A fixed point theorem with applications to truncated lattices

2005

Discrete Mathematics

“…Theorem 4.10 (cf. Rutkowski and Schröder [18]). Let A be a chain-complete ordered set that satisfies the descending chain condition and let P A be as in Definition 4.9.…”

Section: Remark 48mentioning

confidence: 96%

Section: The D + -Relational Fixed Point Propertymentioning

confidence: 99%

Section: Remark 48mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

See 2 more Smart Citations

Isotone relations revisited

Roddy

2005

Discrete Mathematics

“…We have already seen P2 "in action" with the counterexample P2 a . On the positive side, P2 also inspired the paper [42], which discusses the fixed point property for ordered sets that are built like P2.…”

Section: Rutkowski's Small Sets (1989)mentioning

confidence: 99%

The fixed point property for ordered sets

2012

Arab. J. Math.

Self Cite

An ordered set P has the fixed point property iff every order-preserving self-map of P has a fixed point. This paper traces the chronological development of research on this property, including most recent developments and open questions. Mathematics Subject Classification 06A06 IntroductionAn ordered set, or, a partially ordered set consists of a set P and a reflexive, antisymmetric and transitive relation ≤, the order relation. Unless there is the possibility of confusing several order relations, we will refer to the underlying set P as the ordered set. Subsets S ⊆ P inherit the order relation from P by restriction to S. Familiar examples of ordered sets include the number systems N, Z, Q and R, as well as spaces of real valued functions, ordered by the pointwise order. Finite ordered sets are typically represented with a Hasse diagram as in Fig. 2. In a Hasse diagram, elements x and y satisfy the relation x ≤ y iff there is a path from x to y that may go through other elements of the set, but for which all segments are traversed in the upward direction.The homomorphisms between ordered sets P and Q are the order-preserving functions, that is, functions. We will say that an ordered set P has the fixed point property iff every order-preserving self-map f : P → P has a fixed point x = f (x).This survey presents the development of the fixed point property as a research topic in a by-and-large chronological fashion. Topics are presented in the order in which the first results occurred in the literature and they are discussed as fully as possible (regardless of chronology) before proceeding to the next topic. Readers who are interested in presentations from a different point-of-view could consider [44] or [46]. Definitions are presented when they are first needed. Finally, we will focus mainly on finite ordered sets and request the reader's patience when infinite arguments are unavoidable. 123 530 Arab J Math (2012) 1:529-547 2 Knaster's proof (1928)The earliest fixed point result for ordered sets that the author is aware of occurs in Knaster's 1928 proof of the Bernstein-Cantor-Schröder 1 Theorem from set theory in [25]. The Bernstein-Cantor-Schröder Theorem guarantees that when a set A is "smaller than or of equal size as" the set B (meaning there is an injective function from A to B) and "greater than or of equal size as" the set B (meaning there is an injective function from B to A), then A and B have in fact "the same size" (meaning there is a bijective function between them).Theorem 2.

Minimal automorphic superpositions of crowns

2022

Arab. J. Math.