Volodymyr Gavrylkiv

Vasyl Stefanyk Precarpathian National University · Mathematics and Computer Science

A doppelsemigroup (G,⊣,⊢) is calledcyclic if (G,⊣) is a cyclic group. In the paper, we describe up to isomorphism all cyclic (strong) doppelsemigroups. We prove that up to isomorphism there exist τ(n) finite cyclic (strong) doppelsemigroups of order n, where τ is the number of divisors function. Also there exist infinite countably many pairwise non...

A subset $B$ of a group $G$ is called a basis of $G$ if each element $g\in G$ can be written as $g=ab$ for some elements $a,b\in B$. The smallest cardinality $|B|$ of a basis $B\subseteq G$ is called the basis size of $G$ and is denoted by $r[G]$. We prove that each finite group $G$ has $r[G]>\sqrt{|G|}$. If $G$ is Abelian, then $r[G]\ge \sqrt{2|G|...

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 doppelsemig...

A b s t r ac t. In the paper we characterize all interassociates of some non-inverse semigroups and describe up to isomorphism all three-element (strong) doppelsemigroups and their automor-phism groups. We prove that there exist 75 pairwise non-isomorphic three-element doppelsemigroups among which 41 doppelsemigroups are commutative. Non-commutativ...

A family L of subsets of a set X is called linked if A ∩ B = ∅ for any A, B ∈ L. A linked family M of subsets of X is maximal linked if M coincides with each linked family L on X that contains M. The superextension λ(X) of X consists of all maximal linked families on X. Any associative binary operation * : X × X → X can be extended to an associativ...

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 $\lambda(X)$ of $X$ consists of all maximal linked f...

A family 𝒰 of nonempty subsets of a set X is called an upfamily if for each set U∈𝒰 any set F⊃U belongs to 𝒰. The extension υ(X) of X consists of all upfamilies on X. Any associative binary operation ∗:X×X→X can be extended to an associative binary operation ∗:υ(X)×υ(X)→υ(X). In the paper, we study automorphisms of extensions of groups, finite mono...

A family \( \mathcal{A} \) of non-empty subsets of a set X is called an upfamily, if, for each set \( A\in \mathcal{A} \); any set B ⊃ A belongs to \( \mathcal{A} \). An upfamily \( \mathrm{\mathcal{L}} \) is called k-linked, if \( \cap \mathrm{\mathcal{F}}\ne \varnothing \) for any subfamily \( \mathrm{\mathcal{F}}\subset \mathrm{\mathcal{L}} \) o...

The through study of various extensions of semigroups was started in [12] and continued in [1]-[10], [13]-[19]. The largest among these extensions is the semigroup υ(S) of all upfamilies on a semigroup S. A family M of non-empty subsets of a set X is called an upfamily if for each set A ∈ M any subset B ⊃ A of X belongs to M. Each family B of non-e...

The superextension $\lambda(X)$ 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 $*: \lambda(X)\times\lambda(X)\to\lambda(X)$. In the paper we study isomorphisms of superextensions of groups and prove that two groups are isomorphic i...

A family A of non-empty subsets of a set X is called an upfamily if for each set A ∈ A any set B ⊃ A belongs to A. An upfamily L of subsets of X is said to be linked if A ∩ B ≠ ∅ for all A, B ∈ L. A linked upfamily M of subsets of X is maximal linked if M coincides with each linked upfamily L on X that contains M. The superextension λ(X) consists o...

A family $\mathcal{A}$ of non-empty subsets of a set $X$ is called an {\em upfamily} if for each set $A\in\mathcal{A}$ any set $B\supset A$ belongs to $\mathcal{A}$. An upfamily $\mathcal L$ of subsets of $X$ is said to be {\em linked} if $A\cap B\ne\emptyset$ for all $A,B\in\mathcal L$. A linked upfamily $\mathcal M$ of subsets of $X$ is {\em maxi...

Given a group G we study right and left zeros, idempotents, the minimal ideal, left cancellable and right cancellable elements of the semigroup N<ω(G) of centered upfamilies and characterize groups G whose extensions N<ω(G) are commutative. We finish the paper with the complete description of the structure of the semigroups N<ω(G) for all groups G...

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 $\Delta[G]$. The fraction $\eth[G]:=\Delta[G]/{\sqrt{|G|}}$ is cal...

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 $\Delta[G]$. The fraction $\eth[G]:=\frac{\Delta[G]}{\sqrt{|G|}}$...

A subset $B$ of an Abelian group $G$ is called a {\em 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 {\em difference size} of $G$ and is denoted by $\Delta[G]$. We prove that for every $n\in\IN$ th...

