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Oleg Gutik

Publications and source records attributed to Oleg Gutik.

At least 55 records · Page 3Linked to original sources
arXiv2020

On a semitopological extended bicyclic semigroup with adjoined zero

In the paper it is shown that every Hausdorff locally compact semigroup topology on the extended bicyclic semigroup with adjoined zero $\mathscr{C}_{\mathbb{Z}}^{\mathbf{0}}$ is discrete, but on $\mathscr{C}_{\mathbb{Z}}^{\mathbf{0}}$ there exist $\mathfrak{c}$ many different Hausdorff locally compact shift-continuous topologies. Also, it is constructed on $\mathscr{C}_{\mathbb{Z}}^{\mathbf{0}}$ the unique minimal shift continuous topology and the unique minimal inverse semigroup topology.

math.GR↗
arXiv2020

On inverse submonoids of the monoid of almost monotone injective co-finite partial selfmaps of positive integers

In this paper we study submonoids of the monoid $\mathscr{I}_\infty^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ of almost monotone injective co-finite partial selfmaps of positive integers $\mathbb{N}$. Let $\mathscr{I}_\infty^{\!\nearrow}(\mathbb{N})$ be a submonoid of $\mathscr{I}_\infty^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ which consists of cofinite monotone partial bijections of $\mathbb{N}$ and $\mathscr{C}_{\mathbb{N}}$ be a subsemigroup $\mathscr{I}_\infty^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ which is generated by the partial shift $n\mapsto n+1$ and its inverse partial map. We show that every automorphism of a full inverse subsemigroup of $\mathscr{I}_\infty^{\!\nearrow}(\mathbb{N})$ which contains the semigroup $\mathscr{C}_{\mathbb{N}}$ is the identity map. We construct a submonoid $\mathbf{I}\mathbb{N}_\infty^{[\underline{1}]}$ of $\mathscr{I}_\infty^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ with the following property: if $S$ is an inverse submonoid of $\mathscr{I}_\infty^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathbf{I}\mathbb{N}_\infty^{[\underline{1}]}$ as a submonoid, then every non-identity congruence $\mathfrak{C}$ on $S$ is a group congruence. We show that if $S$ is an inverse submonoid of $\mathscr{I}_\infty^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathscr{C}_{\mathbb{N}}$ as a submonoid then $S$ is simple and the quotient semigroup $S/\mathfrak{C}_{\mathbf{mg}}$, where $\mathfrak{C}_{\mathbf{mg}}$ is minimum group congruence on $S$, is isomorphic to the additive group of integers. Also, we study topologizations of inverse submonoids of $\mathscr{I}_\infty^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ which contain $\mathscr{C}_{\mathbb{N}}$ and embeddings of such semigroups into compact-like topological semigroups.

math.GR↗
arXiv2019

The Lawson number of a semitopological semilattice

For a Hausdorff topologized semilattice $X$ its $Lawson\;\; number$ $\barΛ(X)$ is the smallest cardinal $κ$ such that for any distinct points $x,y\in X$ there exists a family $\mathcal U$ of closed neighborhoods of $x$ in $X$ such that $|\mathcal U|\leκ$ and $\bigcap\mathcal U$ is a subsemilattice of $X$ that does not contain $y$. It follows that $\barΛ(X)\le\barψ(X)$, where $\barψ(X)$ is the smallest cardinal $κ$ such that for any point $x\in X$ there exists a family $\mathcal U$ of closed neighborhoods of $x$ in $X$ such that $|\mathcal U|\leκ$ and $\bigcap\mathcal U=\{x\}$. We prove that a compact Hausdorff semitopological semilattice $X$ is Lawson (i.e., has a base of the topology consisting of subsemilattices) if and only if $\barΛ(X)=1$. Each Hausdorff topological semilattice $X$ has Lawson number $\barΛ(X)\leω$. On the other hand, for any infinite cardinal $λ$ we construct a Hausdorff zero-dimensional semitopological semilattice $X$ such that $|X|=λ$ and $\barΛ(X)=\barψ(X)=cf(λ)$. A topologized semilattice $X$ is called (i) $ω$-$Lawson$ if $\barΛ(X)\leω$; (ii) $complete$ if each non-empty chain $C\subset X$ has $\inf C\in\overline{C}$ and $\sup C\in\overline{C}$. We prove that for any complete subsemilattice $X$ of an $ω$-Lawson semitopological semilattice $Y$, the partial order $\le_X=\{(x,y)\in X\times X:xy=x\}$ of $X$ is closed in $Y\times Y$ and hence $X$ is closed in $Y$. This implies that for any continuous homomorphism $h:X\to Y$ from a compete topologized semilattice $X$ to an $ω$-Lawson semitopological semilattice $Y$ the image $h(X)$ is closed in $Y$.

