<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN"> <!-- saved from url=(0089)http://prima.franko.lviv.ua/faculty/mechmat/Departments/Topology/seminararchiv_ta15_16.html --> <HTML><HEAD><TITLE>Lviv seminar on Topological Algebra archivta_2015-2016</TITLE> <META content="text/html; charset=unicode" http-equiv=Content-Type> <META content="text/html; charset=iso-8859-1" http-equiv=??????????-???><!-- *************** DHTML Outline (begin) ***************** --> <STYLE type=text/css>LI.oItem { COLOR: #000000; CURSOR: text } LI.oParent { COLOR: #000088; CURSOR: hand } UL UL { DISPLAY: none } </STYLE> <SCRIPT language=Javascript> <!-- // Returns the closest parent tag with tagName containing // the src tag. If no such tag is found - null is returned. function checkParent( src, tagName ) { while ( src != null ) { if (src.tagName == tagName) return src; src = src.parentElement; } return null; } // Returns the first tag with tagName contained by // the src tag. If no such tag is found - null is returned. function checkContent( src, tagName ) { var pos = src.sourceIndex ; while ( src.contains( document.all[++pos] ) ) if ( document.all[pos].tagName == tagName ) return document.all[pos] ; return null ; } // Handle onClick event in the outline box function outlineAction() { var src = event.srcElement ; var item = checkParent( src, "LI" ) ; if ( parent != null ) { var content = checkContent( item, "UL" ) ; if ( content != null ) if ( content.style.display == "" ) content.style.display = "block" ; else content.style.display = "" ; } event.cancelBubble = true; } // --> </SCRIPT> </head> <body background="backgr_3.gif" bgcolor="#FFCC66" text="#660000" vlink="#990000" alink="#990066" link="#990000"leftmargin="10" rightmargin="0" topmargin="1" > <Table width="95%" border=0 align="center"> <tr> <td > <img src="emb_1.gif"> </td> <td align="center"> <b><font size="+2"><a href=http:./seminarta.html> Scientific Seminar</a></font> <font size="+1"><br>at Geometry and Topology Department of <br> Ivan Franko National University of Lviv <br> <A href="seminararchiv_ta14_15.html"><IMG src="left.gif"></A> </b><i>Archive for</i><b> <font size="+2">(2015/2016)</font> </b><i>academic year</i></b></font> <a href="seminararchiv_ta16_17.html"><img src="right.gif"></a> </td> <td align="center"> <p> <b> <font size="+1"> <hr> <hr> Topological <br> Algebra <hr> <hr> </font></b> </p> </tr> </table> <Table width="91%" border=0> <tr><td colspan=3 HEIGHT="3" BORDER="1" ALIGN="left" bgcolor="#000080" ></tr> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> September 9, 2015 </font><td valign="top" width="15%" > <b><font size="+2"> <A href="./Gutik_mine.html">O. Gutik</A> </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> Divertisement <UL type="square"><LI class='oItem'> <font size="+1"> Active participants of the seminar will present and discuss interesting open problems from various branches of the topological algebra. </font></p> </div> </LI></UL></font></ul></tr> <TR> <TD bgColor=#000080 height=3 colSpan=3 align=left BORDER="1"></TD> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> September 16, 2015 </font><td valign="top" width="15%" > <b><font size="+2"> <A href="./bancv.html">T. Banakh</A>, <br> <A href="./guran.html">I. Guran</A>,<br> <A href="https://scholar.google.com/citations?user=_J_WXR0AAAAJ&hl=uk">O. Ravsky</A> </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> Divertisement <UL type="square"><LI class='oItem'> <font size="+1"> Active participants of the seminar will present and discuss interesting open problems from various branches of the topological algebra. </LI></UL></font></ul></tr> <TR> <TD bgColor=#000080 height=3 colSpan=3 align=left BORDER="1"></TD> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> September 23, 2015 </font><td valign="top" width="15%" > <b><font size="+2"> <A href="./Gutik_mine.html">O. Gutik</A> </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> On semitopological bicyclic extensions of linearly ordered groups <UL type="square"><LI class='oItem'> <font size="+1"> We discus on topologizations of the bicyclic extensions <b>B(G)</b> and <b>B<sup><font size=-1>+</font></sup>(G)</b> of a linearly ordered groups <b>G</b>, as semitopological semigroups (the definitions of the semigroups see in [<a href="/Public/"></a>O. Gutik, D. Pagon, and K. Pavlyk, <b><i>Congruences on bicyclic extensions of a linearly ordered group</i>, Acta Comment. Univ. Tartu. Math. </b> 15 (2011), no. 2, 61-80 (MR2961689, Zbl 1257.20059, <a name="arXiv:1111.2401"><a href="http://arxiv.org/pdf/1111.2401">arXiv:1111.2401</a>)]). We show that for an arbitrary countable linearly ordered