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Lecture Notes On Algebraic Structure Of Lattice Ordered Rings

Author: Jingjing Ma
Publisher: World Scientific
ISBN: 981457144X
Size: 67.19 MB
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Algebraic Structure of Lattice-Ordered Rings presents an introduction to the theory of lattice-ordered rings and some new developments in this area in the last 10-15 years. It aims to provide the reader with a good foundation in the subject, as well as some new research ideas and topic in the field. This book may be used as a textbook for graduate and advanced undergraduate students who have completed an abstract algebra course including general topics on group, ring, module, and field. It is also suitable for readers with some background in abstract algebra and are interested in lattice-ordered rings to use as a self-study book. The book is largely self-contained, except in a few places, and contains about 200 exercises to assist the reader to better understand the text and practice some ideas. Contents:Introduction to Ordered Algebraic Systems:LatticesLattice-Ordered Groups and Vector LatticesLattice-Ordered Rings and AlgebrasLattice-Ordered Algebras with a d-Basis:Examples and Basic PropertiesStructure TheoremsPositive Derivations on ℓ-Rings:Examples and Basic Propertiesƒ-Ring and Its GeneralizationsMatrix ℓ-RingsKernel of a Positive DerivationSome Topics on Lattice-Ordered Rings:Recognition of Matrix ℓ-Rings with the Entrywise OrderPositive CyclesNonzero ƒ-Eelements in ℓ-RingsQuotient Rings of Lattice-Ordered Ore DomainsMatrix ℓ-Algebras Over Totally Ordered Integral Domainsd-Elements That are Not PositiveLattice-Ordered Triangular Matrix Algebrasℓ-Ideals of ℓ-Unital Lattice-Ordered Rings:Maximal ℓ-Idealsℓ-Ideals in commutative ℓ-Unital ℓ-Rings Readership: Graduate students in algebra and number theory. Key Features:The book includes new material such as positive derivations on lattice-ordered rings, lattice-ordered triangular matrix algebrasMore details are provided in proofs of the results in the book for beginners to understand the textThe book presents new research ideas, methods and topics suitable to advanced undergraduate students and master studentsKeywords:Lattice-ordered Ring;Lattice-ordered Algebra

Ordered Algebraic Structures

Author: Jorge Martínez
Publisher: Springer Science & Business Media
ISBN: 9401117233
Size: 39.26 MB
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This volume contains a selection of papers presented at the 1991 Conrad Conference, held in Gainesville, Florida, USA, in December, 1991. Together, these give an overview of some recent advances in the area of ordered algebraic structures. The first part of the book is devoted to ordered permutation groups and universal, as well as model-theoretic, aspects. The second part deals with material variously connected to general topology and functional analysis. Collectively, the contents of the book demonstrate the wide applicability of order-theoretic methods, and how ordered algebraic structures have connections with many research disciplines. For researchers and graduate students whose work involves ordered algebraic structures.

Ordered Algebraic Structures

Author: W.C. Holland
Publisher: Springer Science & Business Media
ISBN: 9401156409
Size: 35.51 MB
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The conference on Ordered Algebraic Structures held in Curat;ao, from the 26th of June through the 30th of June, 1995, at the Avila Beach Hotel, marked the eighth year of ac tivities by the Caribbean Mathematics Foundation (abbr. CMF), which was the principal sponsor of this conference. CMF was inaugurated in 1988 with a conference on Ordered Algebraic Structures. During the years between these two conferences the field has changed sufficiently, both from my point of view and, I believe, that of my co-organizer, W. Charles Holland, to make one wonder about the label "Ordered Algebraic Structures" itself. We recognized this from the start, and right away this conference carried a subtitle, or, if one prefers, an agenda: we concentrated on the one hand, on traditional themes in the theory of ordered groups, including model-theoretic aspects, and, on the other hand, on matters in which topology (more precisely C(X)-style topology) and category theory would play a prominent role. Plainly, ordered algebra has many faces, and it is becoming increas ingly difficult to organize an intimate conference, such as the ones encouraged in the series sponsored by CMF, in this area on a broad set of themes. These proceedings reflect, accurately we think, the spirit of the conferees, but it is not a faithful record of the papers presented at the conference.

A Guide To The Literature On Semirings And Their Applications In Mathematics And Information Sciences

Author: K. Glazek
Publisher: Springer Science & Business Media
ISBN: 9401599645
Size: 65.31 MB
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This volume presents a short guide to the extensive literature concerning semir ings along with a complete bibliography. The literature has been created over many years, in variety of languages, by authors representing different schools of mathematics and working in various related fields. In many instances the terminology used is not universal, which further compounds the difficulty of locating pertinent sources even in this age of the Internet and electronic dis semination of research results. So far there has been no single reference that could guide the interested scholar or student to the relevant publications. This book is an attempt to fill this gap. My interest in the theory of semirings began in the early sixties, when to gether with Bogdan W ~glorz I tried to investigate some algebraic aspects of compactifications of topological spaces, semirings of semicontinuous functions, and the general ideal theory for special semirings. (Unfortunately, local alge braists in Poland told me at that time that there was nothing interesting in investigating semiring theory because ring theory was still being developed). However, some time later we became aware of some similar investigations hav ing already been done. The theory of semirings has remained "my first love" ever since, and I have been interested in the results in this field that have been appearing in literature (even though I have not been active in this area myself).

A Course In Constructive Algebra

Author: Ray Mines
Publisher: Springer Science & Business Media
ISBN: 9780387966403
Size: 75.38 MB
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The constructive approach to mathematics has enjoyed a renaissance, caused in large part by the appearance of Errett Bishop's book Foundations of constr"uctiue analysis in 1967, and by the subtle influences of the proliferation of powerful computers. Bishop demonstrated that pure mathematics can be developed from a constructive point of view while maintaining a continuity with classical terminology and spirit; much more of classical mathematics was preserved than had been thought possible, and no classically false theorems resulted, as had been the case in other constructive schools such as intuitionism and Russian constructivism. The computers created a widespread awareness of the intuitive notion of an effecti ve procedure, and of computation in principle, in addi tion to stimulating the study of constructive algebra for actual implementation, and from the point of view of recursive function theory. In analysis, constructive problems arise instantly because we must start with the real numbers, and there is no finite procedure for deciding whether two given real numbers are equal or not (the real numbers are not discrete) . The main thrust of constructive mathematics was in the direction of analysis, although several mathematicians, including Kronecker and van der waerden, made important contributions to construc tive algebra. Heyting, working in intuitionistic algebra, concentrated on issues raised by considering algebraic structures over the real numbers, and so developed a handmaiden'of analysis rather than a theory of discrete algebraic structures.