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A retract characterization of posets with the fixed-point property

Richard A Brualdi, J.A Dias da Silva
1997 Discrete Mathematics  
It has been an open problem to characterize posets P with the property that every order-preserving map on P has a fixed point. We give a characterization of such posets in terms of their retracts.  ...  It has been an open problem for a number of years to characterize posets with the fixed-point property (see the survey I-5] and [2]).  ...  Our characterization of general finite posets with the fixed-point property is of a similar spirit. To explain it we need to introduce a generalization of the notion of a crown.  ... 
doi:10.1016/s0012-365x(97)87040-9 fatcat:yetiwokuhvawzpf6ohd2nifxza

The Fixed Point Property for Posets of Small Width [article]

Jonathan David Farley
1998 arXiv   pre-print
The fixed point property for finite posets of width 3 and 4 is studied in terms of forbidden retracts. The ranked forbidden retracts for width 3 and 4 are determined explicitly.  ...  The ranked forbidden retracts for the width 3 case that are linearly indecomposable are examined to see which are minimal automorphic. Part of a problem of Niederle from 1989 is thus solved.  ...  Nonetheless, for certain classes of ordered sets, the posets with the fixed point property have been characterized, generally in terms of forbidden retracts: A finite, connected rank 1 poset has the fixed  ... 
arXiv:math/9804146v1 fatcat:v5ui5yupkrbdrhfzvfdg45fmse

Page 6180 of Mathematical Reviews Vol. , Issue 90K [page]

1990 Mathematical Reviews  
three with the fixed point property.  ...  It generalizes several important notions connected with posets and their monotone self-mappings such as: fixed point, fixed edge, retract (usually defined by isotone mappings of posets), nontransposing  ... 

About posets for which no lower cover or no upper cover has the fixed point property [article]

Frank a Campo
2022 arXiv   pre-print
For a finite non-empty set X, let 𝔓(X) denote the set of all posets with carrier X, ordered by inclusion of their partial order relations.  ...  We investigate properties of posets P ∈𝔓(X) for which no lower cover or no upper cover in 𝔓(X) has the fixed point property.  ...  In particular, he provides examples for posets having the fixed point property for which none of the upper covers and none of the lower covers has the fixed point property.  ... 
arXiv:2109.09431v2 fatcat:ejfvrloy3zeqpboff4e4bfmovq

Page 7714 of Mathematical Reviews Vol. , Issue 2001K [page]

2001 Mathematical Reviews  
An ordered set P has the fixed point property if every order pre- serving function f: P — P has a fixed point, i.e., a point x € P such that x = f(x).  ...  As observed by the author, it can easily be shown that if P is chain-complete and r: P — P is a comparative retraction, then P has the fixed point property if and only if the image r(P) has the fixed point  ... 

Page 3460 of Mathematical Reviews Vol. , Issue 98F [page]

1998 Mathematical Reviews  
(P-LISBS; Lisbon) A retract characterization of posets with the fixed-point property. (English summary) Discrete Math. 169 (1997), no. 1-3, 199-203.  ...  Theorem: Let P be a finite poset; then P has the fixed point property if and only if no generalized crown is a retract of P.  ... 

A characterization of finite posets of the width at most three with the fixed point property

Josef Niederle
1989 Czechoslovak Mathematical Journal  
remaining two elements of X s ) would be a retraction of R to a proper retract not having the fixed point property.  ...  The set of all retracts in X that fail to have the fixed point property contains at least one minimal element with respect to set inclusion as it is finite.  ...  It is true that any very nice section is nice as it has a retract isomorphic to a tower of nice sections. The converse remains open. Problem. Are there nice sections that are not very nice?  ... 
doi:10.21136/cmj.1989.102283 fatcat:cmtk4p6mpfhc5d6eeoegnfblwq

Page 730 of Mathematical Reviews Vol. , Issue 98B [page]

1998 Mathematical Reviews  
That all retracts of an ordered set P with the fixed point property (FPP) must also have the FPP is a well-known result (Rival, 1976).  ...  The cacce property is preserved by retracts, and the intersection of a decreasing chain of caccc subposets also has this property.  ... 

