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September 14, 2017
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All participants
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September 27, 2017
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M. Khylynskyi
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On the λ-polycyclic estension of a monoid
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We introduce the λ-polycyclic estension of a monoid and discuss its properties.
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October 4, 2017
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M. Khylynskyi
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On the λ-polycyclic estension of a monoid, II
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We introduce the λ-polycyclic estension of a monoid and discuss its algebraic properties.
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October 18, 2017
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T. Mokrytskyi
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The monoid $\mathcal{IPF}(\mathbb{N}^n)$ of order isomorphisms of principal filters of a power of the positive integers
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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.
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October 25, 2017
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T. Mokrytskyi
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The monoid $\mathcal{IPF}(\mathbb{N}^n)$ of order isomorphisms of principal filters of a power of the positive integers, II
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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.
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November 1, 2017
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A. Ravsky
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A few open problems from Mathematics StackExchange
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November 8, 2017
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O. Sobol
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Syntactic semigroup problem for the semigroup reducts
of affine near-semirings over Brandt semigroups
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November 15, 2017
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O. Sobol
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Syntactic semigroup problem for the semigroup reducts
of affine near-semirings over Brandt semigroups, II
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November 22, 2017
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O. Sobol
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Syntactic semigroup problem for the semigroup reducts
of affine near-semirings over Brandt semigroups, III
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November 29, 2017
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O. Sobol
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On some reuslts of J. Kumar and K. V. Krishna
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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].
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December 6, 2017
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S. Bardyla
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On topological semilattices with compact maximal chains
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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.
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December 13, 2017
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S. Bardyla
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On topological semilattices with compact maximal chains, II
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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.
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December 19, 2017
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All participants
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January 3, 2018
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S. Bardyla
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On topological semilattices with compact maximal chains, III
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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.
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January 24, 2018
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A. Ravsky
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Two new algebra working problems, II
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February 14, 2018
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O. Sobol,
O. Gutik
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On feebly compact semitopological semilattice $\exp_n\lambda$
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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.
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February 21, 2018
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O. Sobol,
O. Gutik
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On feebly compact semitopological semilattice $\exp_n\lambda$, II
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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.
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March 14, 2018
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A. Savchuk,
O. Gutik
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The semigroup of partial co-finite isometries of positive integers
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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.
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March 21, 2018
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O. Gutik
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The semigroup of partial co-finite isometries of positive integers, II
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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.
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March 28, 2018
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O. Gutik
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On group congruences of inverse semigroups whose contain the semigroup of partial co-finite isometries of positive integers
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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.
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April 4, 2018
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L. Plachta
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On mesures of nonplanarity of cubic graphs
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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.
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April 11, 2018
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O. Gutik
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On group congruences of inverse semigroups whose contain the semigroup of partial co-finite isometries of positive integers, II
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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.
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April 18, 2018
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S. Bardyla
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Embedding of graph inverse semigroups into compact-like topological semigroups
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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.
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April 25, 2018
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S. Bardyla
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Embedding of graph inverse semigroups into compact-like topological semigroups, II
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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.
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May 2, 2018
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S. Bardyla
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Embedding of graph inverse semigroups into compact-like topological semigroups, III
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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.
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May 16, 2018
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S. Bardyla
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Embedding of graph inverse semigroups into compact-like topological semigroups, IV
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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.
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July 19, 2018
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A. Ravsky
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On the Koch problem
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We discuss on Koch's problem. In particular, we give a positive answer on Koch's problem in some partial cases.
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