Papers Submitted
Abstract:
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
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.