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Volodymyr Gavrylkiv

Publications and source records attributed to Volodymyr Gavrylkiv.

16 recordsLinked to original sources
arXiv2026

On certain classes of commutative g-dimonoids

We study g-dimonoids, which are algebraic structures equipped with two associative binary operations satisfying a prescribed system of axioms. We develop a construction of g-dimonoids based on inflations of semigroups, construct several classes of 2-dinilpotent commutative g-dimonoids, and investigate the isomorphism problem for these classes. As an application of these results, we obtain a structural description of all commutative g-dimonoids of order 3, up to isomorphism.

math.GR↗
arXiv2025

On some classes of noncommutative dimonoids

The present paper is devoted to the study of dimonoids, algebraic structures with two associative binary operations that satisfy a prescribed system of axioms. We investigate the properties of dual dimonoids. In the class of noncommutative dimonoids, we construct a number of abelian, nonabelian, and rectangular dimonoids. The structure of these objects is analyzed, their automorphism groups and halos are computed.

math.GR↗
arXiv2020

Interassociativity and three-element doppelsemigroups

In the paper we characterize all interassociates of some non-inverse semigroups and describe up to isomorphism all three-element (strong) doppelsemigroups and their automorphism groups. We prove that there exist $75$ pairwise non-isomorphic three-element doppelsemigroups among which $41$ doppelsemigroups are commutative. Non-commutative doppelsemigroups are divided into $17$ pairs of dual doppelsemigroups. Also up to isomorphism there are $65$ strong doppelsemigroups of order $3$, and all non-strong doppelsemigroups are not commutative.

math.GR↗
arXiv2019

Automorphism groups of superextensions of finite monogenic semigroups

A family $\mathcal L$ of subsets of a set $X$ is called linked if $A\cap B\ne\emptyset$ for any $A,B\in\mathcal L$. A linked family $\mathcal M$ of subsets of $X$ is maximal linked if $\mathcal M$ coincides with each linked family $\mathcal L$ on $X$ that contains $\mathcal M$. The superextension $λ(X)$ of $X$ consists of all maximal linked families on $X$. Any associative binary operation $* : X\times X \to X$ can be extended to an associative binary operation $*: λ(X)\timesλ(X)\toλ(X)$. In the paper we study automorphisms of the superextensions of finite monogenic semigroups and characteristic ideals in such semigroups. In particular, we describe the automorphism groups of the superextensions of finite monogenic semigroups of cardinality $\leq 5$.

math.GR↗
arXiv2018

Automorphism groups of superextensions of groups

The superextension $λ(X)$ of a set $X$ consists of all maximal linked families on $X$. Any associative binary operation $*: X\times X \to X$ can be extended to an associative binary operation $*: λ(X)\timesλ(X)\toλ(X)$. In the paper we study isomorphisms of superextensions of groups and prove that two groups are isomorphic if and only if their superextensions are isomorphic. Also we describe the automorphism groups of superextensions of all groups of cardinality $\leq 5$.

math.GR↗
arXiv2017

Difference bases in finite Abelian groups

A subset $B$ of a group $G$ is called a difference basis of $G$ if each element $g\in G$ can be written as the difference $g=ab^{-1}$ of some elements $a,b\in B$. The smallest cardinality $|B|$ of a difference basis $B\subset G$ is called the difference size of $G$ and is denoted by $Δ[G]$. The fraction $ð[G]:=\frac{Δ[G]}{\sqrt{|G|}}$ is called the difference characteristic of $G$. Using properies of the Galois rings, we prove recursive upper bounds for the difference sizes and characteristics of finite Abelian groups. In particular, we prove that for a prime number $p\ge 11$, any finite Abelian $p$-group $G$ has difference characteristic $ð[G]<\frac{\sqrt{p}-1}{\sqrt{p}-3}\cdot\sup_{k\in\mathbb N}ð[C_{p^k}]<\sqrt{2}\cdot\frac{\sqrt{p}-1}{\sqrt{p}-3}$. Also we calculate the difference sizes of all Abelian groups of cardinality $<96$.

math.CO↗
arXiv2017

Difference bases in dihedral groups

A subset $B$ of a group $G$ is called a difference basis of $G$ if each element $g\in G$ can be written as the difference $g=ab^{-1}$ of some elements $a,b\in B$. The smallest cardinality $|B|$ of a difference basis $B\subset G$ is called the difference size of $G$ and is denoted by $Δ[G]$. The fraction $ð[G]:=Δ[G]/{\sqrt{|G|}}$ is called the difference characteristic of $G$. We prove that for every $n\in\mathbb N$ the dihedral group $D_{2n}$ of order $2n$ has the difference characteristic $\sqrt{2}\leð[D_{2n}]\leq\frac{48}{\sqrt{586}}\approx1.983$. Moreover, if $n\ge 2\cdot 10^{15}$, then $ð[D_{2n}]<\frac{4}{\sqrt{6}}\approx1.633$. Also we calculate the difference sizes and characteristics of all dihedral groups of cardinality $\le80$.