math.GN↗
arXiv2019

On the monoid of cofinite partial isometries of $\mathbb{N}^n$ with the usual metric

In this paper we study the structure of the monoid $\mathbf{I}\mathbb{N}_{\infty}^n$ of cofinite partial isometries of the $n$-th power of the set of positive integers $\mathbb{N}$ with the usual metric for a positive integer $n\geqslant 2$. We describe the elements of the monoid $\mathbf{I}\mathbb{N}_{\infty}^n$ as partial transformation of $\mathbb{N}^n$, the group of units and the subset of idempotents of the semigroup $\mathbf{I}\mathbb{N}_{\infty}^n$, the natural partial order and Green's relations on $\mathbf{I}\mathbb{N}_{\infty}^n$. In particular we show that the quotient semigroup $\mathbf{I}\mathbb{N}_{\infty}^n/\mathfrak{C}_{\textsf{mg}}$, where $\mathfrak{C}_{\textsf{mg}}$ is the minimum group congruence on $\mathbf{I}\mathbb{N}_{\infty}^n$, is isomorphic to the symmetric group $\mathscr{S}_n$ and $\mathscr{D}=\mathscr{J}$ in $\mathbf{I}\mathbb{N}_{\infty}^n$. Also, we prove that for any integer $n\geqslant 2$ the semigroup $\mathbf{I}\mathbb{N}_{\infty}^n$ is isomorphic to the semidirect product ${\mathscr{S}_n\ltimes_\mathfrak{h}(\mathscr{P}_{\infty}(\mathbb{N}^n),\cup)}$ of the free semilattice with the unit $(\mathscr{P}_{\infty}(\mathbb{N}^n),\cup)$ by the symmetric group $\mathscr{S}_n$.

math.GR↗
arXiv2019

On the lattice of weak topologies on the bicyclic monoid with adjoined zero

A Hausdorff topology $τ$ on the bicyclic monoid with adjoined zero $\mathcal{C}^0$ is called {\em weak} if it is contained in the coarsest inverse semigroup topology on $\mathcal{C}^0$. We show that the lattice $\mathcal{W}$ of all weak shift-continuous topologies on $\mathcal{C}^0$ is isomorphic to the lattice of all shift-invariant filters on $ω$ with an attached element $1$ endowed with the following partial order: $\mathcal{F}\leq \mathcal{G}$ iff $\mathcal{G}=1$ or $\mathcal{F}\subset \mathcal{G}$. Also, we investigate cardinal characteristics of the lattice $\mathcal{W}$. In particular, we proved that $\mathcal{W}$ contains an antichain of cardinality $2^{\mathfrak{c}}$ and a well-ordered chain of cardinality $\mathfrak{c}$. Moreover, there exists a well-ordered chain of first-countable weak topologies of order type $\mathfrak{t}$.

math.GN↗
arXiv2019

Extensions of semigroups by symmetric inverse semigroups of a bounded finite rank

We study the semigroup extension $\mathscr{I}_λ^n(S)$ of a semigroup $S$ by symmetric inverse semigroups of a bounded finite rank. We describe idempotents and regular elements of the semigroups $\mathscr{I}_λ^n(S)$ and $\overline{\mathscr{I}_λ^n}(S)$ show that the semigroup $\mathscr{I}_λ^n(S)$ ($\overline{\mathscr{I}_λ^n}(S)$) is regular, orthodox, inverse or stable if and only if so is $S$. Green's relations are described on the semigroup $\mathscr{I}_λ^n(S)$ for an arbitrary monoid $S$. We introduce the conception of a semigroup with strongly tight ideal series, and proved that for any infinite cardinal $λ$ and any positive integer $n$ the semigroup $\mathscr{I}_λ^n(S)$ has a strongly tight ideal series provides so has $S$. At the finish we show that for every compact Hausdorff semitopological monoid $(S,τ_S)$ there exists a unique its compact topological extension $\left(\mathscr{I}_λ^n(S),τ_{\mathscr{I}}^\mathbf{c}\right)$ in the class of Haudorff semitopological semigroups.

math.GR↗
arXiv2019

The semigroup of star partial homeomorphisms of a finite deminsional Euclidean space

