The weak normalization of an affine semigroup

In this article, we define and explore the weak normalization of an affine semigroup. In particular, for a fixed prime integer, we provide a geometric description of the weak normalization of an affine semigroup with respect to that prime, which corresponds to the weak normalization of the affine semigroup ring over a field of that prime characteristic, similar to the description of the seminormalization of an affine semigroup given by Reid–Roberts. We then use this description to understand the singularities of an affine semigroup ring defined over a field of prime characteristic and provide several examples. In particular, we demonstrate that all affine semigroup rings defined over fields of prime characteristic have a uniform upper bound on the Frobenius test exponent of all ideals, which provides a large and important class of examples with a positive answer to a question of Katzman–Sharp on uniformity of Frobenius test exponents. Finally, we provide an algorithm and implementation to compute the weak normalization of an affine semigroup, and the Frobenius test exponent and Frobenius closures of ideals in the affine semigroup ring.