group <b>G</b> every Hausdorff topology <b>&#964;</b> on <b>B(G)</b> <b>(B<sup><font size=-1>+</font></sup>(G))</b> such that <b>(B(G),&#964;)</b> <b>((B<sup><font size=-1>+</font></sup>(G),&#964;))</b> is a Baire topological semigroup is discrete. Also we prove that for an arbitrary linearly ordered group <b>G</b> which is not densely ordered every Hausdorff topology<b>&#964;</b> on the semigroup <b>B(G)</b> <b>(B<sup><font size=-1>+</font></sup>(G))</b> such that <b>(B(G),&#964;)</b> <b>((B<sup><font size=-1>+</font></sup>(G),&#964;))</b> is a semitopological semigroup is discrete, and hence <b>B(G)</b> <b>(B<sup><font size=-1>+</font></sup>(G))</b> is a discrete subspace of any semitopological semigroup which contains <b>B(G)</b> <b>(B<sup><font size=-1>+</font></sup>(G))</b> as a subsemigroup. </LI></UL></font></ul></tr> <TR> <TD bgColor=#000080 height=3 colSpan=3 align=left BORDER="1"></TD> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> September 30, 2015 </font><td valign="top" width="15%" > <b><font size="+2"> <A href="./Gutik_mine.html">O. Gutik</A> </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> On semitopological bicyclic extensions of linearly ordered groups, II <UL type="square"><LI class='oItem'> <font size="+1"> We discus on topologizations of the bicyclic extensions <b>B(G)</b> and <b>B<sup><font size=-1>+</font></sup>(G)</b> of a linearly ordered groups <b>G</b>, as semitopological semigroups (the definitions of the semigroups see in [<a href="/Public/"></a>O. Gutik, D. Pagon, and K. Pavlyk, <b><i>Congruences on bicyclic extensions of a linearly ordered group</i>, Acta Comment. Univ. Tartu. Math. </b> 15 (2011), no. 2, 61-80 (MR2961689, Zbl 1257.20059, <a name="arXiv:1111.2401"><a href="http://arxiv.org/pdf/1111.2401">arXiv:1111.2401</a>)]). We show that for an arbitrary countable linearly ordered group <b>G</b> every Hausdorff topology <b>&#964;</b> on <b>B(G)</b> <b>(B<sup><font size=-1>+</font></sup>(G))</b> such that <b>(B(G),&#964;)</b> <b>((B<sup><font size=-1>+</font></sup>(G),&#964;))</b> is a Baire topological semigroup is discrete. Also we prove that for an arbitrary linearly ordered group <b>G</b> which is not densely ordered every Hausdorff topology<b>&#964;</b> on the semigroup <b>B(G)</b> <b>(B<sup><font size=-1>+</font></sup>(G))</b> such that <b>(B(G),&#964;)</b> <b>((B<sup><font size=-1>+</font></sup>(G),&#964;))</b> is a semitopological semigroup is discrete, and hence <b>B(G)</b> <b>(B<sup><font size=-1>+</font></sup>(G))</b> is a discrete subspace of any semitopological semigroup which contains <b>B(G)</b> <b>(B<sup><font size=-1>+</font></sup>(G))</b> as a subsemigroup. </LI></UL></font></ul></tr> <TR> <TD bgColor=#000080 height=3 colSpan=3 align=left BORDER="1"></TD> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> October 7, 2015 </font><td valign="top" width="15%" > <b><font size="+2"> S. Bardyla </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> On a topological <b>&#945;</b>-bicyclic semigroup <UL type="square"><LI class='oItem'> <font size="+1"> We give a survey on results of Joseph W. Hogan and Annie A. Selden about topologizations of the <b>&#945;</b>-bicyclic semigroup obtained in the papers: <ol> <li>J. W. Hogan, <A href="http://dx.doi.org/10.1007/BF02572488">The <b>&#945;</b>-bicyclic semigroup as a topological semigroup</A>, Semigroup Forum <b>28</b> (1984), <A href="https://eudml.org/doc/134654">265-271</A>, (<A href="https://zbmath.org/?q=an:03842148">Zbl 0531.22003</A>). </li> <li>A. A. Selden, <A href="http://dx.doi.org/10.1007/BF02572664">A nonlocally compact nondiscrete topology for the <b>&#945;</b>-bicyclic semigroup</A>, Semigroup Forum <b>31</b> (1985), <A href="https://eudml.org/doc/134761">372-374</A>, (<A href="https://zbmath.org/?q=an:03904864">Zbl 0567.22002</A>). </li> <li>J. W. Hogan, <A href="http://dx.doi.org/10.1007/BF02575016">Hausdorff topologies on the <b>&#945;</b>-bicyclic semigroup</A>, Semigroup Forum <b>36</b> (1987), <A href="https://eudml.org/doc/134896">189-209</A>, (<A href="https://zbmath.org/?q=an:04017208">Zbl 0626.22002</A>). </li> </ol> Some questions on this topics will be posed. </LI></UL></font></ul></tr> </tr><tr><td colspan="3" border="1" align="left" height="3" bgcolor="#000080"></td></tr> <tr> <td colspan="3" height="12"> </td></tr> <tr> <td valign="top" width="10%"> <font size="+2"> October 21, 2015<br> </font> </td><td valign="top" width="15%"> <b><font size="+2"> <A href="./bancv.html">T. Banakh</A>, </br> <A href="./guran.html">I. Guran</A>, </br> <A href="./Gutik_mine.html">O. Gutik</A>, </br> <A href="./zarichnyi.html">M. Zarichny</A></font></b> </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> A