This monograph contains the results of our joint research over the last ten years on the logic of the fixed point operation. The intended au­ dience consists of graduate students and research scientists interested in mathematical treatments of semantics. We assume the reader has a good mathematical background, although we provide some prelimi­ nary facts in Chapter 1. Written both for graduate students and research scientists in theoret­ ical computer science and mathematics, the book provides a detailed investigation of the properties of the fixed point or iteration operation. Iteration plays a fundamental role in the theory of computation: for example, in the theory of automata, in formal language theory, in the study of formal power series, in the semantics of flowchart algorithms and programming languages, and in circular data type definitions. It is shown that in all structures that have been used as semantical models, the equational properties of the fixed point operation are cap­ tured by the axioms describing iteration theories. These structures include ordered algebras, partial functions, relations, finitary and in­ finitary regular languages, trees, synchronization trees, 2-categories, and others.

1 Mathematical Motivation.- 2 Why Iteration Theories?.- 3 Suggestions for the Impatient Reader.- 4 A Disclaimer.- 5 Numbering.- 1 Preliminary Facts.- 1 Sets and Functions.- 2 Posets.- 3 Categories.- 4 2-Categories.- 4.1 Cellc is a 2-Category, Too.- 5 ?-Trees.- 2 Varieties and Theories.- 1 S-Algebras.- 2 Terms and Equations.- 3 Theories.- 4 The Theory of a Variety..- 3 Theory Facts.- 1 Pairing and Separated Sum.- 2 Elementary Properties of TH.- 3 Theories as N x N-Sorted Algebras.- 4 Special Coproducts.- 5 Matrix and Matricial Theories.- 6 Pullbacks and Pushouts of Base Morphisms.- 7 2-Theories.- 4 Algebras.- 1 T-algebras.- 2 Free Algebras in Tb.- 3 Subvarieties of Tb.- 4 The Categories TH and var.- 5 Notes.- 5 Iterative Theories.- 1 Ideal Theories.- 2 Iterative Theories Defined.- 3 Properties of Iteration in Iterative Theories.- 4 Free Iterative Theories.- 5 Notes.- 6 Iteration Theories.- 1 Iteration Theories Defined.- 2 Other Axiomatizations of Iteration Theories.- 3 Theories with a Functorial Dagger.- 4 Pointed Iterative Theories.- 5 Free Iteration Theories.- 6 Constructions on Iteration Theories.- 7 Feedback Theories.- 8 Summary of the Axioms.- 9 Notes.- 7 Iteration Algebras.- 1 Definitions.- 2 Free Algebras in T¿.- 3 The Retraction Lemma.- 4 Some Categorical Facts.- 5 Properties of T¿.- 6 A Characterization Theorem.- 7 Strong Iteration Algebras.- 8 Notes.- 8 Continuous Theories.- 1 Ordered Algebraic Theories.- 2 ?-Continuous Theoriesx.- 3 Rational Theories.- 4 Initiality and Iteration in 2-Theories.- 5 ?-Continuous 2-Theories.- 6 Notes.- 9 Matrix Iteration Theories.- 1 Notation.- 2 Properties of the Star Operation.- 3 Matrix Iteration Theories Defined.- 4 Presentations in Matrix Iteration Theories.- 5 The Initial Matrix Iteration Theory.- 6 An ExtensionTheorem.- 7 Matrix Iteration Theories of Regular Sets.- 8 Notes.- 10 Matricial Iteration Theories.- 1 From Dagger to Star and Omega, and Back.- 2 Matricial Iteration Theories Defined.- 3 Examples.- 4 Additively Closed Subiteration Theories.- 5 Presentations in Matricial Iteration Theories.- 6 The Initial Matricial Iteration Theory.- 7 The Extension Theorem.- 8 Additively Closed Theories of Regular Languages.- 9 Closed Regular (?-Languages.- 10 Notes.- 11 Presentations.- 1 Presentations in Iteration Theories.- 2 Simulations of Presentations.- 3 Coproducts Revisited.- 4 Notes.- 12 Flowchart Behaviors.- 1 Axiomatizing Sequacious Functions.- 2 Axiomatizing Partial Functions.- 3 Diagonal Theories.- 4 Sequacious Functions with Predicates.- 5 Partial Functions with Predicates.- 6 Notes.- 13 Synchronization Trees.- 1 Theories of Synchronization Trees.- 2 Grove Iteration Theories.- 3 Axiomatizing Synchronization Trees.- 4 Bisimilarity.- 5 Notes.- 14 Floyd-Hoare Logic.- 1 Guards.- 2 Partial Correctness Assertions.- 3 The Standard Example.- 4 Rules for Partial Correctness.- 5 Soundness.- 6 The Standard Example, Continued.- 7 A Floyd-Hoare Calculus for Iteration Theories.- 8 The Standard Example, Again.- 9 Completeness.- 10 Examples.- 11 Notes.- List of Symbols.