In the paper the notion of a star partial homeomorphism of a finite dimensional Euclidean space $\mathbb{R}^n$ is introduced. We describe the structure of the semigroup $\mathbf{PStH}_{\mathbb{R}^n}$ of star partial homeomorphisms of the space $\mathbb{R}^n.$ The structure of the band of $\mathbf{PStH}_{\mathbb{R}^n}$ and Green's relations on $\mathbf{PStH}_{\mathbb{R}^n}$ are described. We show that $\mathbf{PStH}_{\mathbb{R}^n}$ is a bisimple inverse semigroup and every non-unit congruence on $\mathbf{PStH}_{\mathbb{R}^n}$ is a group congruence.

math.GR↗
arXiv2019

On the semigroup $\textbf{ID}_{\infty}$

We study the semigroup $\textbf{ID}_{\infty}$ of all partial isometries of the set of integers $\mathbb{Z}$. It is proved that the quotient semigroup $\textbf{ID}_{\infty}/\mathfrak{C}_{\textsf{mg}}$, where $\mathfrak{C}_{\textsf{mg}}$ is the minimum group congruence, is isomorphic to the group ${\textsf{Iso}}(\mathbb{Z})$ of all isometries of $\mathbb{Z}$, $\textbf{ID}_{\infty}$ is an $F$-inverse semigroup, and $\textbf{ID}_{\infty}$ is isomorphic to the semidirect product ${\textsf{Iso}}(\mathbb{Z})\ltimes_\mathfrak{h}\mathscr{P}_{\!\infty}(\mathbb{Z})$ of the free semilattice with unit $(\mathscr{P}_{\!\infty}(\mathbb{Z}),\cup)$ by the group ${\textsf{Iso}}(\mathbb{Z})$. We give the sufficient conditions on a shift-continuous topology $τ$ on $\textbf{ID}_{\infty}$ when $τ$ is discrete. A non-discrete Hausdorff semigroup topology on $\textbf{ID}_{\infty}$ is constructed. Also, the problem of an embedding of the discrete semigroup $\textbf{ID}_{\infty}$ into Hausdorff compact-like topological semigroups is studied.

math.GR↗
arXiv2019

The semigroup of monotone co-finite partial homeomorphisms of the real line

In the paper we investigate the semigroup of monotone co-finite partial homeomorphisms of the space of the usual real line $\mathbb{R}$. We prove that the inverse semigroup $\mathscr{P\!\!H}^+_{\!\!\operatorname{\textsf{cf}}}\!(\mathbb{R})$ is factorizable and $F$-inverse. We describe the structure of the band of the semigroup $\mathscr{P\!\!H}^+_{\!\!\operatorname{\textsf{cf}}}\!(\mathbb{R})$, its two-sided ideals, maximal subgroups and Green's relations. We prove that the quotient semigroup $\mathscr{P\!\!H}^+_{\!\!\operatorname{\textsf{cf}}}\!(\mathbb{R})/\mathfrak{C}_{\textsf{mg}}$, where $\mathfrak{C}_{\textsf{mg}}$ is the maximum group congruence on $\mathscr{P\!\!H}^+_{\!\!\operatorname{\textsf{cf}}}\!(\mathbb{R})/\mathfrak{C}_{\textsf{mg}}$, is isomorphic to the group of all oriental homeomorphisms of the space $\mathbb{R}$, and showe that the semigroup $\mathscr{P\!\!H}^+_{\!\!\operatorname{\textsf{cf}}}\!(\mathbb{R})$ is isomorphic to a semidirect product $\mathscr{H}^+\!(\mathbb{R})\ltimes_\mathfrak{h}\mathscr{P}_{\!\infty}(\mathbb{R})$ of the free semilattice with unit $(\mathscr{P}_{\!\infty}(\mathbb{R}),\cup)$ by the group $\mathscr{H}^+\!(\mathbb{R})$.

math.GR↗
arXiv2019

The semigroup of partial co-finite isometries of positive integers

The semigroup $\mathbf{I}\mathbb{N}_{\infty}$ of all partial co-finite isometries of positive integers is studied. We describe Green's relations on the semigroup $\mathbf{I}\mathbb{N}_{\infty}$, its band and proved that $\mathbf{I}\mathbb{N}_{\infty}$ is a simple $E$-unitary $F$-inverse semigroup. We described the least group congruence $\mathfrak{C}_{\mathbf{mg}}$ on $\mathbf{I}\mathbb{N}_{\infty}$ and proved that the quotient-semigroup $\mathbf{I}\mathbb{N}_{\infty}/\mathfrak{C}_{\mathbf{mg}}$ is isomorphic to the additive group of integers. An example of a non-group congruence on the semigroup $\mathbf{I}\mathbb{N}_{\infty}$ is presented. Also we proved that a congruence on the semigroup $\mathbf{I}\mathbb{N}_{\infty}$ is a group congruence if and only if its restriction onto an isomorphic copy of the bicyclic semigroup in $\mathbf{I}\mathbb{N}_{\infty}$ is a group congruence.

