Lattice Theory Special Topics And Applications

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Lattice Theory: Special Topics and Applications

George Grätzer,Friedrich Wehrung
Lattice Theory Special Topics and Applications Book PDF

Author : George Grätzer,Friedrich Wehrung
Publisher : Birkhäuser
Page : 616 pages
File Size : 55,7 Mb
Release : 2016-10-08
Category : Mathematics
ISBN : 9783319442365

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Lattice Theory: Special Topics and Applications by George Grätzer,Friedrich Wehrung Pdf

George Grätzer's Lattice Theory: Foundation is his third book on lattice theory (General Lattice Theory, 1978, second edition, 1998). In 2009, Grätzer considered updating the second edition to reflect some exciting and deep developments. He soon realized that to lay the foundation, to survey the contemporary field, to pose research problems, would require more than one volume and more than one person. So Lattice Theory: Foundation provided the foundation. Now we complete this project with Lattice Theory: Special Topics and Applications, in two volumes, written by a distinguished group of experts, to cover some of the vast areas not in Foundation. This second volume is divided into ten chapters contributed by K. Adaricheva, N. Caspard, R. Freese, P. Jipsen, J.B. Nation, N. Reading, H. Rose, L. Santocanale, and F. Wehrung.

Lattice Theory: Special Topics and Applications

George Grätzer,Friedrich Wehrung
Lattice Theory Special Topics and Applications Book PDF

Author : George Grätzer,Friedrich Wehrung
Publisher : Springer
Page : 472 pages
File Size : 41,8 Mb
Release : 2014-08-27
Category : Mathematics
ISBN : 9783319064130

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Lattice Theory: Special Topics and Applications by George Grätzer,Friedrich Wehrung Pdf

George Grätzer's Lattice Theory: Foundation is his third book on lattice theory (General Lattice Theory, 1978, second edition, 1998). In 2009, Grätzer considered updating the second edition to reflect some exciting and deep developments. He soon realized that to lay the foundation, to survey the contemporary field, to pose research problems, would require more than one volume and more than one person. So Lattice Theory: Foundation provided the foundation. Now we complete this project with Lattice Theory: Special Topics and Applications, written by a distinguished group of experts, to cover some of the vast areas not in Foundation. This first volume is divided into three parts. Part I. Topology and Lattices includes two chapters by Klaus Keimel, Jimmie Lawson and Ales Pultr, Jiri Sichler. Part II. Special Classes of Finite Lattices comprises four chapters by Gabor Czedli, George Grätzer and Joseph P. S. Kung. Part III. Congruence Lattices of Infinite Lattices and Beyond includes four chapters by Friedrich Wehrung and George Grätzer.

Lattice Theory

George Grätzer,Friedrich Wehrung
Lattice Theory Book PDF

Author : George Grätzer,Friedrich Wehrung
Publisher : Birkhäuser
Page : 0 pages
File Size : 50,5 Mb
Release : 2016-10-20
Category : Mathematics
ISBN : 3319482629

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Lattice Theory by George Grätzer,Friedrich Wehrung Pdf

This three-volume-set comprises the complete lattice theory project. Volume 1 of the set, Lattice Theory: Foundation, is the revised and enlarged third edition of General Lattice Theory. It focuses on introducing the field and covers the fundamental concepts and results. The two Special Topics and Applications volumes (volumes 2 and 3 of the set), jointly edited by George Grätzer and Friedrich Wehrung, update the reader on some of the vast areas not in Foundation. Volume 1 is divided into three parts. Part I. Topology and Lattices includes two chapters by Klaus Keimel, Jimmie Lawson and Ales Pultr, Jiri Sichler. Part II. Special Classes of Finite Lattices comprises four chapters by Gabor Czedli, George Grätzer and Joseph P. S. Kung. Part III. Congruence Lattices of Infinite Lattices and Beyond includes four chapters by Friedrich Wehrung and George Grätzer. Volume 2 is divided into ten chapters contributed by K. Adaricheva, N. Caspard, R. Freese, P. Jipsen, J.B. Nation, N. Reading, H. Rose, L. Santocanale, and F. Wehrung.

