Inna Pozdniakova | Searcharxiv

arXiv2025

On injective endomorphisms of the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^2}$ with the two-element family $\mathscr{F}^2$ of inductive nonempty subsets of $ω$

We describe injective endomorphisms of the semigroup $\boldsymbol{B}_{Z\mathbb{}}^{\mathscr{F}^2}$ with the two-element family $\mathscr{F}^2$ of inductive nonempty subsets of $ω$. In particular we show that every injective endomorphism $\mathfrak{e}$ of $\boldsymbol{B}_{Z\mathbb{}}^{\mathscr{F}^2}$ is presented in the form $\mathfrak{e}=\mathfrak{e}_0\mathfrak{a}$, where $\mathfrak{e}_0$ is an injective $(0,0,[0))$-endomorphism of $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^2}$ and $\mathfrak{a}$ is an automorphism $\mathfrak{a}$ of $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^2}$. Also we describe all injective $(0,0,[0))$-endomorphisms $\mathfrak{e}_0$ of $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^2}$, i.e., such that $(0,0,[0))\mathfrak{e}_0=(0,0,[0))$.

math.GR↗

arXiv2024

On the semigroup of monoid endomorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$ with a two-element family $\mathscr{F}$ of inductive nonempty subsets of $ω$

We study the semigroup of non-injective monoid endomorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$ with a two-elements family $\mathscr{F}$ of inductive nonempty subsets of $ω$. We describe the structure of elements of the semigroup $\boldsymbol{End}^*_0(\boldsymbol{B}_ω^{\mathscr{F}})$ of non-injective monoid endomorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$. In particular we show that its subsemigroup $\boldsymbol{End}^*(\boldsymbol{B}_ω^{\mathscr{F}})$ of non-injective non-annihilating monoid endomorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$ is isomorphic to the direct product of the two-element left-zero semigroup and the multiplicative semigroup of positive integers and describe Green's relations on $\boldsymbol{End}^*(\boldsymbol{B}_ω^{\mathscr{F}})$.

math.GR↗

arXiv2023

On the semigroup of injective monoid endomor\-phisms of the monoid $\boldsymbol{B}_ω^{\mathscr{F}}$ with the two-elements family $\mathscr{F}$ of inductive nonempty subsets of $ω$

We study injective endomorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$ with the two-elements family $\mathscr{F}$ of inductive nonempty subsets of $ω$. We describe the elements of the semigroup $\boldsymbol{End}^1_*(\boldsymbol{B}_ω^{\mathscr{F}})$ of all injective monoid endomorphisms of the monoid $\boldsymbol{B}_ω^{\mathscr{F}}$, and show that Green's relations $\mathscr{R}$, $\mathscr{L}$, $\mathscr{H}$, $\mathscr{D}$, and $\mathscr{J}$ on $\boldsymbol{End}^1_*(\boldsymbol{B}_ω^{\mathscr{F}})$ coincide with the relation of equality.

math.GR↗

arXiv2022

On the group of automorphisms of the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}}$ with the family $\mathscr{F}$ of inductive nonempty subsets of $ω$

We study automorphisms of the semigroup $\boldsymbol{B}_{Z\mathbb{}}^{\mathscr{F}}$ with the family $\mathscr{F}$ of inductive nonempty subsets of $ω$ and prove that the group $\mathbf{Aut}(\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}})$ of automorphisms of the semigroup $\boldsymbol{B}_{Z\mathbb{}}^{\mathscr{F}}$ is isomorphic to the additive group integers.

math.GR↗

arXiv2017

On the monoid of monotone injective partial selfmaps of $\mathbb{N}^{2}_{\leqslant}$ with cofinite domains and images, II

Let $\mathbb{N}^{2}_{\leqslant}$ be the set $\mathbb{N}^{2}$ with the partial order defined as the product of usual order $\leq$ on the set of positive integers $\mathbb{N}$. We study the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})$ of monotone injective partial selfmaps of $\mathbb{N}^{2}_{\leqslant}$ having cofinite domain and image. We describe the natural partial order on $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})$ and show that it coincides with the natural partial order which is induced from symmetric inverse monoid $\mathscr{I}_{\mathbb{N}\times\mathbb{N}}$ onto $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})$. We proved that the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})$ is isomorphic to the semidirect product $\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^2_{\leqslant})\rtimes \mathbb{Z}_2$ of the monoid $\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^2_{\leqslant})$ of orientation-preserving monotone injective partial selfmaps of $\mathbb{N}^{2}_{\leqslant}$ with cofinite domains and images by the cyclic group $\mathbb{Z}_2$ of the order two. Also we describe the congruence $σ$ on $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})$ which is generated by the natural order $\preccurlyeq$ on the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})$. We prove that the quotient semigroup $\mathscr{P\!O}\!_{\infty}^{\,+}(\mathbb{N}^2_{\leqslant})/σ$ is isomorphic to the free commutative monoid $\mathfrak{AM}_ω$ over an infinite countable set and show that the quotient semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})/σ$ is isomorphic to the semidirect product of the free commutative monoid $\mathfrak{AM}_ω$ by $\mathbb{Z}_2$.

math.GR↗

arXiv2016

On the monoid of monotone injective partial selfmaps of $\mathbb{N}^{2}_{\leqslant}$ with cofinite domains and images

Let $\mathbb{N}^{2}_{\leqslant}$ be the set $\mathbb{N}^{2}$ with the partial order defined as the product of usual order $\leq$ on the set of positive integers $\mathbb{N}$. We study the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})$ of monotone injective partial selfmaps of $\mathbb{N}^{2}_{\leqslant}$ having cofinite domain and image. We describe properties of elements of the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})$ as monotone partial bijections of $\mathbb{N}^{2}_{\leqslant}$ and show that the group of units of $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})$ is isomorphic to the cyclic group of order two. Also we describe the subsemigroup of idempotents of $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})$ and the Green relations on $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})$. In particular, we show that $\mathscr{D}=\mathscr{J}$ in $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})$.

math.GR↗

arXiv2015

Congruences on the monoid of monotone injective partial selfmaps of $L_n\times_{\operatorname{lex}}\mathbb{Z}$ with co-finite domains and images

We study congruences of the semigroup $\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$ of monotone injective partial selfmaps of the set of $L_n\times_{\operatorname{lex}}\mathbb{Z}$ having co-finite domain and image, where $L_n\times_{\operatorname{lex}}\mathbb{Z}$ is the lexicographic product of $n$-elements chain and the set of integers with the usual linear order. The structure of the sublattice of congruences on $\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$ which contain in the least group congruence is described.

math.GR↗