Lattices and ordered algebraic structures.

*(English)*Zbl 1073.06001
Universitext. London: Springer (ISBN 1-85233-905-5/hbk). ix, 303 p. (2005).

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 horizont of the advanced ungraduate, hence it can be easily read by students.

The treatment of notions is modern, with a slant towards recent developments in the theory of residuated lattices and ordered regular semigroups. Topics covered include: 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; natural ordered regular and inverse Dubreil-Jacotin semigroups.

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 the book, 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. Since the notion of order plays an important role not only throughout mathematics, but also in adjacent disciplines, such as logic and computer science, it will be of interest to a broad category of specialists in the respective fields.

The treatment of notions is modern, with a slant towards recent developments in the theory of residuated lattices and ordered regular semigroups. Topics covered include: 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; natural ordered regular and inverse Dubreil-Jacotin semigroups.

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 the book, 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. Since the notion of order plays an important role not only throughout mathematics, but also in adjacent disciplines, such as logic and computer science, it will be of interest to a broad category of specialists in the respective fields.

Reviewer: Dimitru Busneag (Craiova)

##### MSC:

06-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordered structures |

06-02 | Research exposition (monographs, survey articles) pertaining to ordered structures |

06-XX | Order, lattices, ordered algebraic structures |