On Gao’s conjecture related with finite field high order elements

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Roman Popovych

Department of Specialized Computer Systems,
Lviv Polytechnic National University

December 5, 2017 at 15:05 in Lecture Room 377

Abstract of talk

The talk will be devoted to the Gao’s conjecture connected with the construction of elements of provable high multiplicative order in general finite fields.

Roman Popovych
December 5, 2017

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Semigroups and S-polygons with annihilation conditions

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Yuriy Ishchuk

Department of Algebra and Logic
Faculty of Mechanics and Mathematics
Ivan Franko National University of Lviv

November 21, 2017 at 15:05 in Lecture Room 377

Abstract of talk

The notions of a semi-commutative semigroup and an abelian S-polygon can be introduced by analogy with the notions of semi-commutative, abelian modules and rings.

A semigroup S is called a semi-commutative semigroup if for any x,y\in S, xy=0 implies xSy=0. A right S-polygon A_S is called abelian if for any a\in A_S and any s\in S, any idempotent e\in S, ase = aes. Using the notions of Baer’s conditions for modules I will introduce p.p.-Baer S-polygons and prove that if A_S is a p.p.-Baer S-polygon, then the conditions for A_S to be a reduced, symmetric, semicommutative and an abelian S-polygon are equivalent.

Yuriy Ishchuk
November 21, 2017

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Semiscalar equivalence of third order polynomial matrices with only one characteristic root

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Bogdan Shavarovskii

Department of Algebra, Pidstryhach
Institute for Applied Problems of
Mechanics and Mathematics of
NAS of Ukraine

November 7, 2017 at 15:05 in Lecture Room 377

Abstract of talk

The canonical forms of polynomial matrices of the third order with only one characteristic root are established.

Bogdan Shavarovskii
November 7, 2017

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Matrix linear equations in two variables over rings

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Vasyl’ Petrychkovych

Department of Algebra, Pidstryhach
Institute for Applied Problems of
Mechanics and Mathematics of
NAS of Ukraine

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Nataliia Dzhaliuk

Department of Algebra, Pidstryhach
Institute for Applied Problems of
Mechanics and Mathematics of
NAS of Ukraine

October 17, 2017 at 15:05 in Lecture Room 377

Abstract of talk

We consider matrix linear equations of form AX+BY=C, AX+YB=C over commutative Bezout rings.
The goal is to present the method of solving such equations using the matrix pair standard form
with respect to generalized equivalence and to establish the particular solutions of the
equation, their construction method and the uniqueness criterion. Moreover, for certain classes of matrix equations
one can find their general solutions.

Vasyl’ Petrychkovych
Nataliia Dzhaliuk

October 17, 2017

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Matrix reduction over Bezout rings

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Andriy Sagan

Department of Algebra and Logic
Faculty of Mechanics and Mathematics
Ivan Franko National University of Lviv

May 16, 2017 at 15:05 in Lecture Room 377

Abstract of talk

It this talk the author will present main results of his PhD thesis.

We will discuss an elementary matrix reduction over different classes of commutative and noncommutative rings. In particular, we will specify the necessary and sufficient conditions for a quasi-Euclidean duo-ring being a ring with elementary matrix reduction. Using this criterion we will be able to describe various duo-rings with elementary matrix reduction. Moreover, it will be established that any right Hermite stable range one ring is a right \omega-Euclidean domain.

Additionally, it will be proved that for any pair of n\times n full matrices over elementary divisor ring, there exists a right (left) divisibility chain of length 2(n-1) , and over PID – of length 2.

Among the other results we will introduce the concept of e-atomic commutative ring and describe its main properties. Furthermore, we will show that any e-atomic Bezout domain and locally e-atomic Bezout ring are rings with elementary matrix reduction.

Finally, we will prove that a integral domain R is an \omega-Euclidean ring if and only if a ring of formal Laurent series R [[ x,x^{-1}]] is an \omega-Euclidean ring.

Andriy Sagan
May 16, 2017

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Matrix value on system of matrix diagonal elements roots and its properties

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Volodymyr Shchedryk

Department of Algebra, Pidstryhach
Institute for Applied Problems of
Mechanics and Mathematics of
NAS of Ukraine

April 25, 2017 at 15:05 in Lecture Room 377

Abstract of talk

We are going to describe the properties of polynomial matrix value on the system of matrix diagonal elements roots.

Volodymyr Shchedryk
April 25, 2017

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On the semigroup ID

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Anatolii Savchuk

Department of Algebra and Logic
Faculty of Mechanics and Mathematics
Ivan Franko National University of Lviv

April 04, 2017 at 15:05 in Lecture Room 377

Abstract of talk

There will be demonstrated some results, received in collaboration with O. Gutik and concerning the inverse semigroup ID_\infty of partially defined automorphisms of integers \mathbb{Z}.

In particular, the report includes the structural theorem for the semigroup ID_\infty.

Anatolii Savchuk
April 04, 2017

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On classification of low-dimensional Lie algebras

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Fedorchuk Vasyl’, Fedorchuk Volodymyr


Institute of Mathematics, Pedagogical University of Kraków, Poland;
Department of Algebra, Pidstryhach Institute for Applied
Problems of Mechanics and Mathematics
National Academy of Sciences of Ukraine

March 21, 2017 at 15:05 in Lecture Room 372

Abstract of talk

We plan to present a short review of the results related to the classification of low-dimensional (\mathrm{dim}(L) \leq 4 )
Lie algebras.

Fedorchuk Vasyl’
Fedorchuk Volodymyr

March 21, 2017

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Commutative Bezout domains whose non-trivial finite homomorphic images are semipotent rings

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Bohdan Zabavsky, Sofia Bilavska-Kudlata

Department of Algebra and Logic
Faculty of Mechanics and Mathematics
Ivan Franko National University of Lviv

March 7, 2017 at 15:05 in Lecture Room 377

Abstract of talk

In this talk we will investigate the structure of commutative Bezout domain whose non-trivial finite homorphic images are semipotent rings.

Bohdan Zabavsky
Sofia Bilavska-Kudlata

March 7, 2017

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