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This 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 βG 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 βG contains no nontrivial finite groups. Also the ideal structure of βG is investigated. In particular, one shows that for every infinite Abelian group G, βG contains 22|G| minimal right ideals. In the third part, using the semigroup βG, 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 ω-resolvable, and consequently, can be partitioned into ω 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.

Topological group theory. --- Ultrafilters (Mathematics) --- Filters (Mathematics) --- Almost Maximal Spaces. --- Finite Semigroup. --- Semigroup. --- Topology. --- Ultrafilter.

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The theory of linear algebraic monoids culminates in a coherent blend of algebraic groups, convex geometry, and semigroup theory. The book discusses all the key topics in detail, including classification, orbit structure, representations, universal constructions, and abstract analogues. An explicit cell decomposition is constructed for the wonderful compactification, as is a universal deformation for any semisimple group. A final chapter summarizes important connections with other areas of algebra and geometry. The book will serve as a solid basis for further research. Open problems are discus

Monoids --- Semigroup algebras --- Monoids. --- Semigroup algebras. --- Algebras, Semigroup --- Mathematics. --- Algebra. --- Geometry. --- Combinatorics. --- Linear algebraic groups. --- Algebra --- Algebraic groups, Linear --- Geometry, Algebraic --- Group theory --- Algebraic varieties --- Semigroups --- Linear algebraic groups --- Combinatorics --- Mathematical analysis --- Mathematics --- Euclid's Elements

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A broad range of topics is covered here, including commutative monoid rings, the Jacobson radical of semigroup rings, blocks of modular group algebras, nilpotency index of the radical of group algebras, the isomorphism problem for group rings, inverse semigroup algebras and the Picard group of an abelian group ring. The survey lectures provide an up-to-date account of the current state of the subject and form a comprehensive introduction for intending researchers.

Group rings --- Semigroup rings --- 512.55 --- 512.55 Rings and modules --- Rings and modules --- Rings, Semi group --- Rings, Semigroup --- Semi group rings --- Rings (Algebra) --- Semigroups --- Group theory --- Congresses

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This is the second revised and extended edition of the successful book on the algebraic structure of the Stone-Čech compactification of a discrete semigroup and its combinatorial applications, primarily in the field known as Ramsey Theory. There has been very active research in the subject dealt with by the book in the 12 years which is now included in this edition. This book is a self-contained exposition of the theory of compact right semigroups for discrete semigroups and the algebraic properties of these objects. The methods applied in the book constitute a mosaic of infinite combinatorics, algebra, and topology. The reader will find numerous combinatorial applications of the theory, including the central sets theorem, partition regularity of matrices, multidimensional Ramsey theory, and many more.

Stone-Čech compactification. --- Topological semigroups. --- Semigroups --- Topological groups --- Compactifications --- Ergodic Theory. --- Ramsey Theory. --- Semigroup Compactification. --- Semigroup. --- Stone-Cech Compactification. --- Topological Dynamics.

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Presents the English translation of Kaljulaid's 1979 Tartu/Minsk Candidate thesis. The thesis was devoted to representation theory in the spirit of his thesis advisor BI Plotkin. Through representation theory, Kaljulaid became also interested in automata theory, which at a later phase became his main area of interest.

Semigroups. --- Semigroup algebras. --- Machine theory. --- Mathematics --- Math --- Science --- Abstract automata --- Abstract machines --- Automata --- Mathematical machine theory --- Algorithms --- Logic, Symbolic and mathematical --- Recursive functions --- Robotics --- Algebras, Semigroup --- Algebra --- Group theory --- History.