﻿ Lviv seminar on Topological Algebra archivta_2017-2018
 Scientific Seminar at Geometry and Topology Department of Ivan Franko National University of Lviv Archive for (2017/2018) academic year Topological Algebra
 September 14, 2017 All participants September 27, 2017 M. Khylynskyi On the λ-polycyclic estension of a monoid We introduce the λ-polycyclic estension of a monoid and discuss its properties. October 4, 2017 M. Khylynskyi On the λ-polycyclic estension of a monoid, II We introduce the λ-polycyclic estension of a monoid and discuss its algebraic properties. October 18, 2017 T. Mokrytskyi The monoid $\mathcal{IPF}(\mathbb{N}^n)$ of order isomorphisms of principal filters of a power of the positive integers Let $n$ be any positive integer and $\mathcal{IPF}(\mathbb{N}^n)$ 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 $\mathcal{IPF}(\mathbb{N}^n)$. In particular, we show that $\mathcal{IPF}(\mathbb{N}^n)$ bisimple inverse semigroup, describe Green's relations on $\mathcal{IPF}(\mathbb{N}^n)$ and its maximal subgroups. We prove that every non-identity congruence $\mathfrak{C}$ on the semigroup $\mathcal{IPF}(\mathbb{N}^n)$ is group. October 25, 2017 T. Mokrytskyi The monoid $\mathcal{IPF}(\mathbb{N}^n)$ of order isomorphisms of principal filters of a power of the positive integers, II Let $n$ be any positive integer and $\mathcal{IPF}(\mathbb{N}^n)$ 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 $\mathcal{IPF}(\mathbb{N}^n)$. In particular, we show that $\mathcal{IPF}(\mathbb{N}^n)$ bisimple inverse semigroup, describe Green's relations on $\mathcal{IPF}(\mathbb{N}^n)$ and its maximal subgroups. We prove that every non-identity congruence $\mathfrak{C}$ on the semigroup $\mathcal{IPF}(\mathbb{N}^n)$ is group. November 1, 2017 A. Ravsky A few open problems from Mathematics StackExchange We shall discuss on open problem in Topological Algebra posed in Mathematics StackExchange and MathOverflow. November 8, 2017 O. Sobol Syntactic semigroup problem for the semigroup reducts of affine near-semirings over Brandt semigroups We discuss on the resuls of the paper: [J. Kumar and K. V. Krishna, Syntactic semigroup problem for the semigroup reducts of affine near-semirings over Brandt semigroups, Asian-Eur. J. Math. 9, No. 1 (2016) 1650021 (10 pages)]. November 15, 2017 O. Sobol Syntactic semigroup problem for the semigroup reducts of affine near-semirings over Brandt semigroups, II We discuss on the resuls of the paper: [J. Kumar and K. V. Krishna, Syntactic semigroup problem for the semigroup reducts of affine near-semirings over Brandt semigroups, Asian-Eur. J. Math. 9, No. 1 (2016) 1650021 (10 pages)]. November 22, 2017 O. Sobol Syntactic semigroup problem for the semigroup reducts of affine near-semirings over Brandt semigroups, III We discuss on the resuls of the paper: [J. Kumar and K. V. Krishna, Syntactic semigroup problem for the semigroup reducts of affine near-semirings over Brandt semigroups, Asian-Eur. J. Math. 9, No. 1 (2016) 1650021 (10 pages)]. November 29, 2017 O. Sobol On some reuslts of J. Kumar and K. V. Krishna We give a survey on the resuls of the papers: [J. Kumar and K. V. Krishna, Affine Near-Semirings Over Brandt Semigroups, Communications in Algebra 42, No. 12 (2014) 5252-5169], [J. Kumar and K. V. Krishna, Rank Properties of Multiplicative Semigroup Reduct of Affine Near-Semirings over $B_n$, arXiv:1311.0789v2], [J. Kumar and K. V. Krishna, The Ranks of the Additive Semigroup Reduct of Affine Near-Semiring over Brandt Semigroup, arXiv:1308.4087v1], [J. Kumar and K. V. Krishna, Syntactic semigroup problem for the semigroup reducts of affine near-semirings over Brandt semigroups, Asian-Eur. J. Math. 9, No. 1 (2016) 1650021 (10 pages)], [J. Kumar and K. V. Krishna, Radicals and Ideals of Affine Near-Semirings Over Brandt Semigroups, Romeo P., Meakin J., Rajan A. (eds), Semigroups, Algebras and Operator Theory. Springer Proceedings in Mathematics & Statistics, vol 142. Springer, New Delhi, pp 123-133] and [J. Kumar and K. V. Krishna, Rank properties of the semigroup reducts of affine near-semirings over Brandt semigroups, Semigroup Forum 93, No. 3 (2014) 516-534]. December 6, 2017 S. Bardyla On topological semilattices with compact maximal chains