A Normal Form for Algebraic Constructions

By a " construction " we mean here a function which takes each structure of one type to some structure of another type; " algebraic " means " such as occurs in algebra ". One example of an algebraic construction is the function F which takes each integral domain R to its field of fractions F(R). Word-constructions, defined below, yield a normal form for algebraic constructions; they also constitute a new method of model construction, generalising the notion of construction by generators and relations (cf. Q-word algebras, Cohn [1]). At the end we list a few applications, which range from technical (preservation theorems for infinitary languages) to foundational (degrees of constructivity). By a similarity type we shall mean a set of function symbols, each with an assigned arity ^ 0. It is clear what is meant by an Cl-structure, i.e. a universal algebra of similarity type Q. Let £, Q be fixed similarity types, and let S be a set of symbols not occurring in Q. By a. frame we shall mean a finite sequence, written £(vlt ...,vk), each of whose terms is either a function symbol from Q, a symbol from S, or a variable from among vu ..., vk; we require also that each variable vu...tvk occurs just once in £(vlt ...,vk). If 51 is a E-structure and au...,ak are elements of the domain of % then by £[alt .-.,ak] we mean the sequence which results from £(vit ...,vk) after replacing the occurrence of vt by an occurrence of a,, for each i. By a word-construction from E to Q, we mean a sequence F = , where