9,619 research outputs found
The Order Dimension of the Poset of Regions in a Hyperplane Arrangement
We show that the order dimension of the weak order on a Coxeter group of type
A, B or D is equal to the rank of the Coxeter group, and give bounds on the
order dimensions for the other finite types. This result arises from a unified
approach which, in particular, leads to a simpler treatment of the previously
known cases, types A and B. The result for weak orders follows from an upper
bound on the dimension of the poset of regions of an arbitrary hyperplane
arrangement. In some cases, including the weak orders, the upper bound is the
chromatic number of a certain graph. For the weak orders, this graph has the
positive roots as its vertex set, and the edges are related to the pairwise
inner products of the roots.Comment: Minor changes, including a correction and an added figure in the
proof of Proposition 2.2. 19 pages, 6 figure
A hierarchical structure of transformation semigroups with applications to probability limit measures
The structure of transformation semigroups on a finite set is analyzed by
introducing a hierarchy of functions mapping subsets to subsets. The resulting
hierarchy of semigroups has a corresponding hierarchy of minimal ideals, or
kernels. This kernel hierarchy produces a set of tools that provides direct
access to computations of interest in probability limit theorems; in
particular, finding certain factors of idempotent limit measures. In addition,
when considering transformation semigroups that arise naturally from edge
colorings of directed graphs, as in the road-coloring problem, the hierarchy
produces simple techniques to determine the rank of the kernel and to decide
when a given kernel is a right group. In particular, it is shown that all
kernels of rank one less than the number of vertices must be right groups and
their structure for the case of two generators is described.Comment: 35 pages, 4 figure
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