survey of scientific achievement of <A href="https://scholar.google.com/citations?user=_J_WXR0AAAAJ&hl=uk">Oleksandr Ravsky</A> for the last 40 years</br> (dedicated to his 40th Birthday) <UL type="square"><LI class='oItem'> <font size="+1"> We give a survey on results of <A href="https://scholar.google.com/citations?user=_J_WXR0AAAAJ&hl=uk">Oleksandr Ravsky</A> obtained in the following topics: </br> <b>&#9786; Paratopological Groups; </b> </br> <b>&#9786; General Topology; </b> </br> <b>&#9786; Combinatorics; </b> </br> <b>&#9786; Topological and Semitopological Semigroups. </b> </LI></UL></font></ul></tr> <TR> <TD bgColor=#000080 height=3 colSpan=3 align=left BORDER="1"></TD> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> October 28, 2015 </font><td valign="top" width="15%" > <b><font size="+2"> S. Bardyla </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> On a topological <b>&#945;</b>-bicyclic semigroup, II <UL type="square"><LI class='oItem'> <font size="+1"> We give a survey on results of Joseph W. Hogan and Annie A. Selden about topologizations of the <b>&#945;</b>-bicyclic semigroup obtained in the papers: <ol> <li>J. W. Hogan, <A href="http://dx.doi.org/10.1007/BF02572488">The <b>&#945;</b>-bicyclic semigroup as a topological semigroup</A>, Semigroup Forum <b>28</b> (1984), <A href="https://eudml.org/doc/134654">265-271</A>, (<A href="https://zbmath.org/?q=an:03842148">Zbl 0531.22003</A>). </li> <li>A. A. Selden, <A href="http://dx.doi.org/10.1007/BF02572664">A nonlocally compact nondiscrete topology for the <b>&#945;</b>-bicyclic semigroup</A>, Semigroup Forum <b>31</b> (1985), <A href="https://eudml.org/doc/134761">372-374</A>, (<A href="https://zbmath.org/?q=an:03904864">Zbl 0567.22002</A>). </li> <li>J. W. Hogan, <A href="http://dx.doi.org/10.1007/BF02575016">Hausdorff topologies on the <b>&#945;</b>-bicyclic semigroup</A>, Semigroup Forum <b>36</b> (1987), <A href="https://eudml.org/doc/134896">189-209</A>, (<A href="https://zbmath.org/?q=an:04017208">Zbl 0626.22002</A>). </li> </ol> Some questions on this topics will be posed. </LI></UL></font></ul></tr> <TR> <TD bgColor=#000080 height=3 colSpan=3 align=left BORDER="1"></TD> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> November 4, 2015 </font><td valign="top" width="15%" > <b><font size="+2"> <A href="./guran.html">I. Guran</A> </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> On <b>&#964;</b>-bounded topological spaces, uniform spaces and topological groups <UL type="square"><LI class='oItem'> <font size="+1"> We consider the notion of the <b>&#964;</b>-boundedness in topological spaces and uniform spaces. Let <b><i>U</i></b> be the universal uniformity of a topological space <b>X</b>. A topological spaces <b>X</b> is said to be <b>&#964;</b>-bounded if <b>(X,<i>U</i>)</b> is a <b>&#964;</b>-bounded uniform space. It is prove that the free topological group <b><i>F</i>(X)</b> is <b>&#964;</b>-bounded if and only if the space <b>X</b> is <b>&#964;</b>-bounded. </LI></UL></font></ul></tr> <TR> <TD bgColor=#000080 height=3 colSpan=3 align=left BORDER="1"></TD> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> November 11, 2015 </font><td valign="top" width="15%" > <b><font size="+2"> <A href="./guran.html">I. Guran</A> </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> On <b>&#964;</b>-bounded topological spaces, uniform spaces and topological groups, II <UL type="square"><LI class='oItem'> <font size="+1"> We consider the notion of the <b>&#964;</b>-boundedness in topological spaces and uniform spaces. Let <b><i>U</i></b> be the universal uniformity of a topological space <b>X</b>. A topological spaces <b>X</b> is said to be <b>&#964;</b>-bounded if <b>(X,<i>U</i>)</b> is a <b>&#964;</b>-bounded uniform space. It is prove that the free topological group <b><i>F</i>(X)</b> is <b>&#964;</b>-bounded if and only if the space <b>X</b> is <b>&#964;</b>-bounded. </LI></UL></font></ul></tr> <TR> <TD bgColor=#000080 height=3 colSpan=3 align=left BORDER="1"></TD> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> November 18, 2015 </font><td valign="top" width="15%" > <b><font size="+2"> <A href="https://scholar.google.com/citations?user=_J_WXR0AAAAJ&hl=uk">A. Ravsky</A> </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> Three current open bounty problems from Mathematics Stack Exchange <UL type="square"><LI class='oItem'> <font size="+1"> The speaker talk about his advances in the solution of these problems. The first of them is from <A href="http://math.stackexchange.com/questions/1520722/question-about-quotient-space-regarding-the-left-coset-space-of-group-g-with">topological