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The contraction principle as a particular case of Kleene’s fixed point theorem. (English) Zbl 0755.54019
Summary: 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 ([{\it J. de Bakker}, Mathematical theory of program correctness. Englewood Cliffs, New Jersey, etc.: Prentice-Hall (1980; Zbl 0452.68011)] and [{\it J. E. Stoy}, Denotational semantics: the Scott-Strachey approach to programming language theory. Cambridge, Massachusetts; London: The MIT Press (1981; Zbl 0503.68059)]). 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 ([{\it J. W. de Bakker} and {\it J. I. Zucker}, Processes of the denotational semantics of concurrency. Foundations of computer science IV. Distributed systems. Part 2: Semantics and logic, Math. Cent. Tracts 159, 45--100 (1983; Zbl 0508.68010)] and [{\it E. G. Manes} and {\it M. A. Arbib}, Algebraic approaches to program semantics. New York etc.: Springer (1986; Zbl 0599.68008)]). 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 the author and {\it C. Popescu} [Bull. Math. Soc. Sci. Math. Répub. Soc. Roum., Nouv. Sér. 31(79), 99-106 (1987; Zbl 0639.54015)] and by {\it C. Livercy} [Théorie des programmes. Schémes, preuves, sémantique. Paris: Bordas (1978; Zbl 0416.68007)] (for compact metric spaces and for compact intervals of the real line, respectively).
##### MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces 68Q55 Semantics 68N01 General theory of software
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##### References:
 [1] Baranga, A.; Popescu, C.: Ordering, compactness, continuity. Bull. math. Soc. sci. Math. R. S. roumanie 31, No. 2, 99-106 (1987) · Zbl 0639.54015 [2] de Bakker, J.W.: Mathematical theory of program correctness. (1980) · Zbl 0452.68011 [3] de Bakker, J.W.; Zucker, J.I.: Semantics and logic. Math. centre tracts 159, 45-100 (1983) · Zbl 0508.68010 [4] Livercy, C.: Theorie des programmes. Schémes, preuves, sémantique. (1978) · Zbl 0416.68007 [5] Hanes, E.G.; Arbib, M.A.: Algebraic approaches to program semantics. (1986) · Zbl 0599.68008 [6] Rudin, W.: Principles of mathematical analysis. (1953) · Zbl 0052.05301 [7] Stoy, J.E.: Denotational semantics: the Scott-strachey approach to programming language theory. (1977) · Zbl 0503.68059