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### Fundamental holes and saturation points of a commutative semigroup and their applications to contingency tables.

**Optimization**

Speaker: | Ruriko Yoshida, Duke University |

Location: | 2112 MSB |

Start time: | Fri, Apr 28 2006, 12:10PM |

When does a given system of linear equations with nonnegative constraints have an integer solution? This problem appears in many areas, such as number theory, operations research, and statistics. To study a To study a family of systems with no integer solution, we focus on a commutative semigroup generated by the columns of its defining matrix. In this paper we will study a commutative semigroup generated by a finite subset of $\Z^d$ and its saturation. We show the necessary and sufficient conditions for the given semigroup to have a finite number of elements in the difference between the semigroup and its saturation. Also we define fundamental holes and saturation points of a commutative semigroup. Then, we show the simultaneous finiteness of the difference between the semigroup and its saturation, the set of non-saturation points of the semigroup, and the set of generators for saturation points, which is a set of generator of a monoid. We apply our results to some three and four dimensional contingency tables.