algebra</A>, the second from <A href="http://math.stackexchange.com/questions/1524097/existence-of-a-section-of-non-zero-measure">measure theory</A> and the third from <A href="http://math.stackexchange.com/questions/1515747/can-finite-metric-space-embedded-into-planar-graph">the theory of finite metric spaces and graph theory</A>. </LI></UL></font></ul></tr> <TR> <TD bgColor=#000080 height=3 colSpan=3 align=left BORDER="1"></TD> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> November 25, 2015 </font><td valign="top" width="15%" > <b><font size="+2"> <A href="https://scholar.google.com/citations?user=_J_WXR0AAAAJ&hl=uk">A. Ravsky</A> </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> Three current open bounty problems from Mathematics Stack Exchange, II <UL type="square"><LI class='oItem'> <font size="+1"> The speaker talk about his advances in the solution of these problems. The first of them is from <A href="http://math.stackexchange.com/questions/1520722/question-about-quotient-space-regarding-the-left-coset-space-of-group-g-with">topological algebra</A>, the second from <A href="http://math.stackexchange.com/questions/1524097/existence-of-a-section-of-non-zero-measure">measure theory</A> and the third from <A href="http://math.stackexchange.com/questions/1515747/can-finite-metric-space-embedded-into-planar-graph">the theory of finite metric spaces and graph theory</A>. </LI></UL></font></ul></tr> <TR> <TD bgColor=#000080 height=3 colSpan=3 align=left BORDER="1"></TD> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> December 9, 2015 </font><td valign="top" width="15%" > <b><font size="+2"> S. Bardyla </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> On the semilattices with compact maximal chains <UL type="square"><LI class='oItem'> <font size="+1"> We introduce some separation axioms in the class of topological semilattices and give a sufficient condition of H-closedness in that class. </LI></UL></font></ul></tr> <TR> <TD bgColor=#000080 height=3 colSpan=3 align=left BORDER="1"></TD> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> December 16, 2015 </font><td valign="top" width="15%" > <b><font size="+2"> S. Bardyla </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> On <b>L</b>-separation axioms for topological semilattices <UL type="square"><LI class='oItem'> <font size="+1"> We introduce some <b>L</b>-separation axioms for topological semilattices and construct examples of topological semilattices which show that they are distinct. </LI></UL></font></ul></tr> <TR> <TD bgColor=#000080 height=3 colSpan=3 align=left BORDER="1"></TD> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> January 18, 2016 </font><td valign="top" width="15%" > <b><font size="+2"> S. Bardyla </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> On the closure of the polycyclic monoid in a topological inverse semigroup. <UL type="square"><LI class='oItem'> <font size="+1"> We study the closure of polycyclic monoid in a topological inverse semigroup and find a criterium of H-closedness of a topological inverse polycyclic monoid in the class of topological inverse semigroups. </LI></UL></font></ul></tr> <TR> <TD bgColor=#000080 height=3 colSpan=3 align=left BORDER="1"></TD> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> February 10, 2016 </font><td valign="top" width="15%" > <b><font size="+2"> S. Bardyla </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> On the closure of the polycyclic monoid in a topological inverse semigroup, II. <UL type="square"><LI class='oItem'> <font size="+1"> We study the closure of polycyclic monoid in a topological inverse semigroup and find a criterium of H-closedness of a topological inverse polycyclic monoid in the class of topological inverse semigroups. </LI></UL></font></ul></tr> <TR> <TD bgColor=#000080 height=3 colSpan=3 align=left BORDER="1"></TD> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> February 17, 2016 </font><td valign="top" width="15%" > <b><font size="+2"> I. Pozdniakova </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> On the monoid of monotone injective partial selfmaps of <b><i>&#8469;<sup>2</sup><sub>&#8804;</sub></i></b> with cofinite domains and images <UL type="square"><LI class='oItem'> <font size="+1"> Let <b>&#8469;<sup>2</sup><sub>&#8804;</sub></b> be the set <b>&#8469;<sup>2</sup></b> with the partial order defining as a product of usual order <b>&#8804;</b> on the set of positive integers <b>&#8469;</b>. We study the semigroup <b><i>PO</i><sub>&infin;</sub>(&#8469;<sup>2</sup><sub>&#8804;</sub>)</b> of monotone injective partial