Rational algebraic theories and fixed-point solutions

In a wide variety of situations, computer science has found it convenient to define complex object as (fixed-point) solutions of certain equations. This has been done in both algebraic and order-theoretic settings, and has often been contrasted with other approaches. This paper shows how to formulate such solutions in a setting which encompasses both algebraic and order-theoretic aspects, so that the advantages of both worlds are available. Moreover, we try to show how this is consistent with other approaches to defining complex objects, through a number of applications, including: languages defined by context-free grammars; flow charts and their interpretations; and monadic recursive program schemes. The main mathematical results concern free rational theories and quotients of rational theories. However, the main goal has been to open up what we believe to be a beautiful and powerful new approach to the syntax and semantics of complex recursive specifications.