44 research outputs found
On the dimension of partially ordered sets
AbstractWe study the topic of the title in some detail. The main results are summarized in the first four paragraphs of this paper
Out of Nowhere: Spacetime from causality: causal set theory
This is a chapter of the planned monograph "Out of Nowhere: The Emergence of
Spacetime in Quantum Theories of Gravity", co-authored by Nick Huggett and
Christian W\"uthrich and under contract with Oxford University Press. (More
information at www.beyondspacetime.net.) This chapter introduces causal set
theory and identifies and articulates a 'problem of space' in this theory.Comment: 29 pages, 5 figure
The interval inclusion number of a partially ordered set
AbstractA containment representation for a poset P is a map ƒ such that x<y in P if and only if ƒ(x) ⊂ ƒ(y). We introduce the interval inclusion number (or interval number) i(P) as the smallest t such that P has a containment representation f in which each f(x) is the union of at most t intervals. Trivially, i(P)=1 if and only if dim(P)⩽2. Posets with i(P)=2 include the standard n-dimensional poset and all interval orders; i.e. posets of arbitrarily high dimension. In general we have the upper bound i(P) ⩽ ⌈;dim(P)2⌉, with equality holding for the Boolean algebras. For lexicographic composition, i(P) = k and dim(Q) = 2k + 1 imply i(P[Q]) = k + 1. This result and i(B2k) = k imply that testing i(P) ⩽ k for any fixed k ⩾ 2 is NP-complete. Concerning removal theorems, we prove that i(P − x) ⩾ i(P) − 1 when x is a maximal or minimal element of P, and in general i(P − x) ⩾ i(P)2
Core Rationalizability in Two-Agent Exchange Economies
We provide a characterization of selection correspondences in two-person exchange economies that can be core rationalized in the sense that there exists a preference profile with some standard properties that generates the observed choices as the set of core elements of the economy for any given initial endowment vector. The approach followed in this paper deviates from the standard rational choice model in that a rationalization in terms of a profile of individual orderings rather than in terms of a single individual or social preference relation is analyzed.
Small Superpatterns for Dominance Drawing
We exploit the connection between dominance drawings of directed acyclic
graphs and permutations, in both directions, to provide improved bounds on the
size of universal point sets for certain types of dominance drawing and on
superpatterns for certain natural classes of permutations. In particular we
show that there exist universal point sets for dominance drawings of the Hasse
diagrams of width-two partial orders of size O(n^{3/2}), universal point sets
for dominance drawings of st-outerplanar graphs of size O(n\log n), and
universal point sets for dominance drawings of directed trees of size O(n^2).
We show that 321-avoiding permutations have superpatterns of size O(n^{3/2}),
riffle permutations (321-, 2143-, and 2413-avoiding permutations) have
superpatterns of size O(n), and the concatenations of sequences of riffles and
their inverses have superpatterns of size O(n\log n). Our analysis includes a
calculation of the leading constants in these bounds.Comment: ANALCO 2014, This version fixes an error in the leading constant of
the 321-superpattern siz
Tree-width and dimension
Over the last 30 years, researchers have investigated connections between
dimension for posets and planarity for graphs. Here we extend this line of
research to the structural graph theory parameter tree-width by proving that
the dimension of a finite poset is bounded in terms of its height and the
tree-width of its cover graph.Comment: Updates on solutions of problems and on bibliograph
Boolean dimension and tree-width
The dimension is a key measure of complexity of partially ordered sets. Small
dimension allows succinct encoding. Indeed if has dimension , then to
know whether in it is enough to check whether in each
of the linear extensions of a witnessing realizer. Focusing on the encoding
aspect Ne\v{s}et\v{r}il and Pudl\'{a}k defined a more expressive version of
dimension. A poset has boolean dimension at most if it is possible to
decide whether in by looking at the relative position of and
in only permutations of the elements of . We prove that posets with
cover graphs of bounded tree-width have bounded boolean dimension. This stays
in contrast with the fact that there are posets with cover graphs of tree-width
three and arbitrarily large dimension. This result might be a step towards a
resolution of the long-standing open problem: Do planar posets have bounded
boolean dimension?Comment: one more reference added; paper revised along the suggestion of three
reviewer