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"The text can serve as an introduction to fundamentals in the respective areas from a residuated-maps perspective and with an eye on coordinatization. The historical notes that are interspersed are also worth mentioning....The exposition is thorough and all proofs that the reviewer checked were highly polished....Overall, the book is a well-done introduction from a distinct point of view and with exposure to the author’s research expertise." --MATHEMATICAL REVIEWS
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Lattices and Ordered Algebraic Structures provides a lucid and concise introduction to the basic results concerning the notion of an order. Although as a whole it is mainly intended for beginning postgraduates, the prerequisities are minimal and selected parts can profitably be used to broaden the horizon of the advanced undergraduate.
The treatment is modern, with a slant towards recent developments in the theory of residuated lattices and ordered regular semigroups. Topics covered include:
[bulleted list]
residuated mappings
Galois connections
modular, distributive, and complemented lattices
Boolean algebras
pseudocomplemented lattices
Stone algebras
Heyting algebras
ordered groups
lattice-ordered groups
representable groups
Archimedean ordered structures
ordered semigroups
naturally ordered regular and inverse Dubreil-Jacotin semigroups
[end od bulleted list]
Featuring material that has been hitherto available only in research articles, and an account of the range of applications of the theory, there are also many illustrative examples and numerous exercises throughout, making it ideal for use as a course text, or as a basic introduction to the field for researchers in mathematics, logic and computer science.
T. S. Blyth is Professor Emeritus at St. Andrews University, UK
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Buch. Zustand: Neu. Neuware -The notion of an order plays an important role ¿ not only throughout mat- maticsbutalsoinadjacentdisciplinessuchaslogicandcomputerscience. The purpose of the present text is to provide a basic introduction to the theory of ordered structures. Taken as a whole, the material is mainly designed for a postgraduate course. However, since prerequisites are minimal, selected parts of it may easily be considered suitable to broaden the horizon of the advanced undergraduate. Indeed, this has been the author¿s practice over many years. A basic tool in analysis is the notion of a continuous function, namely a mapping which has the property that the inverse image of an open set is an open set. In the theory of ordered sets there is the corresponding concept of a residuated mapping, this being a mapping which has the property that the inverse image of a principal down-set is a principal down-set. It comes the- fore as no surprise that residuated mappings are important as far as ordered structures are concerned. Indeed, albeit beyond the scope of the present - position, the naturality of residuated mappings can perhaps best be exhibited using categorical concepts. If we regard an ordered set as a small category then an order-preserving mapping f : A B becomes a functor. Then f is + + residuated if and only if there exists a functor f : B A such that (f,f ) is an adjoint pair.Springer Verlag GmbH, Tiergartenstr. 17, 69121 Heidelberg 316 pp. Englisch. Artikel-Nr. 9781852339050
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Lattices and Ordered Algebraic Structures
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Springer London, Springer London, 2005
ISBN 10: 1852339055
ISBN 13: 9781852339050
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Buch. Zustand: Neu. Druck auf Anfrage Neuware - Printed after ordering - The notion of an order plays an important role not only throughout mat- maticsbutalsoinadjacentdisciplinessuchaslogicandcomputerscience. The purpose of the present text is to provide a basic introduction to the theory of ordered structures. Taken as a whole, the material is mainly designed for a postgraduate course. However, since prerequisites are minimal, selected parts of it may easily be considered suitable to broaden the horizon of the advanced undergraduate. Indeed, this has been the author's practice over many years. A basic tool in analysis is the notion of a continuous function, namely a mapping which has the property that the inverse image of an open set is an open set. In the theory of ordered sets there is the corresponding concept of a residuated mapping, this being a mapping which has the property that the inverse image of a principal down-set is a principal down-set. It comes the- fore as no surprise that residuated mappings are important as far as ordered structures are concerned. Indeed, albeit beyond the scope of the present - position, the naturality of residuated mappings can perhaps best be exhibited using categorical concepts. If we regard an ordered set as a small category then an order-preserving mapping f : A B becomes a functor. Then f is + + residuated if and only if there exists a functor f : B A such that (f,f ) is an adjoint pair. Artikel-Nr. 9781852339050
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