We investigate topological semilattices with compact maximal chains. In particular, we prove that a topological semilattice X is multiclosed in the class of topological semigroups iff X is a semilattice with compact maximal chains. Some open problems will be posed. December 13, 2017 S. Bardyla On topological semilattices with compact maximal chains, II We investigate topological semilattices with compact maximal chains. In particular, we prove that a topological semilattice X is multiclosed in the class of topological semigroups iff X is a semilattice with compact maximal chains. Some open problems will be posed. December 19, 2017 All participants January 3, 2018 S. Bardyla On topological semilattices with compact maximal chains, III We investigate topological semilattices with compact maximal chains. In particular, we prove that a topological semilattice X is multiclosed in the class of topological semigroups iff X is a semilattice with compact maximal chains. Some open problems will be posed. January 24, 2018 A. Ravsky Two new algebra working problems, II Inspired by questions from Mathematics StackExchange (http://math.stackexchange.com/), the speaker will pose two new algebra working problems for investigation for the participants of the seminar. One (http:https://math.stackexchange.com/questions/2575974/venn-diagram-with-rectangles-how-many-among-binomnk-regions-created-by-i) has deep roots in computer science and graph drawing formulation, but the speaker will ask about some properties of the semilattice $(\mathbb N^2,\min)$, the other (https://math.stackexchange.com/questions/2363770/having-a-graph-of-complaints-with-10-of-enemies-prove-that-you-always-may-arre) is devoted to finite oriented graphs with out-degree one, that is a monounary algebras and will be proposed for Inna Pozdniakova. February 14, 2018 O. Sobol, O. Gutik On feebly compact semitopological semilattice $\exp_n\lambda$ We discuss on feebly compact shift-continous topologies $\tau$ on the semilattice $\left(\exp_n\lambda,\cap\right)$. It is proved that for any shift-continuous $T_1$-topology $\tau$ on $\exp_n\lambda$ the following conditions are equivalent: $(i)$ $\tau$ is countably pracompact; $(ii)$ $\tau$ is feebly compact; $(iii)$ $\tau$ is $d$-feebly compact; $(iv)$~$\left(\exp_n\lambda,\tau\right)$ is an $H$-closed space; $(v)$~$\left(\exp_n\lambda,\tau\right)$ is sequentially countably pracompact; $(vi)$~$\left(\exp_n\lambda,\tau\right)$ is selectively sequentially feebly compact; $(vii)$~$\left(\exp_n\lambda,\tau\right)$ is selectively feebly compact; $(viii)$~$\left(\exp_n\lambda,\tau\right)$ is sequentially feebly compact; $(ix)$~$\left(\exp_n\lambda,\tau\right)$ is $\mathfrak{D}(\omega)$-compact; $(x)$~$\left(\exp_n\lambda,\tau\right)$ is $\mathbb{R}$-compact; $(xi)$~$\left(\exp_n\lambda,\tau\right)$ is infra H-closed. February 21, 2018 O. Sobol, O. Gutik On feebly compact semitopological semilattice $\exp_n\lambda$, II We discuss on feebly compact shift-continous topologies $\tau$ on the semilattice $\left(\exp_n\lambda,\cap\right)$. It is proved that for any shift-continuous $T_1$-topology $\tau$ on $\exp_n\lambda$ the following conditions are equivalent: $(i)$ $\tau$ is countably pracompact; $(ii)$ $\tau$ is feebly compact; $(iii)$ $\tau$ is $d$-feebly compact; $(iv)$~$\left(\exp_n\lambda,\tau\right)$ is an $H$-closed space; $(v)$~$\left(\exp_n\lambda,\tau\right)$ is sequentially countably pracompact; $(vi)$~$\left(\exp_n\lambda,\tau\right)$ is selectively sequentially feebly compact; $(vii)$~$\left(\exp_n\lambda,\tau\right)$ is selectively feebly compact; $(viii)$~$\left(\exp_n\lambda,\tau\right)$ is sequentially feebly compact; $(ix)$~$\left(\exp_n\lambda,\tau\right)$ is $\mathfrak{D}(\omega)$-compact; $(x)$~$\left(\exp_n\lambda,\tau\right)$ is $\mathbb{R}$-compact; $(xi)$~$\left(\exp_n\lambda,\tau\right)$ is infra H-closed. March 14, 2018 A. Savchuk, O. Gutik 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 discussed. We describe the Green 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. It is 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 $\mathbb{Z}(+)$. An example of a non-group congruence on the semigroup $\mathbf{I}\mathbb{N}_{\infty}$ is presented. March 21, 2018 O. Gutik The semigroup of partial co-finite isometries of positive integers, II The semigroup $\mathbf{I}\mathbb{N}_{\infty}$ of all partial co-finite isometries of positive integers is discussed. We proved that a congruence on the semigroup $\mathbf{I}\mathbb{N}_{\infty}$ is group if and only if its restriction onto an isomorphic copy of the bicyclic semigroup in $\mathbf{I}\mathbb{N}_{\infty}$ is a group congruence. March 28, 2018 O. Gutik On group congruences of inverse semigroups whose contain the semigroup of partial co-finite isometries of positive integers Let $\mathcal{I}_{\infty}^{\!