Lattice Theory: Foundation

George Grätzer
Lattice Theory Foundation Book PDF

Author : George Grätzer
Publisher : Springer Science & Business Media
Page : 639 pages
File Size : 54,7 Mb
Release : 2011-02-14
Category : Mathematics
ISBN : 9783034800181

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Lattice Theory: Foundation by George Grätzer Pdf

This book started with Lattice Theory, First Concepts, in 1971. Then came General Lattice Theory, First Edition, in 1978, and the Second Edition twenty years later. Since the publication of the first edition in 1978, General Lattice Theory has become the authoritative introduction to lattice theory for graduate students and the standard reference for researchers. The First Edition set out to introduce and survey lattice theory. Some 12,000 papers have been published in the field since then; so Lattice Theory: Foundation focuses on introducing the field, laying the foundation for special topics and applications. Lattice Theory: Foundation, based on the previous three books, covers the fundamental concepts and results. The main topics are distributivity, congruences, constructions, modularity and semimodularity, varieties, and free products. The chapter on constructions is new, all the other chapters are revised and expanded versions from the earlier volumes. Almost 40 “diamond sections’’, many written by leading specialists in these fields, provide a brief glimpse into special topics beyond the basics. “Lattice theory has come a long way... For those who appreciate lattice theory, or who are curious about its techniques and intriguing internal problems, Professor Grätzer's lucid new book provides a most valuable guide to many recent developments. Even a cursory reading should provide those few who may still believe that lattice theory is superficial or naive, with convincing evidence of its technical depth and sophistication.” Bulletin of the American Mathematical Society “Grätzer’s book General Lattice Theory has become the lattice theorist’s bible.” Mathematical Reviews

General Lattice Theory

G. Grätzer
General Lattice Theory Book PDF

Author : G. Grätzer
Publisher : Birkhäuser
Page : 392 pages
File Size : 44,8 Mb
Release : 2012-12-06
Category : Science
ISBN : 9783034876339

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General Lattice Theory by G. Grätzer Pdf

In the first half of the nineteenth century, George Boole's attempt to formalize propositional logic led to the concept of Boolean algebras. While investigating the axiomatics of Boolean algebras at the end of the nineteenth century, Charles S. Peirce and Ernst Schröder found it useful to introduce the lattice concept. Independently, Richard Dedekind's research on ideals of algebraic numbers led to the same discov ery. In fact, Dedekind also introduced modularity, a weakened form of distri butivity. Although some of the early results of these mathematicians and of Edward V. Huntington are very elegant and far from trivial, they did not attract the attention of the mathematical community. It was Garrett Birkhoff's work in the mid-thirties that started the general develop ment of lattice theory. In a brilliant series of papers he demonstrated the importance of lattice theory and showed that it provides a unifying framework for hitherto unrelated developments in many mathematical disciplines. Birkhoff himself, Valere Glivenko, Karl Menger, John von Neumann, Oystein Ore, and others had developed enough of this new field for Birkhoff to attempt to "seIl" it to the general mathematical community, which he did with astonishing success in the first edition of his Lattice Theory. The further development of the subject matter can best be followed by com paring the first, second, and third editions of his book (G. Birkhoff [1940], [1948], and [1967]).

Introduction to Lattice Algebra

Gerhard X. Ritter,Gonzalo Urcid
Introduction to Lattice Algebra Book PDF

Author : Gerhard X. Ritter,Gonzalo Urcid
Publisher : CRC Press
Page : 292 pages
File Size : 53,9 Mb
Release : 2021-08-23
Category : Mathematics
ISBN : 9781000412604

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Introduction to Lattice Algebra by Gerhard X. Ritter,Gonzalo Urcid Pdf

Lattice theory extends into virtually every branch of mathematics, ranging from measure theory and convex geometry to probability theory and topology. A more recent development has been the rapid escalation of employing lattice theory for various applications outside the domain of pure mathematics. These applications range from electronic communication theory and gate array devices that implement Boolean logic to artificial intelligence and computer science in general. Introduction to Lattice Algebra: With Applications in AI, Pattern Recognition, Image Analysis, and Biomimetic Neural Networks lays emphasis on two subjects, the first being lattice algebra and the second the practical applications of that algebra. This textbook is intended to be used for a special topics course in artificial intelligence with a focus on pattern recognition, multispectral image analysis, and biomimetic artificial neural networks. The book is self-contained and – depending on the student’s major – can be used for a senior undergraduate level or first-year graduate level course. The book is also an ideal self-study guide for researchers and professionals in the above-mentioned disciplines. Features Filled with instructive examples and exercises to help build understanding Suitable for researchers, professionals and students, both in mathematics and computer science Contains numerous exercises.

