LEADER 04678nam 2200721 a 450 001 9910789997803321 005 20230725031313.0 010 $a1-283-16472-8 010 $a9786613164728 010 $a3-11-021322-2 024 7 $a10.1515/9783110213225 035 $a(CKB)2670000000088742 035 $a(EBL)690579 035 $a(OCoLC)723945447 035 $a(SSID)ssj0000530901 035 $a(PQKBManifestationID)12214237 035 $a(PQKBTitleCode)TC0000530901 035 $a(PQKBWorkID)10571783 035 $a(PQKB)11537515 035 $a(MiAaPQ)EBC690579 035 $a(DE-B1597)35763 035 $a(OCoLC)754713543 035 $a(OCoLC)853269839 035 $a(DE-B1597)9783110213225 035 $a(Au-PeEL)EBL690579 035 $a(CaPaEBR)ebr10486427 035 $a(CaONFJC)MIL316472 035 $a(EXLCZ)992670000000088742 100 $a20110105d2011 uy 0 101 0 $aeng 135 $aurnn#---|u||u 181 $ctxt 182 $cc 183 $acr 200 10$aUltrafilters and topologies on groups$b[electronic resource] /$fYevhen G. Zelenyuk 210 $aBerlin ;$aNew York $cDe Gruyter$dc2011 215 $a1 online resource (228 p.) 225 1 $aDe Gruyter expositions in mathematics,$x0938-6572 ;$v50 300 $aDescription based upon print version of record. 311 0 $a3-11-020422-3 320 $aIncludes bibliographical references and index. 327 $tFront matter --$tPreface --$tContents --$t1 Topological Groups --$t2 Ultrafilters --$t3 Topological Spaces with Extremal Properties --$t4 Left Invariant Topologies and Strongly Discrete Filters --$t5 Topological Groups with Extremal Properties --$t6 The Semigroup ?S --$t7 Ultrafilter Semigroups --$t8 Finite Groups in ?G --$t9 Ideal Structure of ?G --$t10 Almost Maximal Topological Groups --$t11 Almost Maximal Spaces --$t12 Resolvability --$t13 Open Problems --$tBibliography --$tIndex 330 $aThis book presents the relationship between ultrafilters and topologies on groups. It shows how ultrafilters are used in constructing topologies on groups with extremal properties and how topologies on groups serve in deriving algebraic results about ultrafilters. The contents of the book fall naturally into three parts. The first, comprising Chapters 1 through 5, introduces to topological groups and ultrafilters insofar as the semigroup operation on ultrafilters is not required. Constructions of some important topological groups are given. In particular, that of an extremally disconnected topological group based on a Ramsey ultrafilter. Also one shows that every infinite group admits a nondiscrete zero-dimensional topology in which all translations and the inversion are continuous. In the second part, Chapters 6 through 9, the Stone-Cêch compactification ?G of a discrete group G is studied. For this, a special technique based on the concepts of a local left group and a local homomorphism is developed. One proves that if G is a countable torsion free group, then ?G contains no nontrivial finite groups. Also the ideal structure of ?G is investigated. In particular, one shows that for every infinite Abelian group G, ?G contains 22|G| minimal right ideals. In the third part, using the semigroup ?G, almost maximal topological and left topological groups are constructed and their ultrafilter semigroups are examined. Projectives in the category of finite semigroups are characterized. Also one shows that every infinite Abelian group with finitely many elements of order 2 is absolutely ?-resolvable, and consequently, can be partitioned into ? subsets such that every coset modulo infinite subgroup meets each subset of the partition. The book concludes with a list of open problems in the field. Some familiarity with set theory, algebra and topology is presupposed. But in general, the book is almost self-contained. It is aimed at graduate students and researchers working in topological algebra and adjacent areas. 410 0$aDe Gruyter expositions in mathematics ;$v50. 606 $aTopological group theory 606 $aUltrafilters (Mathematics) 610 $aAlmost Maximal Spaces. 610 $aFinite Semigroup. 610 $aSemigroup. 610 $aTopology. 610 $aUltrafilter. 615 0$aTopological group theory. 615 0$aUltrafilters (Mathematics) 676 $a512/.55 686 $aSK 340$qSEPA$2rvk 700 $aZelenyuk$b Yevhen G$01470241 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910789997803321 996 $aUltrafilters and topologies on groups$93681929 997 $aUNINA