Power Algebras over Semirings

This research monograph investigates sets of the form RA where R is a semiring, and A is a set with a certain structure. Such constructs generalise fuzzy and toll algebraic structures, which have shown themselves of importance in engineering and computer science, as well as in new mathematical disciplines like idempotent analysis. This volume puts much of the ad hoc work of the 1990s on a firm mathematical foundation by creating a unified approach which has important implications in such diverse areas as formal language theory, discrete dynamical systems, models of optimal control, and many others. This book also seeks to address the fundamental question posed by Hohle and Sostak as to what extent basic fields of mathematics like algebra and topology are dependent on the underlying set theory. Since the development of a new mathematical theory is basically inductive, this book begins with a large number of instances which appear in various mathematical contexts, and then develops a general abstract framework, with many examples which will encourage further study in greater depth.