161,767 research outputs found
Irreflexive and reflexive dimension
AbstractThere are three equivalent definitions of dimension for partially ordered sets. When generating these three definitions to C-dimension over an arbitrary class of orders C, the three definitions diverge. We compare these three definitions and determine certain requirements under which they are equivalent
An obstruction based approach to the Kochen-Specker theorem
In [1] it was shown that the Kochen Specker theorem can be written in terms
of the non-existence of global elements of a certain varying set over the
partially ordered set of boolean subalgebras of projection operators on some
Hilbert space. In this paper, we show how obstructions to the construction of
such global elements arise, and how this provides a new way of looking at
proofs of the theorem.Comment: 14 pages, 6 figure
Antimatroids and Balanced Pairs
We generalize the 1/3-2/3 conjecture from partially ordered sets to
antimatroids: we conjecture that any antimatroid has a pair of elements x,y
such that x has probability between 1/3 and 2/3 of appearing earlier than y in
a uniformly random basic word of the antimatroid. We prove the conjecture for
antimatroids of convex dimension two (the antimatroid-theoretic analogue of
partial orders of width two), for antimatroids of height two, for antimatroids
with an independent element, and for the perfect elimination antimatroids and
node search antimatroids of several classes of graphs. A computer search shows
that the conjecture is true for all antimatroids with at most six elements.Comment: 16 pages, 5 figure
Dimensions of finite type for representations of partially ordered sets
We consider the dimensions of finite type of representations of a partially
ordered set, i.e. such that there is only finitely many isomorphism classes of
representations of this dimension. We give a criterion for a dimension to be of
finite type. We also characterize those dimensions of finite type, for which
there is an indecomposable representation of this dimension, and show that
there can be at most one indecomposable representation of any dimension of
finite type. Moreover, if such a representation exists, it only has scalar
endomorphisms.Comment: 14 page
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
A half-space approach to order dimension
The aim of the present paper is to investigate the half-spaces in the
convexity structure of all quasiorders on a given set and to use them in an
alternative approach to classical order dimension. The main result states that
linear orders can almost always be replaced by half-space quasiorders in the
definition of the dimension of a partially ordered set.Comment: 13 page
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