ალექსანდრე ლაშხი

We study chain conditions in D-semimodular lattices whose lattices naturally appear as submodule lattices of modules over principal ideal domains

The idea to consider different unions of points, lines, planes, etc., is rather old. Many important configurations of such kinds are geometric (or matroidal) lattices. In this work, we study Desargues, Pappus, and Pasch configurations in D-semimodular lattices

In the present paper the lattice of cosets R(G) is constructed for Hall’s power group G over the ring W. This lattice is called the affine or coset lattice of G. Since in the lattice R(G) only the elements of G cover \emptyset, the isomorphism f : R( G)\arrow R(G) defines the bijection f : G \arrow G_1. Among all possible isomorphisms f we shall select in the sequel those for which f (1)=1 . Such isomorphisms will be called the affine isomorphisms. We prove the following theorem: Let f : R( G)\arrow R(G) be an affine isomorphism between the w-power groups G and G_1 over the fields W and W_1 , respectively, if G is nilpotent of class 2 then f is either a semilinear isomorphism or a semilinear antiisomorphism with respect to the isomorphism \sigma:W\arrow W_1 . The given example shows that the theorem is false for the class of nilpotency \geq 3 .

For a Lie algebra over a ring, the lattice of cosets is constructed. Necessary and sufficient conditions for the distributivity, modularity, and semimodularity of coset lattices are found. The fundamental theorem of affine geometry for nilpotent Lie algebras of class 2 is proved

To obtain the representation (L, R) of Lie algebras over the ring \Lambda, we construct the lattice of subrepresentations L(L, R). Relations between the algebras L and R and the lattice L(L, R) are studied. It turns out that in some cases the isomorphism of the lattice L(L, R) can be continued so as to obtain a wider sublattice L(L\LambdaR) consisting of subalgebras of a semidirect product L\LambdaR.

Some assertions on lattice isomorphisms of nilpotent and free Lie algebras are proved

For a Lie algebra over a ring, the lattice of cosets is constructed. Necessary and sufficient conditions for the distributivity, modularity, and semimodularity of coset lattices are found. The fundamental theorem of affine geometry for nilpotent Lie algebras of class 2 is proved.

The idea to consider different unions of points, lines, planes, etc., is rather old. Many important configurations of such kinds are geometric (or matroidal) lattices. In this work, we study Desargues, Pappus, and Pasch configurations in D-semimodular lattices.

For Hall power groups over a ring, the lattice of cosets is constructed. For class-2 nilpotent groups over the field, the fundamental theorem of affine geometry is proved. The given example shows that the theorem is invalid for the class of nilpotency ≥3.

We study chain conditions in D-semimodular lattices whose lattices naturally appear as submodule lattices of modules over principal ideal domains

We discuss one historical aspect of the mathematical education in Georgia, namely, the textbook on mathematics by Ilya Zhgenti

In the theory of abelian groups the following two results are fundamental: on the representation of finitely generated abelian groups in the form of a direct sum of cyclic subgroups and the classification of locally cyclic groups. The generalization of the first of them for the case of modules over the principal ideal domains is classical. However, there are no published works for locally cyclic modules over the principal ideal domains. The aim of this paper is to fill this gap, namely, classification locally cyclic modules over the principal ideal domains

This paper is a direct continuation of [26], where we proved that every normal lattice isomorphism of supersolvable Lie ring is induced at the isomorphism. In the present paper,we generalize this theorem for rational Lie rings