Semigroups --- A Computational Approach

TL;DR

This paper explores algorithms for understanding semigroups generated by matrices, focusing on computing 'holes' to solve integer linear systems with non-negative constraints, with applications in combinatorics and polytope properties.

Contribution

It implements an algorithm for computing semigroup holes, applies it to specific models, and analyzes polytope properties up to dimension 7.

Findings

Computed the set of holes for the common diagonal effect model.

Showed the n-th linear ordering polytope has the integer-decomposition property for n ≤ 7.

Provided software implementation for semigroup hole computation.

Abstract

The question whether there exists an integral solution to the system of linear equations with non-negative constraints, $A\x = \b, \, \x \ge 0$, where $A \in \Z^{m\times n}$ and ${\mathbf b} \in \Z^m$, finds its applications in many areas, such as operation research, number theory and statistics. In order to solve this problem, we have to understand the semigroup generated by the columns of the matrix $A$ and the structure of the "holes" which are the difference between the semigroup generated by the columns of the matrix $A$ and its saturation. In this paper, we discuss the implementation of an algorithm by Hemmecke, Takemura, and Yoshida that computes the set of holes of a semigroup, % generated by the columns of $A$ and we discuss applications to problems in combinatorics. Moreover, we compute the set of holes for the common diagonal effect model, and we show that the $n$th linear ordering polytope has the integer-decomposition property for $n\leq 7$. The software is available at \url{http://ehrhart.math.fu-berlin.de/People/fkohl/HASE/}.