Modes

An introduction to the theory and application of modes -structures that capture the common underlying algebra of convex sets, affine spaces and certain ordered sets. Modes appear in many branches of mathematics, particularly geometry and combinatorics, and have been used in computer science, economics, physics and biology. The initial stage of the theory was set out in the authors' research monograph "Modal Theory" (published in 1985). The present text seeks to provide a more complete theory, the result of research conducted during the subsequent 15 years. It contains a clear introduction to selected topics regarding universal algebra, category theory and model theory, and the foundations of the theory of modes, as well as more advanced topics leading to the forefront of research in the field. The authors have included a wide range of exercises, usually placed at the end of the section, and indexed alphabetically. Some exercises are simply designed to familiarise readers with the notation and concepts. Others are more difficult, extending the content of the sections in which they appear, and providing a foretaste of further research.