On some types of radical classes

Let be an infinite cardinal. We denote by the collection of all -representable Boolean algebras. Further, let be the collection of all generalized Boolean algebras B such that for each b ∈ B, the interval [0, b] of B belongs to . In this paper we prove that is a radical class of generalized Boolean algebras. Further, we investigate some related questions concerning lattice ordered groups and generalized MV-algebras. This work was supported by Science and Technology Assistance Agency under the contract No. APVT-51-032002. This work has been partially supported by the Slovak Academy of Sciences via the project Center of Excellence-Physics of Information.     

On the distributivity of the lattice of solvable totally local fitting classes

It is proved that the lattice of all solvable totally local Fitting classes is algebraic and distributive. Translated fromMatematicheskie Zametki, Vol. 67, No. 5, pp. 662–673, May, 2000.     

On the characterization of radical and semisimple classes in categories

We define and characterize radical and semisimple classes in a category K which satisfies certain conditions. These conditions are such that K could be any of the categories of associative rings, groupsR-modules, topological spaces or graphs. Among others, the following is proved:.

A class of objects R in K is a radical class if and only if K is a cohereditary component class which is closed under extensions and with T ? R. A class of objects S in K is a semisimple class if and only if S is a hereditary class which is closed under subdirect embed-dings and extensions with T ? S.     

On the lattice of all totally composition formations of finite groups

It is shown that the lattice of all totally composition formations of finite groups is algebraic.

     

On the spatially homogeneous landau equation for hard potentials part i : existence,uniqueness and smoothness

We study the Cauchy problem for the homogeneous Landau equation of kinetic theory, in the case of hard potentials. We prove that for a large class of initial data, there exists a unique weak solution to this problem, which becomes immediately smooth and rapidly decaying at infinity.     

On measures integrating all functions of a given vector lattice

LetE be a vector lattice of real-valued functions defined on a setX, and (E):={{f1}:fE}. Among others, it is shown that, under some additional assumptions onE, every measure that integrates all functionsfE is (E)--smooth iffX is (E)-complete. An application of this general result to various topological situations yields some new measure-theoretic characterizations of realcompact, Borel-complete andN-compact spaces, respectively.     

On the lattice formed by all normal subloops of a finite loop

All normal subloops of a loopG form a modular latticeL _n (G). It is shown that a finite loopG has a complemented normal subloop lattice if and only ifG is a direct product of simple subloops. In particular,L _n (G) is a Boolean algebra if and only if no two isomorphic factors occurring in a decomposition ofG are abelian groups. The normal subloop lattice of a finite loop is a projective geometry if and only ifG is an elementary abelianp-group for some primep.     

On coverings in the lattice of all group topologies of arbitrary Abelian groups

The remainder of the completion of a topological abelian group (G, τ₀) contains a nonzero element of prime order if and only if G admits a Hausdorff group topology τ₁ that precedes the given topology and is such that (G, τ₀) has no base of closed zero neighborhoods in (G, τ₁).     

On base radical and semisimple classes defined by class operators

By using class operators, we define base radical and semisimple classes, within a broad abstract setting due to Puczylowski. The new notions agree with the usual Kurosh–Amitsur ones for associative rings and groups but differ for not necessarily associative rings, in general lying strictly between the Kurosh–Amitsur and torsion theory notions. A study of the class operators for their own sake is initiated, and a connection with modal logic is made.     

Orthonormal bases of symmetry classes with computer-generated examples

A method is given for constructing an orthonormal basis of a symmetry class of tensor whose associated group is the full symmetric group. This construction has been used to write a computer program which efficiently calculates a basis for any such symmetry class. Two examples which illustrate both the method and the program are included.