Bound Sets in Partial Orders and the Fixed Point Property

1987 ◽  
Vol 30 (4) ◽  
pp. 421-428 ◽  
Author(s):  
Hartmut Höft
Keyword(s):  
Fixed Point ◽  
Partially Ordered Set ◽  
Fixed Point Theorems ◽  
Ordered Set ◽  
Extension Property ◽  
Fixed Point Property ◽  
Point Property ◽  
Maximal Elements ◽  
Partially Ordered ◽  
Bound Sets

AbstractIn this paper we introduce several properties closely related to the fixed point property of a partially ordered set P: the comparability property, the fixed point property for cones, and the fixed point extension property. We apply these properties to the sets of common bounds of the minimal (maximal) elements of the partially ordered set P in order to derive fixed point theorems for P.

scholarly journals Retracts and the Fixed Point Problem for Finite Partially Ordered Sets

1980 ◽  
Vol 23 (2) ◽  
pp. 231-236 ◽  
Author(s):  
Dwight Duffus ◽  
Werner Poguntke ◽  
Ivan Rival
Keyword(s):  
Fixed Point ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered Sets ◽  
Fixed Point Problem ◽  
Fixed Point Property ◽  
Point Problem ◽  
Point Property ◽  
Ordered Sets ◽  
Partially Ordered

A partially ordered set P has the fixed point property if every orderpreserving mapping f of P to P has a fixed point, that is, f(a) = a for some aϵP; call P fixed point free if P does not have the fixed point property.

Some Fixed Point Theorems for Partially Ordered Sets

1976 ◽  
Vol 28 (5) ◽  
pp. 992-997 ◽  
Author(s):  
Hartmut Höft ◽  
Margret Höft
Keyword(s):  
Fixed Point ◽  
Complete Lattice ◽  
Fixed Point Theorems ◽  
Ordered Set ◽  
Fixed Point Property ◽  
List Type ◽  
Point Property ◽  
Ordered Sets ◽  
Partially Ordered ◽  
Order Preserving Map

A partially ordered set P has the fixed point property if every order-preserving map f : P → P has a fixed point, i.e. there exists x ∊ P such that f(x) = x. A. Tarski's classical result (see [4]), that every complete lattice has the fixed point property, is based on the following two properties of a complete lattice P:(A)For every order-preserving map f : P → P there exists x ∊ P such that x ≦ f(x).(B)Suprema of subsets of P exist; in particular, the supremum of the set {x|x ≦ f(x)} ⊂ P exists.

scholarly journals Discrete Optimization for Ordered Weak Proximal Kannan Contractions

2015 ◽  
Vol 0 (0) ◽  
Author(s):  
Moosa Gabeleh
Keyword(s):  
Fixed Point ◽  
Discrete Optimization ◽  
Partially Ordered Set ◽  
Fixed Point Theorems ◽  
Ordered Set ◽  
Best Proximity Point ◽  
Existence Results ◽  
Best Proximity Points ◽  
Partially Ordered

Abstract Let us consider a non-self mapping T : A → B, where A and B are two nonempty subsets of a partially ordered set that is equipped a metric. A best proximity point x⋆ for such a mapping T is a point such that d(x⋆, T x⋆) = dist(A,B). In this work, we provide different existence results of best proximity points and so, we establish some new fixed point theorems in the setting of partially ordered set.

scholarly journals Three counterexamples concerning ω-chain completeness and fixed point properties

1981 ◽  
Vol 24 (2) ◽  
pp. 141-146 ◽  
Author(s):  
J. D. Mashburn
Keyword(s):  
Fixed Point ◽  
Upper Bound ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered ◽  
Fixed Point Properties ◽  
Countable Chain ◽  
Chain Complete

A partially ordered set, is ω-chain complete if, for every countable chain, or ω-chain, in P, the least upper bound of C, denoted by sup C, exists. Notice that C could be empty, so an ω-chain complete partially ordered set has a least element, denoted by 0.

