Varieties Of Lattices
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Varieties Of Lattices
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Author : Peter Jipsen
language : en
Publisher: Springer
Release Date : 2006-11-15
Varieties Of Lattices written by Peter Jipsen and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-11-15 with Mathematics categories.
The study of lattice varieties is a field that has experienced rapid growth in the last 30 years, but many of the interesting and deep results discovered in that period have so far only appeared in research papers. The aim of this monograph is to present the main results about modular and nonmodular varieties, equational bases and the amalgamation property in a uniform way. The first chapter covers preliminaries that make the material accessible to anyone who has had an introductory course in universal algebra. Each subsequent chapter begins with a short historical introduction which sites the original references and then presents the results with complete proofs (in nearly all cases). Numerous diagrams illustrate the beauty of lattice theory and aid in the visualization of many proofs. An extensive index and bibliography also make the monograph a useful reference work.
Algebras Lattices Varieties
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Author : Ralph S. Freese
language : en
Publisher: American Mathematical Society
Release Date : 2022-10-28
Algebras Lattices Varieties written by Ralph S. Freese and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-10-28 with Mathematics categories.
This book is the second of a three-volume set of books on the theory of algebras, a study that provides a consistent framework for understanding algebraic systems, including groups, rings, modules, semigroups and lattices. Volume I, first published in the 1980s, built the foundations of the theory and is considered to be a classic in this field. The long-awaited volumes II and III are now available. Taken together, the three volumes provide a comprehensive picture of the state of art in general algebra today, and serve as a valuable resource for anyone working in the general theory of algebraic systems or in related fields. The two new volumes are arranged around six themes first introduced in Volume I. Volume II covers the Classification of Varieties, Equational Logic, and Rudiments of Model Theory, and Volume III covers Finite Algebras and their Clones, Abstract Clone Theory, and the Commutator. These topics are presented in six chapters with independent expositions, but are linked by themes and motifs that run through all three volumes.
General Lattice Theory
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Author : George Grätzer
language : en
Publisher: Springer Science & Business Media
Release Date : 2002-11-21
General Lattice Theory written by George Grätzer 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 2002-11-21 with Mathematics categories.
"Grätzer’s 'General Lattice Theory' has become the lattice theorist’s bible. Now we have the second edition, in which the old testament is augmented by a new testament. The new testament gospel is provided by leading and acknowledged experts in their fields. This is an excellent and engaging second edition that will long remain a standard reference." --MATHEMATICAL REVIEWS
Nonmodular Lattice Varieties
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Author : Henry Rose
language : en
Publisher: American Mathematical Soc.
Release Date : 1984
Nonmodular Lattice Varieties written by Henry Rose 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 1984 with Mathematics categories.
It is shown that there are eight infinite sequences of join irreducible lattice varieties with the following properties: each term of every sequence has the next term as its unique join irreducible cover. The description of these sequences is based on a detailed study of subdirectly irreducible lattices with the unique critical quotients.
The Lattice Of Interpretability Types Of Varieties
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Author : Octavio Carlos García
language : en
Publisher: American Mathematical Soc.
Release Date : 1984
The Lattice Of Interpretability Types Of Varieties written by Octavio Carlos García 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 1984 with Mathematics categories.
We investigate the lattice, invented by W. D. Neumann in 1974, formed by the class of all varieties under the quasi-ordering "[script]V is interpretable in [script]W." The lattice is found to be non-modular and a proper class. Various familiar varieties are found to be [logical conjunction symbol {up arrow}]-irreducible (or prime) and various filters (especially Mal'tsev classes) are found to be indecomposable (or prime). Many familiar varieties are found to be inequivalent in the lattice, using a new technique of SIN algebras. Seven figures are included which document the known relationships between some sixty known or easily describable varieties and varietal families.
The Structure Of Modular Lattices Of Width Four With Applications To Varieties Of Lattices
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Author : Ralph S. Freese
language : en
Publisher: American Mathematical Soc.
Release Date : 1977
The Structure Of Modular Lattices Of Width Four With Applications To Varieties Of Lattices written by Ralph S. Freese 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 1977 with Mathematics categories.
