Radical distributive functors

The properties of distributive functors on the lattice of subnormal subgroups are studied. A class of radical distributive functors is singled out, a relationship between these functors and the Fitting classes is established, and the Fitting classes inducing a radical distributive functor are described. Translated fromMatematicheskie Zametki, Vol. 68, No. 1, pp. 91–97, July, 2000.     

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.     

Closure properties for fitting functors

Fitting functors have been introduced as a generalization of injectors and radicals for Fitting classes of finite (soluble) groups. This paper provides definitions and properties of certain types of Fitting functors which resemble injectors and radicals for Fitting classes with additional closure properties, most prominently subgroup- or quotient-closed Fitting classes and Fischer classes.     

Lattices of partially local fitting classes

This article deals only with finite groups. We prove the surjectivity of the mapping from the lattice of all normal Fitting classes into the lattice of the Lockett section generated by the Fitting classes that are not Lockett classes. Moreover, we find a sufficient surjectivity condition for the mapping of the lattice of the Lockett section generated by arbitrary Fitting classes into the lattice of the Lockett section generated by ω-local Fitting classes. This confirms Lockett’s conjecture for the ω-local Fitting classes of a given characteristic.     

Non-singly-generated multiply ω-fan fitting classes of finite groups

In the paper, we study algebraic lattices of Fitting classes and describe non-singly-generated Fitting classes all of whose proper Fitting subclasses are singly generated. All groups under consideration are assumed finite.     

Products of Fitting Classes of Finite Groups

Fitting classes are considered. In particular, it is shown that a product of n-multiply ω-local Fitting classes is an n-multiply ω-local Fitting class.     

Factorizations of fitting classes

In this paper, we prove that there exists a infinite set of non-trivial local Fitting classes every element in which is decomposable as a non-trivial product of Fitting classes such that every factor in the product is neither local nor a formation. In particular, this gives a positive answer to Problem 11.25 a) in The Kourovka Notebook.     

Fitting functors and radicals of finite groups

We develop methods for recognizing the Fitting classes and radicals of finite groups by Fitting functors and the prescribed properties of Hall π-subgroups.     

Products of fitting classes of finite groups

Using Fitting classes we generalize some well known theorems on centralizers in finite groups.     

On the Dual Ramsey Property for Finite Distributive Lattices

The class of finite distributive lattices, as many other natural classes of structures, does not have the Ramsey property. It is quite common, though, that after expanding the structures with appropriately chosen linear orders the resulting class has the Ramsey property. So, one might expect that a similar result holds for the class of all finite distributive lattices. Surprisingly, Kechris and Soki? have proved in 2012 that this is not the case: no expansion of the class of finite distributive lattices by linear orders satisfies the Ramsey property. In this paper we prove that the class of finite distributive lattices does not have the dual Ramsey property either. However, we are able to derive a dual Ramsey theorem for finite distributive lattices endowed with a particular linear order. Both results are consequences of the recently observed fact that categorical equivalence preserves the Ramsey property.     

Idempotent distributive semirings II

In [2] it is shown that every idempotent distributive semiring is the P?onka sum of a semilattice ordered system of idempotent distributive semirings which satisfy the generalized absorption law x+xyx+x=x. We shall show that an idempotent distributive semiring which satisfies the above absorption law must be a subdirect product of a distributive lattice and a semiring which satisfies the additional identity xyx+x+xyx=xyx. Using this, we construct the lattice of all equational classes of idempotent distributive semirings for which the two reducts are normal bands.     

Amalgamation classes of some distributive varieties

This paper investigates the amalgamation classes of finitely generated varieties with distributive congruence lattices. Necessary and sufficient conditions are given for an algebra to be a member of the amalgamation class of a variety generated by a finite modular lattice or pseudocomplemented distributive lattice and of a filtral variety.Presented by S. Burris.     

Koszul filtrations and finite lattices

In this note, we characterize when a finite lattice is distributive in terms of the existences of some particular classes of Koszul filtrations.     

Finite groups whose noncentral class sizes have the same p-part for some prime p

We describe the structure of the finite groups in which the sizes of noncentral conjugacy classes have the same p-part, for some prime p. We prove that they are solvable, they have a normal p-complement and their Fitting length is at most three.     

Congruence distributive quasivarieties whose finitely subdirectly irreducible members form a universal class

By a congruence distributive quasivariety we mean any quasivarietyK of algebras having the property that the lattices of those congruences of members ofK which determine quotient algebras belonging toK are distributive. This paper is an attempt to study congruence distributive quasivarieties with the additional property that their classes of relatively finitely subdirectly irreducible members are axiomatized by sets of universal sentences. We deal with the problem of characterizing such quasivarieties and the problem of their finite axiomatizability.Presented by Joel Berman.To the memory of Basia Czelakowska.     

The strong endomorphism kernel property for modular p-algebras and for distributive lattices

We study the strong endomorphism kernel property (SEKP) for some classes of universal algebras. Using the Katriňák-Mederly triple construction, we prove a universal equivalent condition under which a modular p-algebra has SEKP. As a consequence, we characterize distributive lattices with top element that enjoy SEKP. Using Priestley duality, we also characterize unbounded distributive lattices that have SEKP.     

On symmetric algebras of fuzzy sets

In this paper all symmetric algebras of fuzzy sets taking values on a distributive directly nondecomposable lattice with universal bounds 1 and 0, and their classes with respect to automorphisms, are studied. Special attention is devoted to the case where fuzzy sets take values on [0, 1].     

Computational complexity for bounded distributive lattices with negation

We study the computational complexity of the universal and quasi-equational theories of classes of bounded distributive lattices with a negation operation, i.e., a unary operation satisfying a subset of the properties of the Boolean negation. The upper bounds are obtained through the use of partial algebras. The lower bounds are either inherited from the equational theory of bounded distributive lattices or obtained through a reduction of a global satisfiability problem for a suitable system of propositional modal logic.