Written both for graduate students and research scientistsin theoretical computer science and mathematics, this bookprovides a detailed investigation of the properties of thefixed point or iteration operation. Iteration plays afundamental role in the theory of computation: for example,in the theory of automata, in formal language theory, in thestudy of formal power series, in the semantics of flowchartalgorithms and programming languages, and in circular datatype definitions.It is shown that in all structures that have beenused assemantic models, the equational properties of the fixedpoint operation are captured by the axioms describingiteration theories. These structures include orderedalgebras, partial functions, relations, finitary andinfinitary regular languages, trees, synchronization trees,2-categories, and others.The book begins with a gentle introduction to the study ofuniversal algebra in the framework of algebraictheories. Aremarkably useful calculus is developed for manipulatingalgebraic theory terms. The reader is then guided through avast terrain of theorems and applications by means ofdetailed proofs,examples, and exercises, with the emphasison equational proofs. The last chapter shows that thefamiliar topic of correctness logic is a special caseof theequational logic of iteration theories. Several significantopen problems are scattered throughout the text.