math.GR↗
arXiv2019

The monoid of order isomorphisms of principal filters of a power of the positive integers

Let $n$ be any positive integer and $\mathscr{I\!\!P\!F}(\mathbb{N}^n)$ be the semigroup of all order isomorphisms between principal filters of the $n$-th power of the set of positive integers $\mathbb{N}$ with the product order. We study algebraic properties of the semigroup $\mathscr{I\!\!P\!F}(\mathbb{N}^n)$. In particular, we show that $\mathscr{I\!\!P\!F}(\mathbb{N}^n)$ is a bisimple, $E$-unitary, $F$-inverse semigroup, describe Green's relations on $\mathscr{I\!\!P\!F}(\mathbb{N}^n)$ and its maximal subgroups. We show that the semigroup $\mathscr{I\!\!P\!F}(\mathbb{N}^n)$ is isomorphic to the semidirect product of the direct $n$-th power of the bicyclic monoid ${\mathscr{C}}^n(p,q)$ by the group of permutation $\mathscr{S}_n$. Also we prove that every non-identity congruence $\mathfrak{C}$ on the semigroup $\mathscr{I\!\!P\!F}(\mathbb{N}^n)$ is group and describe the least group congruence on $\mathscr{I\!\!P\!F}(\mathbb{N}^n)$. We show that every Hausdorff shift-continuous topology on $\mathscr{I\!\!P\!F}(\mathbb{N}^n)$ is discrete and discuss embedding of the semigroup $\mathscr{I\!\!P\!F}(\mathbb{N}^n)$ into compact-like topological semigroups.

math.GR↗
arXiv2019

Positive answers to Koch's problem in special cases

A topological semigroup is monothetic provided it contains a dense cyclic subsemigroup. The Koch problem asks whether every locally compact monothetic monoid is compact. This problem was opened for more than sixty years, till in 2018 Zelenyuk obtained a negative answer. In this paper we obtain a positive answer for Koch's problem for some special classes of topological monoids. Namely, we show that a locally compact monothetic topological monoid is a compact topological group if and only if $S$ is a submonoid of a quasitopological group if and only if $S$ has open shifts if and only if $S$ is non-viscous in the sense of Averbukh. The last condition means that any neighborhood $U$ of the identity $1$ of $S$ and for any element $a\in S$ there exists a neighborhood $V$ of $a$ such that any element $x\in S$ with $(xV\cup Vx)\cap V\ne\emptyset$ belongs to the neighborhood $U$ of 1.

math.GR↗
arXiv2018

On variants of the extended bicyclic semigroup

In the paper we describe the group $\mathbf{Aut}\left(\mathscr{C}_{\mathbb{Z}}\right)$ of automorphisms of the extended bicyclic semigroup $\mathscr{C}_{\mathbb{Z}}$ and study the variants $\mathscr{C}_{\mathbb{Z}}^{m,n}$ of the extended bicycle semigroup $\mathscr{C}_{\mathbb{Z}}$, where $m,n\in\mathbb{Z}$. In particular, we prove that $\mathbf{Aut}\left(\mathscr{C}_{\mathbb{Z}}\right)$ is isomorphic to the additive group of integers, the extended bicyclic semigroup $\mathscr{C}_{\mathbb{Z}}$ and every its variant are not finitely generated, and describe the subset of idempotents $E(\mathscr{C}_{\mathbb{Z}}^{m,n})$ and Green's relations on the semigroup $\mathscr{C}_{\mathbb{Z}}^{m,n}$. Also we show that $E(\mathscr{C}_{\mathbb{Z}}^{m,n})$ is an $ω$-chain and any two variants of the extended bicyclic semigroup $\mathscr{C}_{\mathbb{Z}}$ are isomorphic. At the end we discuss shift-continuous Hausdorff topologies on the variant $\mathscr{C}_{\mathbb{Z}}^{0,0}$. In particular, we prove that if $τ$ is a Hausdorff shift-continuous topology on $\mathscr{C}_{\mathbb{Z}}^{0,0}$ then each of inequalities $a>0$ or $b>0$ implies that $(a,b)$ is an isolated point of $\big(\mathscr{C}_{\mathbb{Z}}^{0,0},τ\big)$ and construct an example of a Hausdorff semigroup topology $τ^*$ on the semigroup $\mathscr{C}_{\mathbb{Z}}^{0,0}$ such that all its points with $ab\leqslant 0$ and $a+b\leqslant 0$ are not isolated in $\big(\mathscr{C}_{\mathbb{Z}}^{0,0},τ^*\big)$.

