Alexander Lashkhi

Doctor of Science

Muskhelishvili Institute of Computational Mathematics

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Research articles in high impact factor and local Scientific Journals


Chain conditions in D-semimodular lattices. Journal of Mathematical Sciences, Vol. 197, No. 6, March, 2014Grant Project

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

doi.org/10.1007/s10958-014-1761-y
Modeling of ring geometry von Neumann's point of view lattices. Journal of Mathematical Sciences, Vol. 191, No. 6, June, 2013Grant Project

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

doi.org/10.1007/s10958-013-1360-3
Affine lattice of power groups. Bull. Georgian Acad. Sci. Vol. 6, 3, (2012), 25-29Grant Project

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 .

http://science.org.ge/old/moambe/6-3/Bokelavadze.pdf
Fundamental theorem of affine geometry for Lie algebras. Journal of Mathematical Sciences, Vol. 191, No. 6, June, 2013Grant Project

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

doi.org/10.1007/s10958-013-1362-1
Lattices of subrepresentations of Lie algebras and their isomorphisms. Journal of Mathematical Sciences, Vol. 153, No. 4, 2008State Target Program

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.

doi.org/10.1007/s10958-008-9135-y
Lattice isomorphisms of nilpotent and free Lie algebras. Bull. Georgian Acad. Sci. Vol. 7, 1, (2013), 16-20State Target Program

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

http://science.org.ge/old/moambe/7-1/Lashkhi%2016-20.pdf
Fundamental theorem of affine geometry for Lie algebras. Journal of Mathematical Sciences, Vol. 191, No. 6, June, 2013Grant Project

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.

http://www.rmi.ge/proceedings/volumes/pdf/v160-2.pdf
Modelling of ring geometry from von Neumnn's point of view. Journal of Mathematical Sciences, Vol. 191, No. 6, June, 2013Grant Project

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.

doi.org/10.1007/s10958-013-1360-3
Affine Lattices of Power Groups. Journal of Mathematical Sciences, Vol. 193, No. 3, September, 2013Grant Project

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.

doi.org/10.1007/s10958-013-1465-8
Chain conditions in D-semi­mo­dular lattices. Journal of Mathematical Sciences, Vol. 197, No. 6, March, 2014Grant Project

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

doi.org/10.1007/s10958-014-1761-y
On the mathematical educa­tion in Georgia: textbook on mathematics by Ilya Zhgenti. Journal of Mathematical Sciences, Vol. 197, No. 6, March, 2014 State Target Program

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

doi.org/10.1007/s10958-014-1758-6
On Locally cyclic and distributive modules and algebras. Bull. Georgian Acad. Sci. Vol. 9, 3, (2015), 20-25Grant Project

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

http://science.org.ge/bnas/t9-n3/03-Lashkhi.pdf
Projection of Rational Lie Rings. Journal of Mathematical Sciences, Vol. 218, No. 6, Month, 2016Grant Project

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

doi.org/10.1007/s10958-016-3066-9

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