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Pruning Processes and a New Characterization of Convex Geometries

Summary: Pruning Processes and a New Characterization of
Convex Geometries
Federico Ardila
Elitza Maneva
We provide a new characterization of convex geometries via a multivariate version of an
identity that was originally proved, in a special case arising from the k-SAT problem, by Maneva,
Mossel and Wainwright. We thus highlight the connection between various characterizations
of convex geometries and a family of removal processes studied in the literature on random
1 Introduction
This article studies a general class of procedures in which the elements of a set are removed one
at a time according to a given rule. We refer to such a procedure as a removal process. If every
element which is removable at some stage of the process remains removable at any later stage,
we call this a pruning process. The subsets that one can reach through a pruning process have
the elegant combinatorial structure of a convex geometry. Our first goal is to highlight the role
of convex geometries in the literature on random structures, where many pruning processes have
been studied without exploiting their connection to these objects. Our second contribution is a
proof that a generalization of a polynomial identity, first obtained for a specific removal process in
[17], provides a new characterization of pruning processes and of convex geometries. To prove this


Source: Ardila, Federico - Department of Mathematics, San Francisco State University


Collections: Mathematics