The Basic Algebraic Structures in Categories of Derivations

The basic algebraic structures within the categories of derivations determined by rewriting systems are presented. The similarity congruence relation in categories of derivations is given in three versions. The syntax category is formed by taking derivations modulo similarity. This category is a free strict monoidal category, a simple form of a 2-category. The syntax category is central to the study of rewriting systems, morphisms in the category generalizing the notion of "derivation tree," so a detailed development is given. Grifflth's interchange operators on derivations form a 2-category over a category of derivations. Representability of a similarity class is defined and shown to imply the existence of group of operators on the class, induced by interchanges. Uniform representability of rewriting systems is defined and shown to imply that the set of left divisors of each derivation in the syntax category is a distributive lattice.