scholarly journals Some Common Fixed Point Theorems in Partially Ordered Sets

2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Khadija Bouzkoura ◽  
Said Benkaddour
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Common Fixed Point ◽  
Point Theorem ◽  
Fixed Point Theorems ◽  
Ordered Set ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Partially Ordered ◽  
Common Fixed Point Theorems

The purpose of this paper is to prove some new fixed point theorem and common fixed point theorems of a commuting family of order-preserving mappings defined on an ordered set, which unify and generalize some relevant fixed point theorems.

scholarly journals The fixed point property and a technique to harness double fixed point combinators

2019 ◽  
Vol 29 (5) ◽  
pp. 831-880
Author(s):  
Giulio Manzonetto ◽  
Andrew Polonsky ◽  
Alexis Saurin ◽  
Jakob Grue Simonsen
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Sufficient Conditions ◽  
Lambda Calculus ◽  
Extension Property ◽  
Fixed Point Property ◽  
Proof Technique ◽  
Point Property ◽  
Negative Solution ◽  
Conservative Extension

Abstract The ${\lambda }$-calculus enjoys the property that each ${\lambda }$-term has at least one fixed point, which is due to the existence of a fixed point combinator. It is unknown whether it enjoys the ‘fixed point property’ stating that each ${\lambda }$-term has either one or infinitely many pairwise distinct fixed points. We show that the fixed point property holds when considering possibly open fixed points. The problem of counting fixed points in the closed setting remains open, but we provide sufficient conditions for a ${\lambda }$-term to have either one or infinitely many fixed points. In the main result of this paper we prove that in every sensible ${\lambda }$-theory there exists a ${\lambda }$-term that violates the fixed point property. We then study the open problem concerning the existence of a double fixed point combinator and propose a proof technique that could lead towards a negative solution. We consider interpretations of the ${\lambda } {\mathtt{Y}}$-calculus into the ${\lambda }$-calculus together with two reduction extension properties, whose validity would entail the non-existence of any double fixed point combinators. We conjecture that both properties hold when typed ${\lambda } {\mathtt{Y}}$-terms are interpreted by arbitrary fixed point combinators. We prove reduction extension property I for a large class of fixed point combinators. Finally, we prove that the ${\lambda }{\mathtt{Y}}$-theory generated by the equation characterizing double fixed point combinators is a conservative extension of the ${\lambda }$-calculus.

On the Number of Maximal Elements in a Partially Ordered Set

1987 ◽  
Vol 30 (3) ◽  
pp. 351-357 ◽  
Author(s):  
John Ginsburg
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Maximal Chain ◽  
Maximal Elements ◽  
Partially Ordered

AbstractLet P be a partially ordered set. For an element x ∊ P, a subset C of P is called a cutset for x in P if every element of C is noncomparable to x and every maximal chain in P meets {x} ∪ C. The following result is established: if every element of P has a cutset having n or fewer elements, then P has at most 2n maximal elements. It follows that, if some element of P covers k elements of P then there is an element x ∊ P such that every cutset for x in P has at least log2k elements.

A Theorem on Partially Ordered Sets, With Applications to Fixed Point Theorems

1961 ◽  
Vol 13 ◽  
pp. 78-82 ◽  
Author(s):  
Smbat Abian ◽  
Arthur B. Brown
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fixed Point Theorems ◽  
Ordered Set ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Partially Ordered ◽  
Prove Theorem ◽  
The Fixed Point Theorem ◽  
The Axiom Of Choice

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.

scholarly journals Another proof of the Browder–Göhde–Kirk theorem via ordering argument

2002 ◽  
Vol 65 (1) ◽  
pp. 105-107 ◽  
Author(s):  
Jacek Jachymski
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Common Fixed Point ◽  
Point Theorem ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Nonexpansive Mappings ◽  
Partially Ordered ◽  
Common Fixed Point Theorem

Using the Zermelo Principle, we establish a common fixed point theorem for two progressive mappings on a partially ordered set. This result yields the Browder–Göhde–Kirk fixed point theorem for nonexpansive mappings.

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