## Advanced Studies in Pure Mathematics

### Semigroups – A computational approach

#### Abstract

The question whether there exists an integral solution to the system of linear equations with non-negativity constraints, $A\mathbf{x} = \mathbf{b}, \, \mathbf{x} \ge 0$, where $A \in \mathbb{Z}^{m\times n}$ and ${\mathbf b} \in \mathbb{Z}^m$, finds its applications in many areas such as operations research, number theory, combinatorics, 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 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 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 $n^\text{th}$ linear ordering polytope has the integer-decomposition property for $n\leq 7$. The software is available at http://ehrhart.math.fu-berlin.de/People/fkohl/HASE/.

#### Article information

Dates
Received: 11 August 2016
First available in Project Euclid: 21 September 2018

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1537499602

Digital Object Identifier
doi:10.2969/aspm/07710155

Mathematical Reviews number (MathSciNet)
MR3839710

Zentralblatt MATH identifier
07034253

#### Citation

Kohl, Florian; Li, Yanxi; Rauh, Johannes; Yoshida, Ruriko. Semigroups – A computational approach. The 50th Anniversary of Gröbner Bases, 155--170, Mathematical Society of Japan, Tokyo, Japan, 2018. doi:10.2969/aspm/07710155. https://projecteuclid.org/euclid.aspm/1537499602