Advanced Studies in Pure Mathematics

Semigroups – A computational approach

Florian Kohl, Yanxi Li, Johannes Rauh, and Ruriko Yoshida

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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

Article information

The 50th Anniversary of Gröbner Bases, T. Hibi, ed. (Tokyo: Mathematical Society of Japan, 2018), 155-170

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

Permanent link to this document euclid.aspm/1537499602

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11P21: Lattice points in specified regions 52B11: $n$-dimensional polytopes 90C08: Special problems of linear programming (transportation, multi-index, etc.) 97K70: Foundations and methodology of statistics

feasibility problem lattice points polytopes software


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.

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