17,647 research outputs found
A convenient category of locally preordered spaces
As a practical foundation for a homotopy theory of abstract spacetime, we
extend a category of certain compact partially ordered spaces to a convenient
category of locally preordered spaces. In particular, we show that our new
category is Cartesian closed and that the forgetful functor to the category of
compactly generated spaces creates all limits and colimits.Comment: 26 pages, 0 figures, partially presented at GETCO 2005; changes:
claim of Prop. 5.11 weakened to finite case and proof changed due to problems
with proof of Lemma 3.26, now removed; Eg. 2.7, statement before Lem. 2.11,
typos, and other minor problems corrected throughout; extensive rewording;
proof of Lem. 3.31, now 3.30, adde
Primary Facets Of Order Polytopes
Mixture models on order relations play a central role in recent
investigations of transitivity in binary choice data. In such a model, the
vectors of choice probabilities are the convex combinations of the
characteristic vectors of all order relations of a chosen type. The five
prominent types of order relations are linear orders, weak orders, semiorders,
interval orders and partial orders. For each of them, the problem of finding a
complete, workable characterization of the vectors of probabilities is
crucial---but it is reputably inaccessible. Under a geometric reformulation,
the problem asks for a linear description of a convex polytope whose vertices
are known. As for any convex polytope, a shortest linear description comprises
one linear inequality per facet. Getting all of the facet-defining inequalities
of any of the five order polytopes seems presently out of reach. Here we search
for the facet-defining inequalities which we call primary because their
coefficients take only the values -1, 0 or 1. We provide a classification of
all primary, facet-defining inequalities of three of the five order polytopes.
Moreover, we elaborate on the intricacy of the primary facet-defining
inequalities of the linear order and the weak order polytopes
Stochastic order on metric spaces and the ordered Kantorovich monad
In earlier work, we had introduced the Kantorovich probability monad on
complete metric spaces, extending a construction due to van Breugel. Here we
extend the Kantorovich monad further to a certain class of ordered metric
spaces, by endowing the spaces of probability measures with the usual
stochastic order. It can be considered a metric analogue of the probabilistic
powerdomain.
The spaces we consider, which we call L-ordered, are spaces where the order
satisfies a mild compatibility condition with the metric itself, rather than
merely with the underlying topology. As we show, this is related to the theory
of Lawvere metric spaces, in which the partial order structure is induced by
the zero distances.
We show that the algebras of the ordered Kantorovich monad are the closed
convex subsets of Banach spaces equipped with a closed positive cone, with
algebra morphisms given by the short and monotone affine maps. Considering the
category of L-ordered metric spaces as a locally posetal 2-category, the lax
and oplax algebra morphisms are exactly the concave and convex short maps,
respectively.
In the unordered case, we had identified the Wasserstein space as the colimit
of the spaces of empirical distributions of finite sequences. We prove that
this extends to the ordered setting as well by showing that the stochastic
order arises by completing the order between the finite sequences, generalizing
a recent result of Lawson. The proof holds on any metric space equipped with a
closed partial order.Comment: 49 pages. Removed incorrect statement (Theorem 6.1.10 of previous
version
Rowmotion and generalized toggle groups
We generalize the notion of the toggle group, as defined in [P. Cameron-D.
Fon-der-Flaass '95] and further explored in [J. Striker-N. Williams '12], from
the set of order ideals of a poset to any family of subsets of a finite set. We
prove structure theorems for certain finite generalized toggle groups, similar
to the theorem of Cameron and Fon-der-Flaass in the case of order ideals. We
apply these theorems and find other results on generalized toggle groups in the
following settings: chains, antichains, and interval-closed sets of a poset;
independent sets, vertex covers, acyclic subgraphs, and spanning subgraphs of a
graph; matroids and convex geometries. We generalize rowmotion, an action
studied on order ideals in [P. Cameron-D. Fon-der-Flaass '95] and [J.
Striker-N. Williams '12], to a map we call cover-closure on closed sets of a
closure operator. We show that cover-closure is bijective if and only if the
set of closed sets is isomorphic to the set of order ideals of a poset, which
implies rowmotion is the only bijective cover-closure map.Comment: 26 pages, 5 figures, final journal versio
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