arXiv subjects

Taras Mokrytskyi

Publications and source records attributed to Taras Mokrytskyi.

3 recordsLinked to original sources

arXiv2023

The monoid of order isomorphisms between principal filters of $σ{\mathbb{N}^κ}$

Consider the following generalization of the bicyclic monoid. Let $κ$ be any infinite cardinal and let $\mathcal{IP\!F}\left(σ{\mathbb{N}^κ}\right)$ be the semigroup of all order isomorphisms between principal filters of the set $σ{\mathbb{N}^κ}$ with the product order. We shall study algebraic properties of the semigroup $\mathcal{IP\!F}\left(σ{\mathbb{N}^κ}\right)$, show that it is bisimple, $E$-unitary, $F$-inverse semigroup, describe Green's relations on $\mathcal{IP\!F}\left(σ{\mathbb{N}^κ}\right)$, describe the group of units $H\left(\mathbb{I}\right)$ of the semigroup $\mathcal{IP\!F}\left(σ{\mathbb{N}^κ}\right)$ and describe its maximal subgroups. We prove that the semigroup $\mathcal{IP\!F}\left(σ{\mathbb{N}^κ}\right)$ is isomorphic to the semidirect product $\mathcal{S}_κ\ltimesσ{\mathbb{B}^κ}$ of the semigroup $σ{\mathbb{B}^κ}$ by the group $\mathcal{S}_κ$, show that every non-identity congruence $\mathfrak{C}$ on the semigroup $\mathcal{IP\!F}\left(σ{\mathbb{N}^κ}\right)$ is a group congruence and describe the least group congruence on $\mathcal{IP\!F}\left(σ{\mathbb{N}^κ}\right)$.

math.GR↗

arXiv2019

On the dichotomy of a locally compact semitopological monoid of order isomorphisms between principal filters of $\mathbb{N}^n$ with adjoined zero

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 prove that a Hausdorff locally compact semitopological semigroup ${\mathscr{I\!P\!F}(\mathbb{N}^n)}$ with an adjoined zero is either compact or discrete.

math.GN↗

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↗