\nearrow}(\mathbb{N})$ be the the semigroup of cofinite monotone partial bijections of the set of pusitive integers $\mathbb{N}$ and $\mathbf{I}\mathbb{N}_{\infty}$ be the semigroup of all partial co-finite isometries of $\mathbb{N}$. We give a criterium when an inverse subsemigroup $S$ of $\mathcal{I}_{\infty}^{\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathbf{I}\mathbb{N}_{\infty}$ has the following property: every non-identity congruence of $S$ is a group congrunce. We describe minimal inviverse subsemigroups with such property. April 4, 2018 L. Plachta On mesures of nonplanarity of cubic graphs We study two measures of nonplanarity of cubic graphs $G$, the genus $\gamma(G)$ and the edge deletion number $\mathbf{ed}(G)$. For cubic graphs of small order these parameters are compared with another measure of nonplanarity, the (rectiliniar) crossing number $\overline{\mathbf{cr}}(G)$. We introduce operations of connected sum, specified for cubic graphs $G$, and show that under certain conditions the parameters $\gamma(G)$ and $ed(G)$ are additive (subadditive) with respect to them. The minimal genus graphs (i.e. the cubic graphs of minimum order with given value of genus $\gamma$) and the minimal edge deletion graphs (i.e. cubic graphs of minimum order with given value of edge deletion number $\mathbf{ed}$) are introduced and studied. We provide upper bounds for the order of minimal genus and minimal edge deletion graphs. April 11, 2018 O. Gutik On group congruences of inverse semigroups whose contain the semigroup of partial co-finite isometries of positive integers, II Let $\mathcal{I}_{\infty}^{\!\nearrow}(\mathbb{N})$ be the the semigroup of cofinite monotone partial bijections of the set of pusitive integers $\mathbb{N}$ and $\mathbf{I}\mathbb{N}_{\infty}$ be the semigroup of all partial co-finite isometries of $\mathbb{N}$. We give a criterium when an inverse subsemigroup $S$ of $\mathcal{I}_{\infty}^{\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathbf{I}\mathbb{N}_{\infty}$ has the following property: every non-identity congruence of $S$ is a group congrunce. We describe minimal inviverse subsemigroups with such property. April 18, 2018 S. Bardyla Embedding of graph inverse semigroups into compact-like topological semigroups We investigate graph inverse semigroups which are subsemigroups of compact-like topological semigroups. More precisely, we characterise graph inverse semigroups which embeds densely into d-compact topological semigroups. April 25, 2018 S. Bardyla Embedding of graph inverse semigroups into compact-like topological semigroups, II We investigate graph inverse semigroups which are subsemigroups of compact-like topological semigroups. More precisely, we characterise graph inverse semigroups which embeds densely into d-compact topological semigroups. May 2, 2018 S. Bardyla Embedding of graph inverse semigroups into compact-like topological semigroups, III We investigate graph inverse semigroups which are subsemigroups of compact-like topological semigroups. More precisely, we characterise graph inverse semigroups which embeds densely into d-compact topological semigroups. May 16, 2018 S. Bardyla Embedding of graph inverse semigroups into compact-like topological semigroups, IV We investigate graph inverse semigroups which are subsemigroups of compact-like topological semigroups. More precisely, we characterise graph inverse semigroups which embeds densely into d-compact topological semigroups. July 19, 2018 A. Ravsky On the Koch problem We discuss on Koch's problem. In particular, we give a positive answer on Koch's problem in some partial cases.