Ordered Sets
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Ordered Sets
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Author : Bernd Siegfried Walter Schröder
language : en
Publisher: Springer Science & Business Media
Release Date : 2003
Ordered Sets written by Bernd Siegfried Walter Schröder and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003 with Mathematics categories.
This work is an introduction to the basic tools of the theory of (partially) ordered sets such as visualization via diagrams, subsets, homomorphisms, important order-theoretical constructions, and classes of ordered sets. Using a thematic approach, the author presents open or recently solved problems to motivate the development of constructions and investigations for new classes of ordered sets. A wide range of material is presented, from classical results such as Dilworth's, Szpilrajn's and Hashimoto's Theorems to more recent results such as the Li--Milner Structure Theorem. Major topics covered include: chains and antichains, lowest upper and greatest lower bounds, retractions, lattices, the dimension of ordered sets, interval orders, lexicographic sums, products, enumeration, algorithmic approaches and the role of algebraic topology. Since there are few prerequisites, the text can be used as a focused follow-up or companion to a first proof (set theory and relations) or graph theory class. After working through a comparatively lean core, the reader can choose from a diverse range of topics such as structure theory, enumeration or algorithmic aspects. Also presented are some key topics less customary to discrete mathematics/graph theory, including a concise introduction to homology for graphs, and the presentation of forward checking as a more efficient alternative to the standard backtracking algorithm. The coverage throughout provides a solid foundation upon which research can be started by a mathematically mature reader. Rich in exercises, illustrations, and open problems, Ordered Sets: An Introduction is an excellent text for undergraduate and graduate students and a good resource for the interested researcher. Readers will discover order theory's role in discrete mathematics as a supplier of ideas as well as an attractive source of applications.
Ordered Sets
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Author : Egbert Harzheim
language : en
Publisher: Springer Science & Business Media
Release Date : 2005-02-17
Ordered Sets written by Egbert Harzheim and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-02-17 with Mathematics categories.
This detailed textbook presents a great deal of material on ordered sets not previously published in the still rather limited textbook literature. It should be suitable as a text for a course on order theory.
Ordered Sets
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Author : Egbert Harzheim
language : en
Publisher: Springer Science & Business Media
Release Date : 2005-07-25
Ordered Sets written by Egbert Harzheim and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-07-25 with Mathematics categories.
This book is written particularly for mathematics students and, of course, for mathematicians interested in set theory. Only some fun- mental parts of naive set theory are presupposed, - not more than is treated in a textbook on set theory, even if this restricts us only to the most basic facts of this field. We have summarized all of this in Chapter 0 without longer discusssions and explanations, because there are s- eral textbooks which can be consulted by the reader, e.g. HrbacekIJech [88], KneeboneIRotman [99], ShenIVereshchagin [159]. Besides this only elementary facts of analysis are used. The theory of ordered sets can be divided into two parts, depending on whether the sets under consideration are finite or infinite. The first part is grounded mainly in combinatorics and graph theory and does not make essential use of set-theorical concepts, whereas the second part presupposes a knowledge of the fundamental notions of set theory, in particular of the system of ordinal and cardinal numbers. In this book we mainly deal with general infinite ordered sets. In this field the textbook literature is still very small. Therefore this book supplements the existing literature.
Ordered Sets
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Author : Bernd Schröder
language : en
Publisher: Birkhäuser
Release Date : 2016-05-11
Ordered Sets written by Bernd Schröder and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-05-11 with Mathematics categories.
An introduction to the basic tools of the theory of (partially) ordered sets such as visualization via diagrams, subsets, homomorphisms, important order-theoretical constructions and classes of ordered sets. Using a thematic approach, the author presents open or recently solved problems to motivate the development of constructions and investigations for new classes of ordered sets. The text can be used as a focused follow-up or companion to a first proof (set theory and relations) or graph theory course.
Finite Ordered Sets
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Author : Nathalie Caspard
language : en
Publisher: Cambridge University Press
Release Date : 2012-01-26
Finite Ordered Sets written by Nathalie Caspard and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-01-26 with Mathematics categories.
Ordered sets are ubiquitous in mathematics and have significant applications in computer science, statistics, biology and the social sciences. As the first book to deal exclusively with finite ordered sets, this book will be welcomed by graduate students and researchers in all of these areas. Beginning with definitions of key concepts and fundamental results (Dilworth's and Sperner's theorem, interval and semiorders, Galois connection, duality with distributive lattices, coding and dimension theory), the authors then present applications of these structures in fields such as preference modelling and aggregation, operational research and management, cluster and concept analysis, and data mining. Exercises are included at the end of each chapter with helpful hints provided for some of the most difficult examples. The authors also point to further topics of ongoing research.
Hausdorff On Ordered Sets
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Author : Felix Hausdorff
language : en
Publisher: American Mathematical Soc.
Release Date : 2005
Hausdorff On Ordered Sets written by Felix Hausdorff and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005 with Mathematics categories.