Page 659 of Mathematical Reviews Vol. , Issue 96b [page]

1996 Mathematical Reviews  
Then for each chain-complete p.o. set @, P x Q has the fixed point property if and only if (p \ A) x Q has the fixed point property and each element of A is contracting.  ...  Various fixed point theorems for ordered sets which contain a retractable set are proved as generalizations of earlier known results for sets with retractable points.  ... 

Isotone relations and the fixed point property for posets

James W Walker
1984 Discrete Mathematics  
A strengthened form of the fixed point property for posets is presented, in which isotone functions are replaced by more general isotone relations.  ...  But an example shows that not every infinite poset with the relational fixed point property is dismantlable. Applications to quotients and direct products are given.  ...  We begin, in Section 2, with some notions, such as order homotopy, retracts, and irreducibles, which are useful in the study of the fixed point property.  ... 
doi:10.1016/0012-365x(84)90188-2 fatcat:z5vlajrvifcv5d55ltdbbodlay

Convex Sublattices of a Lattice and a Fixed Point Property [article]

Dwight Duffus, Claude Laflamme, Maurice Pouzet, Robert Woodrow
2011 arXiv   pre-print
Key words and phrases. posets, lattices, convex sublattice, retracts, fixed point property  ...  We say that T has the fixed point property for convex sublattices (CLFPP for short) if every order preserving map f from T to CL(T) has a fixed point, that is x > f(x) for some x > T.  ...  ), in connection with the fixed point property for the product of posets.  ... 
arXiv:1108.5210v1 fatcat:jpsrod7h4vhmzm2bysptdugeaq

Page 1169 of Mathematical Reviews Vol. , Issue 87c [page]

1987 Mathematical Reviews  
A 21 (1976), no. 3, 309-318; MR 54 #7330] and obtain new fixed point theorems and new proofs of known ones. “Let P be a finite poset. P!?!  ...  Bjérner for fixed point problems in posets [Adv. in Math. 31 (1979), no. 3, 263-287; MR 81c:06001], we generalize the notion of a dismantlable poset [see I. Rival, J. Combin. Theory Ser.  ... 

A Generalization of a Theorem of Erné about the Number of Posets with a Fixed Antichain

Frank a Campo
2021 Order  
We characterize finite, connected posets P having a 4-crown as retract.  ...  We define a multigraph 𝔉(P) whose vertices are the so-called improper 4-crowns contained in the extremal points of P, and we show that P contains a 4-crown as retract iff there exists a graph homomorphism  ...  Crowns can be regarded as the "natural enemies" of the fixed point property, because a crown being a retract of a poset directly delivers a fixed point free homomorphism.  ... 
doi:10.1007/s11083-021-09585-0 fatcat:hnxd4b5kxnab7p4556k7aa5bwq

Generalized metric spaces. Relations with graphs, ordered sets and automata : A survey [article]

Mustapha Kabil, Maurice Pouzet
2020 arXiv   pre-print
In this paper, we report the results obtained over the years with a particular emphasis on the fixed point property.  ...  For example, the characterization of absolute retracts, by Aronszajn and Panitchpakdi (1956), the construction of the injective envelope by Isbell (1965) and the fixed point theorem of Sine and Soardi  ...  ACKNOWLEDGEMENT This paper is a preliminary version of a book chapter of "New Trends in Analysis and Geometry". We thank Mohamed Amine Khamsi for his encouragements.  ... 
arXiv:2002.03019v1 fatcat:2xvave6jkvgvdhmvei6rnf2hbe

Page 3286 of Mathematical Reviews Vol. , Issue 85h [page]

1985 Mathematical Reviews  
The characterization of posets having the (ordinary) fixed point property is still an open problem.  ...  A poset P is said to have the relational fixed point property (=relational FPP) if every isotone relation from P to P has a fixed point, i.e., an element z of P for which z € f(z).  ... 
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