Pattern Recognition on Oriented Matroids: Symmetric Cycles in the Hypercube Graphs. V

Abstract

We consider decompositions of topes of the oriented matroid realizable as the arrangement of coordinate hyperplanes in R 2 t , with respect to a distinguished symmetric 2 • 2 t -cycle in its hypercube graph of topes H(2 t , 2). We seek interpretations of such decompositions in the context of subset families on the ground set Et := {1, . . . , t} and of the families of their blocking sets, in the context of clutters on Et and of their blockers. Contents 1. Introduction Decomposing 2. Topes, their relabeled opposites, and decompositions Blocking 3. Increasing families of blocking sets, and blockers: Set covering problems 4. Families of subsets of the ground set E t : Characteristic vectors and characteristic topes 5. Increasing families of blocking sets, and blockers: Characteristic vectors and characteristic topes 5.1. A clutter {{a}} 5.1.1. The principal increasing family of blocking sets B({{a}}) = {{a}} 5.1.2. The blocker B({{a}}) = {{a}} 5.1.3. More on the principal increasing family B({{a}}) = {{a}} 5.2. A clutter {A} 5.2.1. The increasing family of blocking sets B({A}) = {{a} : a ∈ A} 5.2.2. The blocker B({A}) = {{a} : a ∈ A} 5.2.3. More on the increasing families {A} and B({A}) 5.3. A clutter A := {A 1 , . . . , A α } 5.3.1. The increasing family of blocking sets B(A) 5.3.2. The blocker B(A) 5.3.3. More on the increasing families A and B(A)

The Increasing Families of Sets Generated by Self-dual Clutters

“…The blocker B(A) of the clutter A is defined to be the family of all inclusion-minimal blocking sets of A; see, e.g., the monographs [4,7,9,10,11,12,13,15,16,19,20,21,22,24,26,31,32,33,35,36,38]. Additional references can be found in [27,Pt 2]. The increasing family B(A) is by definition the family of all blocking sets of the clutter A.…”