Normality of an affine semigroup - MathOverflow

Normality of an affine semigroup

2010-01-30T16:06:21Z

An affine monoid is a finitely generated commutative submonoid of $\mathbb Z^k$ for some positive integer k. Let S be an affine monoid and let G(S) be the group generated by S. We say the monoid S is normal if and only if for all $g \in G(S)$ and $n \in \mathbb N \setminus {0}$, $ng \in S$ implies $g \in S$.

Let $S$ be the submonoid generated by the finite set $T= \{(p_0,p_1, \cdots ,p_{n-1}) \in (\mathbb Z_{\geq 0})^n: \sum_{i=0}^{n-1}p_i=n \,\, and \,\, \sum_{i=0}^{n-1}i.p_i \cong 0 \pmod n\}$ . Now my question is how to prove that $S$ is normal ?

Answer by yanzhang for Normality of an affine semigroup

2010-02-02T22:33:00Z

I worked on this for a bit with Ricky Liu, who came up with this very quick solution:

Take your set T. Suppose $\sum p_i = kn$, where $k \geq 2$. Create the following set T': let $i$ appear $p_i$ times. This creates a set $T'$ with at least $2n$ elements (by your first constraint and $k \geq 2$), whose sum is divisible by $n$ by your second constraint.

However, by Erdos-Ginsberg-Ziv, there's a subset of $n$ elements which add to $n$, which exactly corresponds to your generator, so we're done.

Answer by unknown (yahoo) for Normality of an affine semigroup

2010-02-06T18:26:22Z

Sorry, I did not get the above answer. Can somebody please explain me.

Suppose, $q.(p_0,p_1, \cdots ,p_{n-1}) \in S$, where $ q \in \mathbb N \setminus 0$ and $(p_0,p_1, \cdots ,p_{n-1}) \in G(S)$, why is it clear from the above answer that $(p_0,p_1, \cdots ,p_{n-1})$ is already in $S$. Some of the $p_i$'s can be negative as well. It is not at all clear to me. Thanks in advance.