classes and a timetabling application

We consider a combinatorial problem motivated by a special simplified timetabling problem for subway networks. Mathematically the problem is to find (pairwise) disjoint congruence classes modulo certain given integers; each such class corresponds to the arrival times of a subway line of a given frequency. For a large class of instances we characterize when such disjoint congruence classes exist and how they may be determined. We also study a generalization involving a minimum distance requirement between congruence classes, and a comparison of different frequency families in terms of their “efficiency”. Finally, a general method based on integer programming is also discussed.     

The power of programmed grammars with graphs from various classes

Programmed grammars, one of the most important and well investigated classes of grammars with context-free rules and a mechanism controlling the application of the rules, can be described by graphs. We investigate whether or not the restriction to special classes of graphs restricts the generative power of programmed grammars with erasing rules and without appearance checking, too. We obtain that Eulerian, Hamiltonian, planar and bipartite graphs and regular graphs of degree at least three are pr-universal in that sense that any language which can be generated by programmed grammars (with erasing rules and without appearance checking) can be obtained by programmed grammars where the underlying graph belongs to the given special class of graphs, whereas complete graphs, regular graphs of degree 2 and backbone graphs lead to proper subfamilies of the family of programmed languages.