This original research monograph concerns various aspects of how (based on the decompositions of vertices of hypercube graphs with respect to their symmetric cycles) the vertex sets of related discrete hypercubes, as well as the power sets of the corresponding ground sets, emerge from rank 2 oriented matroids, from underlying rank 2 systems of linear inequalities, and thus literally from arrangements of straight lines crossing a common point on a piece of paper. It reveals some beautiful and earlier-hidden fragments in the true foundations of discrete mathematics. The central observation made and discussed in the book from various viewpoints consists in that 2t subsets of a finite t-element set Et, which form in a natural way a cyclic structure (well, just t subsets that are the vertices of a path in the cycle suffice), allow us to construct any of 2t subsets of the set Et by means of a more than elementary voting procedure expressed in basic linear algebraic terms. The monograph will be of interest to researchers, students, and readers in the fields of discrete mathematics, theoretical computer science, Boolean function theory, enumerative combinatorics and combinatorics on words, combinatorial optimization, coding theory, and discrete and computational geometry.
1. Preliminaries and Notational Conventions
2. A 2D Perspective on Higher Dimensional Discrete Hypercubes and the Power Sets of Finite Sets
3. Vertex Decompositions in Hypercube Graphs, and Dehn–Sommerville Type Relations
4. Vertex Decompositions in Hypercube Graphs, and Orthogonality Relations
5. Distinguished Symmetric Cycles in Hypercube Graphs and Computation-free Vertex Decompositions
6. Distinguished Symmetric Cycles in Hypercube Graphs and Pairwise Decompositions of Vertices: Two-member Families of Disjoint Sets
7. Distinguished Symmetric Cycles in Hypercube Graphs and Pairwise Decompositions of Vertices: Arbitrary Two-member Clutters
8. Vertices, Their Relabeled Opposites, and Distinguished Symmetric Cycles in Hypercube Graphs
9. Set Families, Blocking Sets, Blockers, and Distinguished Symmetric Cycles in Hypercube Graphs
10. Vertex Decompositions and Subtope Decompositions in Hypercube Graphs
Biography
Dr. Andrey O. Matveev is the author of the research monographs Pattern Recognition on Oriented Matroids and Farey Sequences: Duality and Maps Between Subsequences (De Gruyter, 2017).