A Primer of Subquasivariety Lattices

Kira Adaricheva,Jennifer Hyndman,J. B. Nation,Joy N. Nishida
A Primer of Subquasivariety Lattices Book PDF

Author : Kira Adaricheva,Jennifer Hyndman,J. B. Nation,Joy N. Nishida
Publisher : Springer Nature
Page : 293 pages
File Size : 47,7 Mb
Release : 2022-08-18
Category : Mathematics
ISBN : 9783030980887

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A Primer of Subquasivariety Lattices by Kira Adaricheva,Jennifer Hyndman,J. B. Nation,Joy N. Nishida Pdf

This book addresses Birkhoff and Mal'cev's problem of describing subquasivariety lattices. The text begins by developing the basics of atomic theories and implicational theories in languages that may, or may not, contain equality. Subquasivariety lattices are represented as lattices of closed algebraic subsets of a lattice with operators, which yields new restrictions on the equaclosure operator. As an application of this new approach, it is shown that completely distributive lattices with a dually compact least element are subquasivariety lattices. The book contains many examples to illustrate these principles, as well as open problems. Ultimately this new approach gives readers a set of tools to investigate classes of lattices that can be represented as subquasivariety lattices.

Lattices and Ordered Sets

Steven Roman
Lattices and Ordered Sets Book PDF

Author : Steven Roman
Publisher : Springer Science & Business Media
Page : 307 pages
File Size : 51,6 Mb
Release : 2008-12-15
Category : Mathematics
ISBN : 9780387789019

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Lattices and Ordered Sets by Steven Roman Pdf

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.

Introduction to Lattice Theory with Computer Science Applications

Vijay K. Garg
Introduction to Lattice Theory with Computer Science Applications Book PDF

Author : Vijay K. Garg
Publisher : John Wiley & Sons
Page : 272 pages
File Size : 41,6 Mb
Release : 2016-03-02
Category : Computers
ISBN : 9781119069737

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Introduction to Lattice Theory with Computer Science Applications by Vijay K. Garg Pdf

A computational perspective on partial order and lattice theory, focusing on algorithms and their applications This book provides a uniform treatment of the theory and applications of lattice theory. The applications covered include tracking dependency in distributed systems, combinatorics, detecting global predicates in distributed systems, set families, and integer partitions. The book presents algorithmic proofs of theorems whenever possible. These proofs are written in the calculational style advocated by Dijkstra, with arguments explicitly spelled out step by step. The author’s intent is for readers to learn not only the proofs, but the heuristics that guide said proofs. Introduction to Lattice Theory with Computer Science Applications: Examines; posets, Dilworth’s theorem, merging algorithms, lattices, lattice completion, morphisms, modular and distributive lattices, slicing, interval orders, tractable posets, lattice enumeration algorithms, and dimension theory Provides end of chapter exercises to help readers retain newfound knowledge on each subject Includes supplementary material at www.ece.utexas.edu/~garg Introduction to Lattice Theory with Computer Science Applications is written for students of computer science, as well as practicing mathematicians.

The Congruences of a Finite Lattice

George Grätzer
The Congruences of a Finite Lattice Book PDF

Author : George Grätzer
Publisher : Springer Nature
Page : 440 pages
File Size : 41,9 Mb
Release : 2023-03-23
Category : Mathematics
ISBN : 9783031290633

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The Congruences of a Finite Lattice by George Grätzer Pdf