selfmaps of <b>&#8469;<sup>2</sup><sub>&#8804;</sub></b> having cofinite domain and image. We describe properties of elements of the semigroup <b><i>PO</i><sub>&infin;</sub>(&#8469;<sup>2</sup><sub>&#8804;</sub>)</b> as monotone partial bijection of <b>&#8469;<sup>2</sup><sub>&#8804;</sub></b> and show that the group of units of <b><i>PO</i><sub>&infin;</sub>(&#8469;<sup>2</sup><sub>&#8804;</sub>)</b> is isomorphic to the cyclic group of the order two. Also we describe the subsemigroup of idempotents of <b><i>PO</i><sub>&infin;</sub>(&#8469;<sup>2</sup><sub>&#8804;</sub>)</b> and the Green relations on <b><i>PO</i><sub>&infin;</sub>(&#8469;<sup>2</sup><sub>&#8804;</sub>)</b>. In particularly we prove that <b><i>D=J</b></i> in <b><i>PO</i><sub>&infin;</sub>(&#8469;<sup>2</sup><sub>&#8804;</sub>)</b>. </LI></UL></font></ul></tr> <TR> <TD bgColor=#000080 height=3 colSpan=3 align=left BORDER="1"></TD> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> February 24, 2016 </font><td valign="top" width="15%" > <b><font size="+2"> S. Bardyla </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> On a topological <b>&#945;</b>-bicyclic monoid <UL type="square"><LI class='oItem'> <font size="+1"> We will consider the <b>&#945;</b>-bicyclic monoid as a semitopological semigroup and construct some non discrete locally compact topologies on it. </LI></UL></font></ul></tr> <TR> <TD bgColor=#000080 height=3 colSpan=3 align=left BORDER="1"></TD> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> March 2, 2016 </font><td valign="top" width="15%" > <b><font size="+2"> S. Bardyla </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> On a topological <b>&#945;</b>-bicyclic monoid, II <UL type="square"><LI class='oItem'> <font size="+1"> We will consider the <b>&#945;</b>-bicyclic monoid as a topological semigroup and construct some non discrete locally compact topologies on it. </LI></UL></font></ul></tr> <TR> <TD bgColor=#000080 height=3 colSpan=3 align=left BORDER="1"></TD> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> March 9, 2016 </font><td valign="top" width="15%" > <b><font size="+2"> <A href="./Gutik_mine.html">O. Gutik</A> </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> On semigroups of bijective monotone partial self-maps of posets with cofinite domains and images <UL type="square"><LI class='oItem'> <font size="+1"> A short survey about results on the topics of the title discussed and reporter put some open problems on this topics. </LI></UL></font></ul></tr> <TR> <TD bgColor=#000080 height=3 colSpan=3 align=left BORDER="1"></TD> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> March 16, 2016 </font><td valign="top" width="15%" > <b><font size="+2"> S. Bardyla </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> On a topological <b>&#945;</b>-bicyclic monoid, III <UL type="square"><LI class='oItem'> <font size="+1"> We will consider the <b>&#945;</b>-bicyclic monoid as a topological semigroup and construct some non discrete locally compact topologies on it. </LI></UL></font></ul></tr> <TR> <TD bgColor=#000080 height=3 colSpan=3 align=left BORDER="1"></TD> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> March 23, 2016 </font><td valign="top" width="15%" > <b><font size="+2"> <A href="https://sites.google.com/site/overbitsky/">O. Verbitsky</A> </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> Universal covers, integral polytopes, and graph isomorphism <UL type="square"><LI class='oItem'> <font size="+1"> I will present two recent results that are motivated by applications in distributed computing and isomorphism testing. Both results concern the classical color refinement algorithm. <br> <br> Given a connected graph <b>G</b> and its vertex <b>x</b>, let <b>U(G,x)</b> denote the universal cover of <b>G</b> obtained by unfolding <b>G</b> into a tree starting from <b>x</b>. Let <b>T=T(n)</b> be the minimum number such that, for graphs <b>G</b> and <b>H</b> with at most <b>n</b> vertices each, the isomorphism of <b>U(G,x)</b> and <b>U(H,y)</b> surely follows from the isomorphism of these rooted trees truncated at depth <b>T</b>. Norris [Discrete Appl. Math. 1995] asked if the value of <b>T(n)</b> is bounded by <b>n</b>. We answer this question in the negative by establishing that <b>T(n)=(2-o(1))n</b>. <br> <br> The graphs <b>G</b> and <b>H</b> we construct for each <b>n</b> to prove the lower bound for <b>T(n)</b> also show some other tight lower bounds. Both having <b>n</b> vertices, <b>G</b> and <b>H</b> can be distinguished in 2-variable counting logic only with quantifier depth <b>(1-o(1))n</b>. It follows that Color Refinement (CR), the classical