The contraction principle as a particular case of Kleene's fixed point theorem

Abstract Fixed point theorems are an important tool in the whole of mathematics. For instance, the denotational semantics of programming languages is essentially based on such theorems. Partially ordered sets and Kleene's fixed point theorem are the mathematical support for a great amount of semantic models (de Bakker (1980) and Stay (1977)). In the last years attempts have been made to replace partially ordered sets by metric spaces, and Kleene's fixed point theorem by the Banach contraction principle (de Bakker and Zucker (1983), and Hanes and Arbib (1986)). The aim of the present paper is to prove that the latter theorem may be regarded as a particular case of the former. This is done by embedding a metric space in an ordered set. We note that the problem of the connection between these two fixed point theorems has been solved, in some special cases only, by Baranga and Popescu (1987), and by Livercy (1978) (for compact metric spaces and for compact intervals of the real line, respectively).