Fixed-points and algebras with infinitely long expressions: Part I. Regular algebras

Abstract

This paper introduces a class of algebras (called the class of regular algebras), in which the algebra of regular trees (unfoldments of monadic program schemes) is an initial algebra. This means that we have for the above-mentioned class “semantics” of monadic program schemes. We show how to treat, in a unified way, such concepts as: monadic and recursive monadic program schemes, regular and context-free languages.

On the other hand, the investigation of the properties of regular algebras may be of intrinsic interest, in particular this leads to a very nice generalization of the notion of a polynomial in an algebra. These “new” polynomials, in general, are determined by infinitely long expressions, and existence of such polynomials in the class of regular algebras is closely connected with the property that every finite tuple of algebraic mappings has a least fixed-point which is obtainable as a least upper bound of a denumerable chain of “approximations”.