Download or read book Topics in Ordered Topological Spaces Including a Representation Theory for Distributive Lattices written by Hilary A. Priestley and published by . This book was released on 1970 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Lattices and Ordered Sets written by Steven Roman and published by Springer Science & Business Media. This book was released on 2008-12-15 with total page 307 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Introduction to Lattices and Order written by B. A. Davey and published by . This book was released on 1990 with total page 248 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the first introductory textbook on ordered sets and lattices, and covers both the basic theory and its applications. The importance of ordered structures has been increasingly recognised in recent years due to an explosion of interest in computer science and all areas of discrete mathematics. The authors provide a thorough introduction to ordered sets, lattices, distributive lattices and Boolean algebras. Ordered sets, and in particular lattices, can be represented pictorially, and this key feature is emphasised throughout. Lattices are also considered as algebraic structures and their study from this viewpoint reinforces ideas encountered in the theory of groups and rings. The representation of distributive lattices by ordered topological spaces is presented; a self-contained treatment of the requisite topology is included. Two chapters are devoted to topics with application to computer science. These cover complete partial orders, domains (including their relation to information systems), and fixpoint theory. Another chapter deals with formal concept analysis - a new and important application of lattice theory of interest to mathematicians and social scientists. Prerequisites are minimal; all that is assumed is exposure to the notation of set theory and elementary abstract algebra. The numerous classroom-tested exercises will make the book especially useful for course accompaniment, but it will also be valuable as a background reference for mathematicians, logicians and computer scientists.
Download or read book Topics in Logic Philosophy and Foundations of Mathematics and Computer Science written by Stanisław Krajewski and published by IOS Press. This book was released on 2007 with total page 380 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume honors Professor Andrzej Grzegorczyk, the nestor of Polish logicians, on his 85th anniversary. The editors would like to express the respect and sympathy they have for him. His textbook The Outline of Mathematical Logic has been published in many editions and translated into several languages. It was this textbook that introduced many of us into the world of mathematical logic. Professor Grzegorczyk has made fundamental contributions to logic and to philosophy. His results, mainly on hierarchy of primitive recursive functions, known as the Grzegorczyk hierarchy, are of fundamental importance to theoretical computer science. In particular, they were precursory for the computational complexity theory. The editors would like to stress that this special publication celebrates a scientist who is still actively pursuing genuinely innovative directions of research. Quite recently, Andrzej Grzegorczyk gave a new proof of undecidability of the first order functional calculus. His proof does not use the arithmetization of Kurt Gödel. In recognition of his merits, the University of Clermont-Ferrand conferred to Professor Andrzej Grzegorczyk the Doctorat Honoris Causa. The work and life of Professor Andrzej Grzegorczyk is presented in the article by Professors Stanislaw Krajewski and Jan Wolenski. The papers in this collection have been submitted on invitational basis.
Download or read book Ordered Sets and Lattices II written by and published by American Mathematical Soc.. This book was released on with total page 262 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Selected Papers on Algebra and Topology by Garrett Birkhoff written by J.S. Oliveira and published by Springer Science & Business Media. This book was released on 1987-01-01 with total page 648 pages. Available in PDF, EPUB and Kindle. Book excerpt: The present volume of reprints are what I consider to be my most interesting and influential papers on algebra and topology. To tie them together, and to place them in context, I have supplemented them by a series of brief essays sketching their historieal background (as I see it). In addition to these I have listed some subsequent papers by others which have further developed some of my key ideas. The papers on universal algebra, lattice theory, and general topology collected in the present volume concern ideas which have become familiar to all working mathematicians. It may be helpful to make them readily accessible in one volume. I have tried in the introduction to each part to state the most significant features of ea ch paper reprinted there, and to indieate later developments. The background that shaped and stimulated my early work on universal algebra, lattice theory, and topology may be of some interest. As a Harvard undergraduate in 1928-32, I was encouraged to do independent reading and to write an original thesis. My tutorial reading included de la Vallee-Poussin's beautiful Cours d'Analyse Infinitesimale, Hausdorff's Grundzüge der Mengenlehre, and Frechet's Espaces Abstraits. In addition, I discovered Caratheodory's 1912 paper "Vber das lineare Mass von Punktmengen" and Hausdorff's 1919 paper on "Dimension und Ausseres Mass," and derived much inspiration from them. A fragment of my thesis, analyzing axiom systems for separable metrizable spaces, was later published [2]. * This background led to the work summarized in Part IV.
Download or read book Spectral Spaces written by Max Dickmann and published by Cambridge University Press. This book was released on 2019-03-21 with total page 652 pages. Available in PDF, EPUB and Kindle. Book excerpt: Spectral spaces are a class of topological spaces. They are a tool linking algebraic structures, in a very wide sense, with geometry. They were invented to give a functional representation of Boolean algebras and distributive lattices and subsequently gained great prominence as a consequence of Grothendieck's invention of schemes. There are more than 1,000 research articles about spectral spaces, but this is the first monograph. It provides an introduction to the subject and is a unified treatment of results scattered across the literature, filling in gaps and showing the connections between different results. The book includes new research going beyond the existing literature, answering questions that naturally arise from this comprehensive approach. The authors serve graduates by starting gently with the basics. For experts, they lead them to the frontiers of current research, making this book a valuable reference source.