math.GR↗
arXiv2018

On locally compact semitopological $0$-bisimple inverse $ω$-semigroups

We describe the structure of Hausdorff locally compact semitopological $0$-bisimple inverse $ω$-semigroups with compact maximal subgroups. In particular, we show that a Hausdorff locally compact semitopological $0$-bisimple inverse $ω$-semigroup with a compact maximal subgroup is either compact or topologically isomorphic to the topological sum of its $\mathscr{H}$-classes. We describe the structure of Hausdorff locally compact semitopological $0$-bisimple inverse $ω$-semigroups with a monothetic maximal subgroups. In particular we prove the dichotomy for $T_1$ locally compact semitopological Reilly semigroup $\left(\textbf{B}(\mathbb{Z}_{+},θ)^0,τ\right)$ with adjoined zero and with a non-annihilating homomorphism $θ\colon \mathbb{Z}_{+}\to \mathbb{Z}_{+}$: $\left(\textbf{B}(\mathbb{Z}_{+},θ)^0,τ\right)$ is either compact or discrete. At the end we discuss on the remainder under the closure of the discrete Reilly semigroup $\textbf{B}(\mathbb{Z}_{+},θ)^0$ in a semitopological semigroup.

math.GR↗
arXiv2018

On feebly compact semitopological symmetric inverse semigroups of a bounded finite rank

We study feebly compact shift-continuous $T_1$-topologies on the symmetric inverse semigroup $\mathscr{I}_λ^n$ of finite transformations of the rank $\leqslant n$. For any positive integer $n\geqslant2$ and any infinite cardinal $λ$ a Hausdorff countably pracompact non-compact shift-continuous topology on $\mathscr{I}_λ^n$ is constructed. We show that for an arbitrary positive integer $n$ and an arbitrary infinite cardinal $λ$ for a $T_1$-topology $τ$ on $\mathscr{I}_λ^n$ the following conditions are equivalent: $(i)$ $τ$ is countably pracompact; $(ii)$ $τ$ is feebly compact; $(iii)$ $τ$ is $d$-feebly compact; $(iv)$ $\left(\mathscr{I}_λ^n,τ\right)$ is H-closed; $(v)$ $\left(\mathscr{I}_λ^n,τ\right)$ is $\mathbb{N}_{\mathfrak{d}}$-compact for the discrete countable space $\mathbb{N}_{\mathfrak{d}}$; $(vi)$ $\left(\mathscr{I}_λ^n,τ\right)$ is $\mathbb{R}$-compact; $(vii)$ $\left(\mathscr{I}_λ^n,τ\right)$ is infra H-closed. Also we prove that for an arbitrary positive integer $n$ and an arbitrary infinite cardinal $λ$ every shift-continuous semiregular feebly compact $T_1$-topology $τ$ on $\mathscr{I}_λ^n$ is compact.

math.GR↗
arXiv2017

On semitopological bicyclic extensions of linearly ordered groups

For a linearly ordered group $G$ let us define a subset $A\subseteq G$ to be a \emph{shift-set} if for any $x,y,z\in A$ with $y < x$ we get $x\cdot y^{-1}\cdot z\in A$. We describe the natural partial order and solutions of equations on the semigroup $\mathscr{B}(A)$ of shifts of positive cones of $A$. We study topologizations of the semigroup $\mathscr{B}(A)$. In particular, we show that for an arbitrary countable linearly ordered group $G$ and a non-empty shift-set $A$ of $G$ every Baire shift-continuous $T_1$-topology $τ$ on $\mathscr{B}(A)$ is discrete. Also we prove that for an arbitrary linearly non-densely ordered group $G$ and a non-empty shift-set $A$ of $G$, every shift-continuous Hausdorff topology $τ$ on the semigroup $\mathscr{B}(A)$ is discrete, and hence $\left(\mathscr{B}(A),τ\right)$ is a discrete subspace of any Hausdorff semitopological semigroup which contains $\mathscr{B}(A)$ as a subsemigroup.

math.GR↗