Georg Cantor, the founder of set theory, published his last paper on sets in 1897. In 1900, David Hilbert made Cantor's Continuum Problem and the challenge of well-ordering the real numbers the first problem in his famous Paris lecture. It was time for the appearance of the second generation of Cantorians. They emerged in the decade 1900-1909, and foremost among them were Ernst Zermelo and Felix Hausdorff. Zermelo isolated the Choice Principle, proved that every set could be well-ordered, and axiomatized the concept of set. He became the father of abstract set theory. Hausdorff eschewed foundations and pursued set theory as part of the mathematical arsenal. He was recognized as the era's leading Cantorian. From 1901-1909, Hausdorff published seven articles in which he created a representation theory for ordered sets and investigated sets of real sequences partially ordered by eventual dominance, together with their maximally ordered subsets. These papers are translated and appear in this volume. Each is accompanied by an introductory essay. These highly accessible works are of historical significance, not only for set theory, but also for model theory, analysis and algebra.
Ordered Sets And Lattices Ii
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Author :
language : en
Publisher: American Mathematical Soc.
Release Date :
Ordered Sets And Lattices Ii written by and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on with categories.
This indispensable reference source contains a wealth of information on lattice theory. The book presents a survey of virtually everything published in the fields of partially ordered sets, semilattices, lattices, and Boolean algebras that was reviewed in Referativnyi Zhurnal Matematika from mid-1982 to the end of 1985. A continuation of a previous volume (the English translation of which was published by the AMS in 1989, as volume 141 in Translations - Series 2), this comprehensive work contains more than 2200 references. Many of the papers covered here were originally published in virtually inaccessible places. The compilation of the volume was directed by Milan Kolibiar of Comenius University at Bratislava and Lev A. Skornyakov of Moscow University. Of interest to mathematicians, as well as to philosophers and computer scientists in certain areas, this unique compendium is a must for any mathematical library.
Linear Representations Of Partially Ordered Sets And Vector Space Categories
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Author : Daniel Simson
language : en
Publisher: CRC Press
Release Date : 1993-01-01
Linear Representations Of Partially Ordered Sets And Vector Space Categories written by Daniel Simson and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1993-01-01 with Mathematics categories.
This volume provides an elementary yet comprehensive introduction to representations of partially ordered sets and bimodule matrix problems, and their use in representation theory of algebras. It includes a discussion of representation types of algebras and partially ordered sets. Various characterizations of representation-finite and representation-tame partially ordered sets are offered and a description of their indecomposable representations is given. Auslander-Reiten theory is demonstrated together with a computer accessible algorithm for determining in decomposable representations and the Auslander-Reiten quiver of any representation-finite partially ordered set.
Lattices And Ordered Sets
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Author : Steven Roman
language : en
Publisher: Springer Science & Business Media
Release Date : 2008-12-15
Lattices And Ordered Sets written by Steven Roman and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-12-15 with Mathematics categories.
This book is intended to be a thorough introduction to the subject of order and lattices, with an emphasis on the latter. It can be used for a course at the graduate or advanced undergraduate level or for independent study. Prerequisites are kept to a minimum, but an introductory course in abstract algebra is highly recommended, since many of the examples are drawn from this area. This is a book on pure mathematics: I do not discuss the applications of lattice theory to physics, computer science or other disciplines. Lattice theory began in the early 1890s, when Richard Dedekind wanted to know the answer to the following question: Given three subgroups EF , and G of an abelian group K, what is the largest number of distinct subgroups that can be formed using these subgroups and the operations of intersection and sum (join), as in E?FßÐE?FÑ?GßE?ÐF?GÑ and so on? In lattice-theoretic terms, this is the number of elements in the relatively free modular lattice on three generators. Dedekind [15] answered this question (the answer is #)) and wrote two papers on the subject of lattice theory, but then the subject lay relatively dormant until Garrett Birkhoff, Oystein Ore and others picked it up in the 1930s. Since then, many noted mathematicians have contributed to the subject, including Garrett Birkhoff, Richard Dedekind, Israel Gelfand, George Grätzer, Aleksandr Kurosh, Anatoly Malcev, Oystein Ore, Gian-Carlo Rota, Alfred Tarski and Johnny von Neumann.
Problems And Theorems In Classical Set Theory
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Author : Peter Komjath
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-05-02
Problems And Theorems In Classical Set Theory written by Peter Komjath and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-05-02 with Mathematics categories.
This volume contains a variety of problems from classical set theory and represents the first comprehensive collection of such problems. Many of these problems are also related to other fields of mathematics, including algebra, combinatorics, topology and real analysis. Rather than using drill exercises, most problems are challenging and require work, wit, and inspiration. They vary in difficulty, and are organized in such a way that earlier problems help in the solution of later ones. For many of the problems, the authors also trace the history of the problems and then provide proper reference at the end of the solution.