procedure used in isomorphism testing and other areas for computing the coarsest equitable partition of a graph, needs <b>(1-o(1))n</b> rounds to achieve color stabilization on each of <b>G</b> and <b>H</b>. Somewhat surprisingly, this number of rounds is not enough for color stabilization on the disjoint union of <b>G</b> and <b>H</b>, where <b>(2-o(1))n</b> rounds are needed. <br> <br> Exploring a linear programming approach to Graph Isomorphism, Tinhofer defined the concept of a compact graph: A graph is compact if the polytope of its fractional automorphisms is integral. Tinhofer noted that isomorphism testing for compact graphs can be done quite efficiently by linear programming. However, the problem of characterizing and recognizing compact graphs in polynomial time remains an open question. <br> <br> We relate this approach to the color refinement algorithm. We call a graph CR-definable if the CR procedure distinguishes it from any non-isomorphic graph. Babai, ErdQs, and Selkow showed that random graphs are CR-definable with high probability. We suggest an efficient characterization of the class of all CR-definable graphs. Using the last result, we prove that all CR-definable graphs are compact. In other words, the applicability range for Tinhofer's linear programming approach to isomorphism testing is at least as large as for the combinatorial approach based on color refinement. <br> <br> This is joint work with A.Krebs [1] and V.Arvind, J.K&#246;bler, and G.Rattan [2,3]. <br> <br> [1] A.Krebs and O.Verbitsky. Universal covers, color refinement, and twovariable counting logic: Lower bounds for the depth. Proc. LICS, pp. 689-700. IEEE Press, 2015. <br> <br> [2] V.Arvind, J.K&#246;bler, G.Rattan, and O.Verbitsky. On the power of color refinement. Proc. FCT'15, LNCS 9210, pp. 339-350. Springer, 2015. <br> <br> [3] V.Arvind, J.K&#246;bler, G.Rattan, and O.Verbitsky. On Tinhofer's linear programming approach to isomorphism testing. Proc. MFCS'15, LNCS 9235, pp. 26-37. Springer, 2015. </LI></UL></font></ul></tr> <TR> <TD bgColor=#000080 height=3 colSpan=3 align=left BORDER="1"></TD> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> April 6, 2016 </font><td valign="top" width="15%" > <b><font size="+2"> <A href="./Gutik_mine.html">O. Gutik</A> </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> Divertissement <UL type="square"><LI class='oItem'> <font size="+1"> Active participants of the seminar will present and discuss interesting open problems from various branches of topological algebra and its applications. </LI></UL></font></ul></tr> <TR> <TD bgColor=#000080 height=3 colSpan=3 align=left BORDER="1"></TD> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> April 13, 2016 </font><td valign="top" width="15%" > <b><font size="+2"> <A href="./guran.html">I. C@0=</A> </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> @> ?@>1;5<C B>B>6=>ABV A;V2 C 3@C?0E <UL type="square"><LI class='oItem'> <font size="+1"> &8:; 4>?>2V459 1C45 ?@8A2OG5=8< ?@>1;5<V B>B>6=>ABV A;V2 C 3@C?0E. </LI></UL></font></ul></tr> <TR> <TD bgColor=#000080 height=3 colSpan=3 align=left BORDER="1"></TD> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> April 20, 2016 </font><td valign="top" width="15%" > <b><font size="+2"> <A href="./Bardyla.html">S. Bardyla</A> </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> Zariski topology on the topological semilattices <UL type="square"><LI class='oItem'> <font size="+1"> We will introduce a notion of Zariski topology on topological semilattices and give necessary and sufficient conditions for Zariski topology to be compact. </LI></UL></font></ul></tr> <TR> <TD bgColor=#000080 height=3 colSpan=3 align=left BORDER="1"></TD> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> April 27, 2016 </font><td valign="top" width="15%" > <b><font size="+2"> <A href="./Gutik_mine.html">O. Gutik</A> </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> Compact-like semilattices topologies on the semilattice </i> <b>exp<sub><font size=-1>n</font></sub>(&#955)</b> <i></i> <UL type="square"><LI class='oItem'> <font size="+1"> We discuss about compact-like semilattices topologies on the semilattice <b>exp<sub><font size=-1>n</font></sub>(&#955)</b>, i.e., the semilattice of finite subsets of a bounded rank <b>n</b> of an arbirary infinite cardinal <b>&#955</b> with the operation <b>&cap;</b>. Particularly, we describe all feebly compact Hausdorff semilattice topologies on <b>exp<sub><font size=-1>n</font></sub>(&#955)</b>. </LI></UL></font></ul></tr> <TR> <TD bgColor=#000080 height=3 colSpan=3 align=left BORDER="1"></TD> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> May 