The group of $n\times n$ matrices with integer entries and determinant equal to 1, $SL(n,Z)$, is a finitely generated group (in fact, it is generated by 2 matrices). I am interested to know if the semigroup of the matrices in $SL(n,Z)$ where all the entries are nonnegative is also finitely generated. This is true at least in dimension $n=2$. 



Is the semigroup of nonnegative integer matrices with determinant 1 finitely generated?
The question has been amply answered, but perhaps it's worth explaining the geometry of the situation. The positive monoid acts by projective transformations on the $n-1$-simplex (the view of the positve orthant as seen from the origin, and the partially order on the monoid is the order by inclusion of the image under this action. In $R^n$, the image of the simplex spanned by the origin together with the basis vectors cannot contain any interior lattice points. This implies that in the projective action, no proper image can contain its barycenter. This, with its ramifications, gives a strong limitation on images of the entire simplex under the positive monoid.


On the other hand, it's not hard to prove that for $n \ge 3$, the set of images of any face are dense. In fact, using elementary and well-known theory, there is a positive density of $(n-1)$-tuples of lattice points that extend to a basis for $Z^n$, and any such $n$-tuple of positive elements can be extended to a positive basis.


This implies that there is no finite set of generators, because a single image of the $n-1$-simplex can contain at most simplices near the barycenter over a small range of angles. Here's a picture for the 2-dimensional case:
[alt text http://dl.dropbox.com/u/5390048/MatrixSemigroup.jpg](http://dl.dropbox.com/u/5390048/MatrixSemigroup.jpg)


The line segment is the projection of the pair of lattice points 
$L\_1=(285, 684, 112)$ and $L\_2 = (764, 318, 949)$ found, by pseudorandom selection from lattice cubes that projected near the targeted segment, subject to the condition that they extend to a free basis.


A third element of a basis can be transformed to be project arbitrarily close to any desired point on the line through the image of $L\_1$ and $L\_2$ by adding $N\*L1 + M\*L2$, for suitable $N$ and $M$.


Moreover, this characterize the closure (with respect to the Hausdorff topology) of the set of images of the triangle under the positive monoid: it consists of all triangles spanned by positive free bases, together with set of all line segments contained in the triangle.