A subset $A$ of a group $G$ is called self-linked if $A\cap gA\ne\emptyset$ for every $g\in G$. The smallest cardinality $|A|$ of a self-linked subset $A\subset G$ is called the self-linked number $sl(G)$ of $G$. In the paper we find lower and upper bounds for the self-linked number $sl(G)$ of a finite group $G$ and prove that $$\frac{1+\sqrt{4|G|+...

Given a group (Formula presented.), we study right and left zeros, idempotents, the minimal ideal, left cancelable and right cancelable elements of the semigroup (Formula presented.) of (Formula presented.)-linked upfamilies and characterize groups (Formula presented.) whose extensions (Formula presented.) are commutative. We finish the paper with...

A subset B of a group G is called a basis of G if G = B². The smallest cardinality of a basis of G is called the basis size of G. We prove upper bounds for basis sizes of dihedral and Boolean groups. We find a lower bound for the basis size of a Boolean group. We also calculate basis sizes for dihedral and Boolean groups of small orders.

Given a finite monogenic semigroup S, we study the minimal ideal, the center, left cancelable, and right cancelable elements of the extension () S N ω < consisting of centered upfamilies on S and characterize monogenic semigroups whose extensions are commutative.

We characterize semigroups X whose semigroups of filters ϕ(X), maximal linked systems λ(X), linked upfamilies N2(X), and upfamilies υ(X) are commutative.

12 липня 2013 року, у розквiтi творчих сил та планiв, помер вiдомий український математик-алгебраїст, доктор фiзико-математичних наук, професор, завiдувач кафедри математики Нацiонального унiверситету "Києво-Могилянська академiя"-Юрiй Вiкто-рович Боднарчук. Юрiй Вiкторович Боднарчук народився 13 жовтня 1955 року в Черкасах у родинi ма-тематикiв, Бо...

Given a cyclic semigroup S we study right and left zeros, singleton left ideals, the minimal ideal, left cancelable and right cancelable elements of superextensions λ(S) and characterize cyclic semi-groups whose superextensions are commutative.

We characterize semigroups $X$ whose semigroups of filters $\varphi(X)$, maximal linked systems $\lambda(X)$, linked upfamilies $N_2(X)$, and upfamilies $\upsilon(X)$ are commutative.

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

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 $\beta(X)$, the superextension $\lambda(X)$, and the space of filters $\phi(X)$ on $X$ as closed subsemigroups. We prove that $\upsilon(X)$ is a semilattice iff $\lambda(X)$ is...

Given a group $X$ we study the algebraic structure of the compact right-topological semigroup $G(X)$ consisting of inclusion hyperspaces on $X$. This semigroup contains the semigroup $\lambda(X)$ of maximal linked systems as a closed subsemigroup. We construct a faithful representation of the semigroups $G(X)$ and $\lambda(X)$ in the semigroup $\ma...

Гаврилків Володимир Михайлович УДК 512.53 Алгебро-топологiчнi структури на суперрозширеннях Спецiальнiсть 01.01.06 алгебра i теорiя чисел Дисертацiя на здобуття наукового ступеня кандидата фiзико-математичних наук Науковий керiвник д.ф.-м.н., професор Банах Тарас Онуфрiйович Iвано-Франкiвськ 2009

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.

In the talk we shall discuss the structure of minimal (left) ideals of the superex- tensions �(G) of Abelian groups G. By definition, a familyL of subsets of a set X is called a linked system on X if A\ B is nonempty for all A,B 2 L. Such a linked system is maximal linked if it coincides with any linked system M on X that containsL. The space �(X)...

Given a group $X$ we study the algebraic structure of the compact right-topological semigroup $\lambda(X)$ consisting of maximal linked systems on $X$. This semigroup contains the semigroup $\beta(X)$ of ultrafilters as a closed subsemigroup. We construct a faithful representation of the semigroup $\lambda(X)$ in the semigroup of all self-maps of t...

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

Given a group $X$ we study the algebraic structure of its superextension $\lambda(X)$. 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 characterize rig...

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...

Given a group X we study the algebraic structure of the compact right-topological semigroup $\lambda(X)$ consisting of maximal linked systems on X. This semigroup, called the superextension of X, contains the semigroup $\beta(X)$ of ultrafilters as a closed subsemigroup. We construct a faithful representation of $\lambda(X)$ in the semigroup P(X)P(...

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 A • B = {C ⊂ X : {x ∈ X : x −1 C ∈ B} ∈ A} that extends the group operation of X. We show that the subsemigroup λ • (X) of free maximal linked systems c...

We extend the construction of the space G(X) of inclusion hyperspaces to non-compact spaces, prove the supercompactness of G(X) for any T 1 -space X, study the algebraic structure of G(X), and define some important subspaces of G(X).