A variety (equational class) of lattices is said to be finitely based if there exists a finite set of identities defining the variety. Let [capital script]M [infinity symbol] [over][subscript italic]n denote the lattice variety generated by all modular lattices of width not exceeding [subscript italic]n. [capital script]M [infinity symbol] [over]1 and [capital script]M [infinity symbol] [over]2 are both the class of all distributive lattices and consequently finitely based. B. Jónsson has shown that [capital script]M [infinity symbol] [over]3 is also finitely based. On the other hand, K. Baker has shown that [capital script]M [infinity symbol] [over][subscript italic]n is not finitely based for 5 [less than or equal to symbol] [italic]n [less than] [lowercase Greek]Omega. This paper settles the finite bases problem for [capital script]M [infinity symbol] [over]4.
Epimorphisms And Dominions In Varieties Of Lattices
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Author : David Robert Wasserman
language : en
Publisher:
Release Date : 2001
Epimorphisms And Dominions In Varieties Of Lattices written by David Robert Wasserman and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001 with categories.
Axioms For Lattices And Boolean Algebras
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Author : Ranganathan Padmanabhan
language : en
Publisher: World Scientific
Release Date : 2008
Axioms For Lattices And Boolean Algebras written by Ranganathan Padmanabhan and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with Mathematics categories.
The importance of equational axioms emerged initially with the axiomatic approach to Boolean algebras, groups, and rings, and later in lattices. This unique research monograph systematically presents minimal equational axiom-systems for various lattice-related algebras, regardless of whether they are given in terms of ?join and meet? or other types of operations such as ternary operations. Each of the axiom-systems is coded in a handy way so that it is easy to follow the natural connection among the various axioms and to understand how to combine them to form new axiom systems. A new topic in this book is the characterization of Boolean algebras within the class of all uniquely complemented lattices. Here, the celebrated problem of E V Huntington is addressed, which ? according to G Gratzer, a leading expert in modern lattice theory ? is one of the two problems that shaped a century of research in lattice theory. Among other things, it is shown that there are infinitely many non-modular lattice identities that force a uniquely complemented lattice to be Boolean, thus providing several new axiom systems for Boolean algebras within the class of all uniquely complemented lattices. Finally, a few related lines of research are sketched, in the form of appendices, including one by Dr Willian McCune of the University of New Mexico, on applications of modern theorem-proving to the equational theory of lattices.
Semimodular Lattices
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Author : Manfred Stern
language : en
Publisher: Cambridge University Press
Release Date : 1999-05-13
Semimodular Lattices written by Manfred Stern 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 1999-05-13 with Mathematics categories.
A survey of semimodularity that presents theory and applications in discrete mathematics, group theory and universal algebra.
Semigroups And Their Subsemigroup Lattices
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Author : L.N. Shevrin
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-09
Semigroups And Their Subsemigroup Lattices written by L.N. Shevrin 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 2013-03-09 with Mathematics categories.
0.1. General remarks. For any algebraic system A, the set SubA of all subsystems of A partially ordered by inclusion forms a lattice. This is the subsystem lattice of A. (In certain cases, such as that of semigroups, in order to have the right always to say that SubA is a lattice, we have to treat the empty set as a subsystem.) The study of various inter-relationships between systems and their subsystem lattices is a rather large field of investigation developed over many years. This trend was formed first in group theory; basic relevant information up to the early seventies is contained in the book [Suz] and the surveys [K Pek St], [Sad 2], [Ar Sad], there is also a quite recent book [Schm 2]. As another inspiring source, one should point out a branch of mathematics to which the book [Baer] was devoted. One of the key objects of examination in this branch is the subspace lattice of a vector space over a skew field. A more general approach deals with modules and their submodule lattices. Examining subsystem lattices for the case of modules as well as for rings and algebras (both associative and non-associative, in particular, Lie algebras) began more than thirty years ago; there are results on this subject also for lattices, Boolean algebras and some other types of algebraic systems, both concrete and general. A lot of works including several surveys have been published here.