The Fixed Point Property for Posets of Small Width

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

Convex sublattices of a lattice and a fixed point property

The collection CL(T) of nonempty convex sublattices of a lattice T ordered by bi-domination is a lattice. 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. We examine which lattices may have CLFPP. We introduce the selection property for convex sublattices (CLSP); we observe that a complete lattice with CLSP must have CLFPP, and that this property implies that CL(T) is complete. We show that for a lattice T, the fact that CL(T) is complete is equivalent to the fact that T is complete and the lattice of all subsets of a countable set, ordered by containment, is not order embeddable into T. We show that for the lattice T = I(P) of initial segments of a poset P, the implications above are equivalences and that these properties are equivalent to the fact that P has no infinite antichain. A crucial part of this proof is a straightforward application of a wonderful Hausdor type result due to Abraham, Bonnet, Cummings, Dzamondja and Thompson [2010]

Hom complexes and homotopy theory in the category of graphs

We investigate a notion of $\times$-homotopy of graph maps that is based on the internal hom associated to the categorical product in the category of graphs. It is shown that graph $\times$-homotopy is characterized by the topological properties of the \Hom complex, a functorial way to assign a poset (and hence topological space) to a pair of graphs; \Hom complexes were introduced by Lov\'{a}sz and further studied by Babson and Kozlov to give topological bounds on chromatic number. Along the way, we also establish some structural properties of \Hom complexes involving products and exponentials of graphs, as well as a symmetry result which can be used to reprove a theorem of Kozlov involving foldings of graphs. Graph $\times$-homotopy naturally leads to a notion of homotopy equivalence which we show has several equivalent characterizations. We apply the notions of $\times$-homotopy equivalence to the class of dismantlable graphs to get a list of conditions that again characterize these. We end with a discussion of graph homotopies arising from other internal homs, including the construction of `$A$-theory' associated to the cartesian product in the category of reflexive graphs.Comment: 28 pages, 13 figures, final version, to be published in European J. Com

Inverting weak dihomotopy equivalence using homotopy continuous flow

A flow is homotopy continuous if it is indefinitely divisible up to S-homotopy. The full subcategory of cofibrant homotopy continuous flows has nice features. Not only it is big enough to contain all dihomotopy types, but also a morphism between them is a weak dihomotopy equivalence if and only if it is invertible up to dihomotopy. Thus, the category of cofibrant homotopy continuous flows provides an implementation of Whitehead's theorem for the full dihomotopy relation, and not only for S-homotopy as in previous works of the author. This fact is not the consequence of the existence of a model structure on the category of flows because it is known that there does not exist any model structure on it whose weak equivalences are exactly the weak dihomotopy equivalences. This fact is an application of a general result for the localization of a model category with respect to a weak factorization system.Comment: 22 pages; LaTeX2e ; v2 : corrected bibliography + improvement of the statement of the main theorems ; v3 final version published in http://www.tac.mta.ca/tac

A Few Notes on Formal Balls

Using the notion of formal ball, we present a few new results in the theory of quasi-metric spaces. With no specific order: every continuous Yoneda-complete quasi-metric space is sober and convergence Choquet-complete hence Baire in its $d$-Scott topology; for standard quasi-metric spaces, algebraicity is equivalent to having enough center points; on a standard quasi-metric space, every lower semicontinuous $\bar{\mathbb{R}}_+$-valued function is the supremum of a chain of Lipschitz Yoneda-continuous maps; the continuous Yoneda-complete quasi-metric spaces are exactly the retracts of algebraic Yoneda-complete quasi-metric spaces; every continuous Yoneda-complete quasi-metric space has a so-called quasi-ideal model, generalizing a construction due to K. Martin. The point is that all those results reduce to domain-theoretic constructions on posets of formal balls

Fiber polytopes for the projections between cyclic polytopes

The cyclic polytope $C(n,d)$ is the convex hull of any $n$ points on the moment curve ${(t,t^2,...,t^d):t \in \reals}$ in $\reals^d$. For $d' >d$, we consider the fiber polytope (in the sense of Billera and Sturmfels) associated to the natural projection of cyclic polytopes $\pi: C(n,d') \to C(n,d)$ which "forgets" the last $d'-d$ coordinates. It is known that this fiber polytope has face lattice indexed by the coherent polytopal subdivisions of $C(n,d)$ which are induced by the map $\pi$. Our main result characterizes the triples $(n,d,d')$ for which the fiber polytope is canonical in either of the following two senses: - all polytopal subdivisions induced by $\pi$ are coherent, - the structure of the fiber polytope does not depend upon the choice of points on the moment curve. We also discuss a new instance with a positive answer to the Generalized Baues Problem, namely that of a projection $\pi:P\to Q$ where $Q$ has only regular subdivisions and $P$ has two more vertices than its dimension.Comment: 28 pages with 1 postscript figur