This book provides a lucid and concise introduction to the basic results concerning the notion of an order. Although 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:- 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 semigroupsFeaturing 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.