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Some fixed point theorems of the mappings of partially ordered sets

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Some fixed point theorems of the mappings of partially ordered sets RENDICONTI del SEMINARIO MATEMATICO della UNIVERSITÀ DI PADOVA ANTONIO PASINI Some fixed point theorems of themappings of partially ordered sets Rendiconti del Seminario Matematico della Università di Padova, tome 51 (1974), p. 167-177. <http://www.numdam.org/item?id=RSMUP_1974__51__167_0> © Rendiconti del Seminario Matematico della Università di Padova, 1974, tous droits réservés. L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) implique l’ac- cord avec les conditions générales d’utilisation (http://www.numdam.org/legal. php). Toute utilisation commerciale ou impression systématique est consti- tutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Some Fixed Point Theorems of the Mappings of Partially Ordered Sets. ANTONIO PASINI (Firenze) (*) 1. Introduction. In this paper we give new simple proofs of some fixed point theorems, and strengthen others. The methods we shall use base them- selves on two « strong » induction principles, we stated and utilized in [5]. We shall show, moreover, that one of them is equivalent to Axiom of Choice. Let’s now recall some results on the fixed points of a function de- fined from a partially ordered set P; &#x3E; into itself. PROPOSITION A. Let P; nonempty partially ordered set every well ordered subset of which has an upper bound. And let f be a functions P such that x ~ f (x) for every x; then f has a point. The preceding result is proved in [2] by using Axiom of Choice. As a corollary we get: PROPOSITION P; &#x3E; be a nonempty partially ordered set ezery well ordered subsets of which has a least upper bound, and let f be ac just like propositon A’s one; then f has a fixed point.

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