A Theorem on Partially Ordered Sets, With Applications to Fixed Point Theorems
| Autor: | Smbat Abian, Arthur B. Brown |
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| Rok vydání: | 1961 |
| Předmět: | |
| Zdroj: | Canadian Journal of Mathematics. 13:78-82 |
| ISSN: | 1496-4279 0008-414X |
| DOI: | 10.4153/cjm-1961-007-5 |
| Popis: | In this paper the authors prove Theorem 1 on maps of partially ordered sets into themselves, and derive some fixed point theorems as corollaries.Here, for any partially ordered set P, and any mapping f : P → P and any point a ∈ P, a well ordered subset W(a) ⊂ P is constructed. Except when W(a) has a last element ε greater than or not comparable to f(ε), W(a), although constructed differently, is identical with the set A of Bourbaki (3) determined by a, f , and P1: {x|x ∈ P, x ≤ f(x)}.Theorem 1 and the fixed point Theorems 2 and 4, as well as Corollaries 2 and 4, are believed to be new.Corollaries 1 and 3 are respectively the well-known theorems given in (1, p. 54, Theorem 8, and Example 4).The fixed point Theorem 3 is that of (1, p. 44, Example 4); and has as a corollary the theorem given in (2) and (3).The proofs are based entirely on the definitions of partially and well ordered sets and, except in the cases of Theorem 4 and Corollary 4, make no use of any form of the axiom of choice. |
| Databáze: | OpenAIRE |
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