All Graduate Plan B and other Reports

5-2011

Report

Degree Name

Master of Science (MS)

Department

Mathematics and Statistics

David Brown

Abstract

Given , a tournament on vertices, Landau derived a method to determine how close is to being transitive or regular. This comparison is based on the tournament’s hierarchy number, ̅, a value derived from its score vector ̅ ( ). Let be the set of all score vectors of tournaments on vertices with the entries listed in non-decreasing order. A partial order, poset, exists on the set using the following binary relation. Given ̅ ̅ such that ̅ ̅, let ̅ ̅ if Σ Σ for and Σ Σ . Let this poset be represented as ( ) ( ) where {( ̅ ̅) ̅ ̅}. The value ̅ can also be used to define a partial order on the set where ̅ ̅ if ̅ ̅ . I propose that this new poset is an extension of the poset ( ). This can be proven using a method of comparing score vectors algebraically equivalent to Landau’s hierarchy method. Specifically, Σ Σ if and only if ̅ ̅. Furthermore, I conjecture that extending this method to compare ̅ and ̅ using Σ for will always yield an extension of ( ), and that there exists some integer dependent on which will result in a linear extension of ( ).