Download or read book A Compendium of Continuous Lattices written by G. Gierz and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 390 pages. Available in PDF, EPUB and Kindle. Book excerpt: A mathematics book with six authors is perhaps a rare enough occurrence to make a reader ask how such a collaboration came about. We begin, therefore, with a few words on how we were brought to the subject over a ten-year period, during part of which time we did not all know each other. We do not intend to write here the history of continuous lattices but rather to explain our own personal involvement. History in a more proper sense is provided by the bibliography and the notes following the sections of the book, as well as by many remarks in the text. A coherent discussion of the content and motivation of the whole study is reserved for the introduction. In October of 1969 Dana Scott was lead by problems of semantics for computer languages to consider more closely partially ordered structures of function spaces. The idea of using partial orderings to correspond to spaces of partially defined functions and functionals had appeared several times earlier in recursive function theory; however, there had not been very sustained interest in structures of continuous functionals. These were the ones Scott saw that he needed. His first insight was to see that - in more modern terminology - the category of algebraic lattices and the (so-called) Scott-continuous functions is cartesian closed.
Download or read book Representations of Distributive Topological Lattices of Finite Breadth written by Henry Forrest Bunch and published by . This book was released on 1981 with total page 144 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Catalogue of the University of Michigan written by University of Michigan and published by . This book was released on 1949 with total page 954 pages. Available in PDF, EPUB and Kindle. Book excerpt: Announcements for the following year included in some vols.
Download or read book Distributive Lattices written by Raymond Balbes and published by Ray Balbes. This book was released on 1975 with total page 320 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Ordered Sets and Lattices written by Milan Kolibiar and published by American Mathematical Soc.. This book was released on 1989 with total page 224 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is another publication in the recent surveys of ordered sets and lattices. The papers, which might be characterized as ``reviews of reviews,'' are based on articles reviewed in the Referativnyi Zhurnal: Matematika from 1978 to 1982. For the sake of completeness, the authors also attempted to integrate information from other relevant articles from that period. The bibliography of each paper provides references to the reviews in RZhMat and Mathematical Reviews where one can seek more detailed information. Specifically excluded from consideration in this volume were such topics as algebras of logic, Boolean functions, vector lattices, ordered algebraic systems (including ordered topological spaces), and semigroup properties of semilattices, as well as papers in which such topics as nonlattice-theoretical properties of congruence lattices and subalgebra lattices were considered.
Download or read book Topological Duality for Distributive Lattices written by Mai Gehrke and published by Cambridge University Press. This book was released on 2024-02-29 with total page 369 pages. Available in PDF, EPUB and Kindle. Book excerpt: Introducing Stone-Priestley duality theory and its applications to logic and theoretical computer science, this book equips graduate students and researchers with the theoretical background necessary for reading and understanding current research in the area. After giving a thorough introduction to the algebraic, topological, logical, and categorical aspects of the theory, the book covers two advanced applications in computer science, namely in domain theory and automata theory. These topics are at the forefront of active research seeking to unify semantic methods with more algorithmic topics in finite model theory. Frequent exercises punctuate the text, with hints and references provided.
Download or read book Introduction to Lattices and Order written by B. A. Davey and published by Cambridge University Press. This book was released on 2002-04-18 with total page 316 pages. Available in PDF, EPUB and Kindle. Book excerpt: This new edition of Introduction to Lattices and Order presents a radical reorganization and updating, though its primary aim is unchanged. The explosive development of theoretical computer science in recent years has, in particular, influenced the book's evolution: a fresh treatment of fixpoints testifies to this and Galois connections now feature prominently. An early presentation of concept analysis gives both a concrete foundation for the subsequent theory of complete lattices and a glimpse of a methodology for data analysis that is of commercial value in social science. Classroom experience has led to numerous pedagogical improvements and many new exercises have been added. As before, exposure to elementary abstract algebra and the notation of set theory are the only prerequisites, making the book suitable for advanced undergraduates and beginning graduate students. It will also be a valuable resource for anyone who meets ordered structures.
Download or read book University of Michigan Official Publication written by and published by UM Libraries. This book was released on 1949 with total page 978 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Introduction to the Theory of Partially Ordered Spaces written by Boris Zakharovich Vulikh and published by Routledge. This book was released on 1967 with total page 416 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Probabilistic Lattices With Applications To Psychology written by Louis E Narens and published by World Scientific. This book was released on 2014-12-30 with total page 206 pages. Available in PDF, EPUB and Kindle. Book excerpt: There are many books on lattice theory in the field, but none interfaces with the foundations of probability. This book does. It also develops new probability theories with rigorous foundations for decision theory and applies them to specific well-known problematic examples. There is only one other book that attempts this. It uses quantum probability theory from physics. The new probability theories developed in this book are different; they are not borrowed from physics but are explicitly designed for decision theory.