4, 2016 </font><td valign="top" width="15%" > <b><font size="+2"> <A href="./Gutik_mine.html">O. Gutik</A> </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> The extension <b>I<sub><font size=-1>&#955</font></sub><sup><font size=-1>n</font></sup>(S)</b> of a semigroup <b>S</b> by symmetric inverse semigroup of a bounded finite rank <b>n</b> preserves the property to have tight ideal series <UL type="square"><LI class='oItem'> <font size="+1"> Let <b>I<sub><font size=-1>&#955</font></sub><sup><font size=-1>n</font></sup>(S)</b> be the extension of a semigroup <b>S</b> by symmetric inverse semigroup of a bounded finite rank <b>n</b>, where <b>&#955</b> is any non-zero cardinal and <b>n</b> is an arbitrary positive integer <b>&le; &#955</b>. We describe all ideals of the semigroup <b>I<sub><font size=-1>&#955</font></sub><sup><font size=-1>n</font></sup>(S)</b> up to modulo of a monoid <b>S</b> and show if the semigroup <b>S</b> has tight ideal series then <b>I<sub><font size=-1>&#955</font></sub><sup><font size=-1>n</font></sup>(S)</b> has tight ideal series too. </LI></UL></font></ul></tr> <TR> <TD bgColor=#000080 height=3 colSpan=3 align=left BORDER="1"></TD> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> May 11, 2016 </font><td valign="top" width="15%" > <b><font size="+2"> . 02AL:89,</br> <A href="./Gutik_mine.html">O. CBV:</A> </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> V4?>2V4L =0 ?8B0==O Salvo Tringali 7 <b>MathOwerflow</b> <UL type="square"><LI class='oItem'> <font size="+1"> >1C4>20=> ?@8:;04 30CA4>@D>2>W 015;L>2>W B>?>;>3VG=>W =0?V2B@C?8 <b>S</b>, O:0 =5 4>?CA:0T 3><><>@D=>3> CIV;L=5==O C 30CA4>@D>2C (?0@0)B>?>;>3VG=C 3@C?C, B0:8< G8=>< 2V4?>2V4L =0 ?8B0==O Salvo Tringali 7 <A href="http://mathoverflow.net/questions/222828/embedding-abelian-cancellative-hausdorff-topological-semigroups-into-abelian-hau/223539#223539"><b>MathOwerflow</b></A>. # O:>ABV =0?V23@C?8 <b>S</b> <>6=0 27OB8 2V;L=C 015;L>2C =0?V23@C?C =04 30CA4>@D>28< ?@>AB>@><, O:89 =5 T DC=:FV>=0;L=> 30CA4>@D>28<. </LI></UL></font></ul></tr> <TR> <TD bgColor=#000080 height=3 colSpan=3 align=left BORDER="1"></TD> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> May 18, 2016 </font><td valign="top" width="15%" > <b><font size="+2"> <A href="./guran.html">I. C@0=</A> </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> @> ?@>1;5<C B>B>6=>ABV A;V2 C 3@C?0E, II <UL type="square"><LI class='oItem'> <font size="+1"> &8:; 4>?>2V459 1C45 ?@8A2OG5=8< ?@>1;5<V B>B>6=>ABV A;V2 C 3@C?0E. </LI></UL></font></ul></tr> <TR> <TD bgColor=#000080 height=3 colSpan=3 align=left BORDER="1"></TD> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> May 25, 2016 </font><td valign="top" width="15%" > <b><font size="+2"> <A href="./Bardyla.html">S. Bardyla</A> </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> On the dychotomy of a locally comapct semitopological <b>&#955</b>-polycyclic monoid <UL type="square"><LI class='oItem'> <font size="+1"> For every non-zero cardinal <b>&#955</b> we will consider the <b>&#955</b>-polycyclic monoid <b>P<sub><font size=-1>&#955</font></sub></b> as a semitopological semigroup and describe all locally compact topologies on <b>P<sub><font size=-1>&#955</font></sub></b>. </LI></UL></font></ul></tr> <TR> <TD bgColor=#000080 height=3 colSpan=3 align=left BORDER="1"></TD> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> June 1, 2016 </font><td valign="top" width="15%" > <b><font size="+2"> <A href="./Bardyla.html">S. Bardyla</A> </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> On the dychotomy of a locally comapct semitopological <b>&#955</b>-polycyclic monoid, II <UL type="square"><LI class='oItem'> <font size="+1"> For every non-zero cardinal <b>&#955</b> we will consider the <b>&#955</b>-polycyclic monoid <b>P<sub><font size=-1>&#955</font></sub></b> as a semitopological semigroup and describe all locally compact topologies on <b>P<sub><font size=-1>&#955</font></sub></b>. </LI></UL></font></ul></tr> <TR> <TD bgColor=#000080 height=3 colSpan=3 align=left BORDER="1"></TD> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> June 22, 2016 </font><td valign="top" width="15%" > <b><font size="+2"> <A href="./Bardyla.html">S. Bardyla</A> </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> Some topological properties of a semitopological <b>&#945;</b>-bicyclic semigroup <UL type="square"><LI class='oItem'> <font size="+1"> We will consider locally compact topologizations of the <b>&#945;</b>-bicyclic monoid as a semitopological semigroup and investigate some completions of a topological inverse <b>&#945;</b>-bicyclic