The congruences of a lattice form the congruence lattice. Over the last several decades, the study of congruence lattices has established itself as a large and important field with a great number of interesting and deep results, as well as many open problems. Written by one of the leading experts in lattice theory, this text provides a self-contained introduction to congruences of finite lattices and presents the major results of the last 90 years. It features the author’s signature “Proof-by-Picture” method, which is used to convey the ideas behind formal proofs in a visual, more intuitive manner. Key features include: an insightful discussion of techniques to construct "nice" finite lattices with given congruence lattices and "nice" congruence-preserving extensions complete proofs, an extensive bibliography and index, and over 180 illustrations additional chapters covering new results of the last seven years, increasing the size of this edition to 430 pages, 360 statements, and 262 references This text is appropriate for a one-semester graduate course in lattice theory, and it will also serve as a valuable reference for researchers studying lattices. Reviews of previous editions: “[This] monograph...is an exceptional work in lattice theory, like all the contributions by this author. The way this book is written makes it extremely interesting for the specialists in the field but also for the students in lattice theory. — Cosmin Pelea, Studia Universitatis Babes-Bolyai Mathematica LII (1), 2007 "The book is self-contained, with many detailed proofs presented that can be followed step-by-step. I believe that this book is a much-needed tool for any mathematician wishing a gentle introduction to the field of congruences representations of finite lattices, with emphasis on the more 'geometric' aspects." — Mathematical Reviews

Mordell–Weil Lattices

Matthias Schütt,Tetsuji Shioda
Mordell Weil Lattices Book PDF

Author : Matthias Schütt,Tetsuji Shioda
Publisher : Springer Nature
Page : 431 pages
File Size : 45,7 Mb
Release : 2019-10-17
Category : Mathematics
ISBN : 9789813293014

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Mordell–Weil Lattices by Matthias Schütt,Tetsuji Shioda Pdf

This book lays out the theory of Mordell–Weil lattices, a very powerful and influential tool at the crossroads of algebraic geometry and number theory, which offers many fruitful connections to other areas of mathematics. The book presents all the ingredients entering into the theory of Mordell–Weil lattices in detail, notably, relevant portions of lattice theory, elliptic curves, and algebraic surfaces. After defining Mordell–Weil lattices, the authors provide several applications in depth. They start with the classification of rational elliptic surfaces. Then a useful connection with Galois representations is discussed. By developing the notion of excellent families, the authors are able to design many Galois representations with given Galois groups such as the Weyl groups of E6, E7 and E8. They also explain a connection to the classical topic of the 27 lines on a cubic surface. Two chapters deal with elliptic K3 surfaces, a pulsating area of recent research activity which highlights many central properties of Mordell–Weil lattices. Finally, the book turns to the rank problem—one of the key motivations for the introduction of Mordell–Weil lattices. The authors present the state of the art of the rank problem for elliptic curves both over Q and over C(t) and work out applications to the sphere packing problem. Throughout, the book includes many instructive examples illustrating the theory.

Relational and Algebraic Methods in Computer Science

Jules Desharnais,Walter Guttmann,Stef Joosten
Relational and Algebraic Methods in Computer Science Book PDF

Author : Jules Desharnais,Walter Guttmann,Stef Joosten
Publisher : Springer
Page : 385 pages
File Size : 41,8 Mb
Release : 2018-10-22
Category : Mathematics
ISBN : 9783030021498

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Relational and Algebraic Methods in Computer Science by Jules Desharnais,Walter Guttmann,Stef Joosten Pdf

This book constitutes the proceedings of the 17th International Conference on Relational and Algebraic Methods in Computer Science, RAMiCS 2018, held in Groningen, The Netherlands, in October/November 2018. The 21 full papers and 1 invited paper presented together with 2 invited abstracts and 1 abstract of a tutorial were carefully selected from 31 submissions. The papers are organized in the following topics: Theoretical foundations; reasoning about computations and programs; and applications and tools.

