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.

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.

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 Retracts of partially ordered sets

1979 ◽  
Vol 27 (4) ◽  
pp. 495-506 ◽  
Author(s):  
Dwight Duffus ◽  
Ivan Rival
Keyword(s):  
Fixed Points ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Partially Ordered

AbstractLet P be a finite, connected partially ordered set containing no crowns and let Q be a subset of P. Then the following conditions are equivalent: (1) Q is a retract of P; (2) Q is the set of fixed points of an order-preserving mapping of P to P; (3) Q is obtained from P by dismantling by irreducibles.

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.

CAYLEY GRAPHS OF PARTIALLY ORDERED SETS

2013 ◽  
Vol 12 (04) ◽  
pp. 1250184 ◽  
Author(s):  
MOJGAN AFKHAMI ◽  
ZAHRA BARATI ◽  
KAZEM KHASHYARMANESH
Keyword(s):  
Cayley Graph ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Cayley Graphs ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Partially Ordered ◽  
Basic Properties ◽  
Vertex Set

In this paper, we introduce the Cayley graph of a partially ordered set (poset). Let (P, ≤) be a poset, and let S be a subset of P. We define the undirected Cayley graph of P, denoted by Cay (P, S), as a graph with vertex-set P and edge-set E consisting of those sets {x, y} such that y ∈ {x, s}ℓ or x ∈ {y, s}ℓ for some s ∈ S, where for a subset T of P, Tℓ is the set of all x ∈ P such that x ≤ t, for all t ∈ T. We study some basic properties of Cay (P, S) such as connectivity, diameter and girth.

The Ordering of Spec R

1976 ◽  
Vol 28 (4) ◽  
pp. 820-835 ◽  
Author(s):  
William J. Lewis ◽  
Jack Ohm
Keyword(s):  
Commutative Ring ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered Sets ◽  
Prime Ideals ◽  
Ordered Sets ◽  
Partially Ordered

Let Specie denote the set of prime ideals of a commutative ring with identity R, ordered by inclusion; and call a partially ordered set spectral if it is order isomorphic to Spec R for some R. What are some conditions, necessary or sufficient, for a partially ordered set X to be spectral? The most desirable answer would be the type of result that would allow one to stare at the diagram of a given X and then be able to say whether or not X is spectral. For example, it is known that finite partially ordered sets are spectral (see [2] or [5]).

The superintuitionistic predicate logic of finite Kripke frames is not recursively axiomatizable

2005 ◽  
Vol 70 (2) ◽  
pp. 451-459 ◽  
Author(s):  
Dmitrij Skvortsov
Keyword(s):  
Partially Ordered Set ◽  
Possible Worlds ◽  
Ordered Set ◽  
Predicate Logic ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Partially Ordered ◽  
Kripke Frames ◽  
Finite Partially Ordered Set

AbstractWe prove that an intermediate predicate logic characterized by a class of finite partially ordered sets is recursively axiomatizable iff it is “finite”, i.e., iff it is characterized by a single finite partially ordered set. Therefore, the predicate logic LFin of the class of all predicate Kripke frames with finitely many possible worlds is not recursively axiomatizable.

Endomorphisms of Partially Ordered Sets

1998 ◽  
Vol 7 (1) ◽  
pp. 33-46
Author(s):  
DWIGHT DUFFUS ◽  
TOMASZ ŁUCZAK ◽  
VOJTĚCH RÖDL ◽  
ANDRZEJ RUCIŃSKI
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Partially Ordered

It is shown that every partially ordered set with n elements admits an endomorphism with an image of a size at least n1/7 but smaller than n. We also prove that there exists a partially ordered set with n elements such that each of its non-trivial endomorphisms has an image of size O((n log n)1/3).

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.

Decidability and finite axiomatizability of theories of ℵ0-categorical partially ordered sets

1981 ◽  
Vol 46 (1) ◽  
pp. 101-120 ◽  
Author(s):  
James H. Schmerl
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered Sets ◽  
Finite Width ◽  
Ordered Sets ◽  
Partially Ordered ◽  
Decidable Theory ◽  
Axiomatizable Theory ◽  
Finitely Axiomatizable Theory

AbstractEvery ℵ0-categorical partially ordered set of finite width has a finitely axiomatizable theory. Every ℵ0-categorical partially ordered set of finite weak width has a decidable theory. This last statement constitutes a major portion of the complete (with three exceptions) characterization of those finite partially ordered sets for which any ℵ0-categorical partially ordered set not embedding one of them has a decidable theory.

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