math.CO↗
arXiv2017

Difference bases in cyclic groups

A subset $B$ of an Abelian group $G$ is called a difference basis of $G$ if each element $g\in G$ can be written as the difference $g=a-b$ of some elements $a,b\in B$. The smallest cardinality $|B|$ of a difference basis $B\subset G$ is called the difference size of $G$ and is denoted by $Δ[G]$. We prove that for every $n\in\mathbb N$ the cyclic group $C_n$ of order $n$ has difference size $\frac{1+\sqrt{4|n|-3}}2\le Δ[C_n]\le\frac32\sqrt{n}$. If $n\ge 9$ (and $n\ge 2\cdot 10^{15}$), then $Δ[C_n]\le\frac{12}{\sqrt{73}}\sqrt{n}$ (and $Δ[C_n]<\frac2{\sqrt{3}}\sqrt{n}$). Also we calculate the difference sizes of all cyclic groups of cardinality $\le 100$.

math.CO↗
arXiv2012

Algebra in superextensions of inverse semigroups

We find necessary and sufficient conditions on an (inverse) semigroup $X$ under which its semigroups of maximal linked systems $λ(X)$, filters $ϕ(X)$, linked upfamilies $N_2(X)$, and upfamilies $\upsilon(X)$ are inverse.

math.GR↗
arXiv2011

Algebra in the superextensions of groups: minimal left ideals

We prove that the minimal left ideals of the superextension $λ(Z)$ of the discrete group $Z$ of integers are metrizable topological semigroups, topologically isomorphic to minimal left ideals of the superextension $λ(Z_2)$ of the compact group $Z_2$ of integer 2-adic numbers.

math.GN↗
arXiv2011

Algebra in the superextensions of semilattices

Given a semilattice $X$ we study the algebraic properties of the semigroup $\upsilon(X)$ of upfamilies on $X$. The semigroup $\upsilon(X)$ contains the Stone-Cech extension $β(X)$, the superextension $λ(X)$, and the space of filters $ϕ(X)$ on $X$ as closed subsemigroups. We prove that $\upsilon(X)$ is a semilattice iff $λ(X)$ is a semilattice iff $ϕ(X)$ is a semilattice iff the semilattice $X$ is finite and linearly ordered. We prove that the semigroup $β(X)$ is a band if and only if $X$ has no infinite antichains, and the semigroup $λ(X)$ is commutative if and only if $X$ is a bush with finite branches.

math.GR↗
arXiv2010

Algebra in superextensions of twinic groups

Given a group $X$ we study the algebraic structure of the compact right-topological semigroup $λ(X)$ consisting of maximal linked systems on $X$. This semigroup contains the semigroup $β(X)$ of ultrafilters as a closed subsemigroup. We construct a faithful representation of the semigroup $λ(X)$ in the semigroup of all self-maps of the power-set of $X$ and using this representation describe the structure of minimal ideal and minimal left ideals of $λ(X)$ for each twinic group $X$. The class of twinic groups includes all amenable groups and all groups with periodic commutators but does not include the free group with two generators.

math.GR↗
arXiv2010

Extending binary operations to funtor-spaces

Given a continuous monadic functor T in the category of Tychonov spaces for each discrete topological semigroup X we extend the semigroup operation of X to a right-topological semigroup operation on TX whose topological center contains the dense subsemigroup of all elements of TX that have finite support.

math.CT↗
arXiv2008

Algebra in superextension of groups, II: cancelativity and centers

Given a countable group $X$ we study the algebraic structure of its superextension $λ(X)$. This is a right-topological semigroup consisting of all maximal linked systems on $X$ endowed with the operation $$\mathcal A\circ\mathcal B=\{C\subset X:\{x\in X:x^{-1}C\in\mathcal B\}\in\mathcal A\}$$ that extends the group operation of $X$. We show that the subsemigroup $λ^\circ(X)$ of free maximal linked systems contains an open dense subset of right cancelable elements. Also we prove that the topological center of $λ(X)$ coincides with the subsemigroup $λ^\bullet(X)$ of all maximal linked systems with finite support. This result is applied to show that the algebraic center of $λ(X)$ coincides with the algebraic center of $X$ provided $X$ is countably infinite. On the other hand, for finite groups $X$ of order $3\le|X|\le5$ the algebraic center of $λ(X)$ is strictly larger than the algebraic center of $X$.

math.GN↗
arXiv2008

Right-topological semigroup operations on inclusion hyperspaces

We show that for any discrete semigroup $X$ the semigroup operation can be extended to a right-topological semigroup operation on the space $G(X)$ of inclusion hyperspaces on $X$. We detect some important subsemigroups of $G(X)$, study the minimal ideal, the (topological) center, left cancelable elements of $G(X)$, and describe the structure of the semigroups $G(\IZ_n)$ for small numbers $n$.

math.GN↗