The Lattice of Subquasivarieties of a Locally Finite Quasivariety

Jennifer Hyndman,J. B. Nation
The Lattice of Subquasivarieties of a Locally Finite Quasivariety Book PDF

Author : Jennifer Hyndman,J. B. Nation
Publisher : Springer
Page : 162 pages
File Size : 54,6 Mb
Release : 2018-08-28
Category : Computers
ISBN : 9783319782355

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The Lattice of Subquasivarieties of a Locally Finite Quasivariety by Jennifer Hyndman,J. B. Nation Pdf

This book discusses the ways in which the algebras in a locally finite quasivariety determine its lattice of subquasivarieties. The book starts with a clear and comprehensive presentation of the basic structure theory of quasivariety lattices, and then develops new methods and algorithms for their analysis. Particular attention is paid to the role of quasicritical algebras. The methods are illustrated by applying them to quasivarieties of abelian groups, modular lattices, unary algebras and pure relational structures. An appendix gives an overview of the theory of quasivarieties. Extensive references to the literature are provided throughout.

Lattice Path Combinatorics and Applications

George E. Andrews,Christian Krattenthaler,Alan Krinik
Lattice Path Combinatorics and Applications Book PDF

Author : George E. Andrews,Christian Krattenthaler,Alan Krinik
Publisher : Springer
Page : 418 pages
File Size : 45,8 Mb
Release : 2019-03-02
Category : Mathematics
ISBN : 9783030111021

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Lattice Path Combinatorics and Applications by George E. Andrews,Christian Krattenthaler,Alan Krinik Pdf

The most recent methods in various branches of lattice path and enumerative combinatorics along with relevant applications are nicely grouped together and represented in this research contributed volume. Contributions to this edited volume will be mainly research articles however it will also include several captivating, expository articles (along with pictures) on the life and mathematical work of leading researchers in lattice path combinatorics and beyond. There will be four or five expository articles in memory of Shreeram Shankar Abhyankar and Philippe Flajolet and honoring George Andrews and Lajos Takács. There may be another brief article in memory of Professors Jagdish Narayan Srivastava and Joti Lal Jain. New research results include the kernel method developed by Flajolet and others for counting different classes of lattice paths continues to produce new results in counting lattice paths. The recent investigation of Fishburn numbers has led to interesting counting interpretations and a family of fascinating congruences. Formulas for new methods to obtain the number of Fq-rational points of Schubert varieties in Grassmannians continues to have research interest and will be presented here. Topics to be included are far reaching and will include lattice path enumeration, tilings, bijections between paths and other combinatoric structures, non-intersecting lattice paths, varieties, Young tableaux, partitions, enumerative combinatorics, discrete distributions, applications to queueing theory and other continuous time models, graph theory and applications. Many leading mathematicians who spoke at the conference from which this volume derives, are expected to send contributions including. This volume also presents the stimulating ideas of some exciting newcomers to the Lattice Path Combinatorics Conference series; “The 8th Conference on Lattice Path Combinatorics and Applications” provided opportunities for new collaborations; some of the products of these collaborations will also appear in this book. This book will have interest for researchers in lattice path combinatorics and enumerative combinatorics. This will include subsets of researchers in mathematics, statistics, operations research and computer science. The applications of the material covered in this edited volume extends beyond the primary audience to scholars interested queuing theory, graph theory, tiling, partitions, distributions, etc. An attractive bonus within our book is the collection of special articles describing the top recent researchers in this area of study and documenting the interesting history of who, when and how these beautiful combinatorial results were originally discovered.

Formal Concept Analysis

Diana Cristea,Florence Le Ber,Baris Sertkaya
Formal Concept Analysis Book PDF

Author : Diana Cristea,Florence Le Ber,Baris Sertkaya
Publisher : Springer
Page : 355 pages
File Size : 42,6 Mb
Release : 2019-06-14
Category : Computers
ISBN : 9783030214623

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Formal Concept Analysis by Diana Cristea,Florence Le Ber,Baris Sertkaya Pdf

This book constitutes the proceedings of the 15th International Conference on Formal Concept Analysis, ICFCA 2019, held in Frankfurt am Main, Germany, in June 2019. The 15 full papers and 5 short papers presented in this volume were carefully reviewed and selected from 36 submissions. The book also contains four invited contributions in full paper length. The field of Formal Concept Analysis (FCA) originated in the 1980s in Darmstadt as a subfield of mathematical order theory, with prior developments in other research groups. Its original motivation was to consider complete lattices as lattices of concepts, drawing motivation from philosophy and mathematics alike. FCA has since then developed into a wide research area with applications much beyond its original motivation, for example in logic, data mining, learning, and psychology.