### Unique fixed points vs. least fixed points

Jerzy Tiuryn
1980 Theoretical Computer Science
The aim of this paper is to compare two approaches to the semantics of programming languages: the least fixed point approach, and the unique fixed point aypproach. Briefly speaking, we investigate here the problem of existence of extensions of algebras with the unique fixed point property to ordered algebras with the least fixed point property, that preserve the fixed point solutions. We prove that such extensions always exist, the construction of a free extension is given. It is also shown
more » ... in some cases there is no 'faithful' extension, i.e. some elements of a carrier are always collapsed. 230 J. Tiuryn algebraic system of fixed point equations has a unique solution. The notion of an iterative algebra is introduced in this paper, and it is shown here that it corresponds to the notion of iterative algebraic theory. Here we would like to justify our choice of algebras rather than algebraic theories. Beside of author's preference of algebras there are some sound reasons for our choice. It seems that despite of a strong tendency to work within a language of category theory there is still a remarkable number of mathematicians better understanding (or prefering) results formulated in algebraic terms rather than in categorical ones. Secondly, and perhaps in connection with the previous argument, it seems that the methods of proof applicable to a single structure, as this is the case in this paper, are more transparent when using a traditional language of algebra rather than that of category theory. (We are not discussing here a useful role of category theory in the process of generalizing and/or comparing results.) On the other hand, having clear relationships between the algebraic structures we are dealing in this paper with and their categorical counterparts, it is a routine matter to reformulate our results in terms of categorical notions. Let k, n E W, and let p = (pO, . . . , pn -1) be a vector of n + kary polynomial symbols (i.e. finite trees) over the signature C. Assume moreover that none of pi'