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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 |
Description
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.