semigroup. </LI></UL></font></ul></tr> <TR> <TD bgColor=#000080 height=3 colSpan=3 align=left BORDER="1"></TD> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> July 27, 2016 </font><td valign="top" width="15%" > <b><font size="+2"> <A href="./Gutik_mine.html">O. Gutik</A> </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> On locally compact semitopological <b>0</b>-bisimple inverse <b>&omega;</b>-semigroups <UL type="square"><LI class='oItem'> <font size="+1"> We discuss on the structure of Hausdorff locally compact semitopological <b>0</b>-bisimple inverse <b>&omega;</b>-semigroups with a compact maximal subgroup. In particular, we show that a Hausdorff locally compact semitopological <b>0</b>-bisimple inverse <b>&omega;</b>-semigroup with a compact maximal subgroup is either compact or topologically isomorphic to the topological sum of its <b><i>H</i></b>-classes. </LI></UL></font></ul></tr> <TR> <TD bgColor=#000080 height=3 colSpan=3 align=left BORDER="1"></TD> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> July 29, 2016 </font><td valign="top" width="15%" > <b><font size="+2"> <A href="./Gutik_mine.html">O. Gutik</A> </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> On locally compact semitopological <b>0</b>-bisimple inverse <b>&omega;</b>-semigroups, II <UL type="square"><LI class='oItem'> <font size="+1"> We discuss on the structure of Hausdorff locally compact semitopological <b>0</b>-bisimple inverse <b>&omega;</b>-semigroups with a compact maximal subgroup. In particular, we show that a Hausdorff locally compact semitopological <b>0</b>-bisimple inverse <b>&omega;</b>-semigroup with a compact maximal subgroup is either compact or topologically isomorphic to the topological sum of its <b><i>H</i></b>-classes. </LI></UL></font></ul></tr> <TR> <TD bgColor=#000080 height=3 colSpan=3 align=left BORDER="1"></TD> <tr> <td colspan=3 HEIGHT="12"> </td></tr> <tr> <td valign="top" width="25%"> <font size="+2"> August 17, 2016 </font><td valign="top" width="15%" > <b><font size="+2"> <A href="./Gutik_mine.html">O. Gutik</A> </font></b> <td valign="center"><DIV onClick="JavaScript: outlineAction();"> <ul type="square"><LI class='oParent' value="30"> <font size="+2"> On locally compact semitopological <b>0</b>-bisimple inverse <b>&omega;</b>-semigroups, III <UL type="square"><LI class='oItem'> <font size="+1"> We discuss on the structure of Hausdorff locally compact semitopological <b>0</b>-bisimple inverse <b>&omega;</b>-semigroups with a compact maximal subgroup. In particular, we show that a Hausdorff locally compact semitopological <b>0</b>-bisimple inverse <b>&omega;</b>-semigroup with a compact maximal subgroup is either compact or topologically isomorphic to the topological sum of its <b><i>H</i></b>-classes. </LI></UL></font></ul></tr> <TR> <TD bgColor=#000080 height=3 colSpan=3 align=left BORDER="1"></TD> </table> <a href="https://plus.google.com/u/0/+OlegGutik"><img src="Rgplus-Gutik.jpg"></a> <a href="https://plus.google.com/u/0/+OlGutikPallady"><img src="Rgplus-Gutik.jpg"></a> <a href="https://plus.google.com/u/0/+TopologicalAlgebraSeminar"><img src="Rgplus-Gutik.jpg"></a> <br> <br> <!-- Start of StatCounter Code for Default Guide --> <script type="text/javascript"> var sc_project=9174828; var sc_invisible=0; var sc_security="d274098c"; var scJsHost = (("https:" == document.location.protocol) ? "https://secure." : "http://www."); document.write("<sc"+"ript type='text/javascript' src='" + scJsHost+ "statcounter.com/counter/counter.js'></"+"script>"); </script> <noscript><div class="statcounter"><a title="web analytics" href="http://statcounter.com/" target="_blank"><img class="statcounter" src="http://c.statcounter.com/9174828/0/d274098c/0/" alt="web analytics"></a></div></noscript> <!-- End of StatCounter Code for Default Guide --> <br> <!-- HotLog --> <script type="text/javascript"> hotlog_r=""+Math.random()+"&s=2303117&im=214&r="+ escape(document.referrer)+"&pg="+escape(window.location.href); hotlog_r+="&j="+(navigator.javaEnabled()?"Y":"N"); hotlog_r+="&wh="+screen.width+"x"+screen.height+"&px="+ (((navigator.appName.substring(0,3)=="Mic"))?screen.colorDepth:screen.pixelDepth); hotlog_r+="&js=1.3"; document.write('<a href="http://click.hotlog.ru/?2303117" target="_blank"><img '+ 'src="http://hit3.hotlog.ru/cgi-bin/hotlog/count?'+ hotlog_r+'" border="0" width="88" height="100" title="" alt="HotLog"><\/a>'); </script> <noscript> <a href="http://click.hotlog.ru/?2303117" target="_blank"><img src="http://hit3.hotlog.ru/cgi-bin/hotlog/count?s=2303117&im=214" border="0" width="88" height="100" title="" alt="HotLog"></a> </noscript> <!-- /HotLog --> </body> </html>