A numerical semigroup S is a set of nonnegative integers such that S contains 0, S is closed under addition, and the complement of S is finite. This paper considers pairs (S,I) of a given numerical semigroup S and corresponding relative ideal I such that \mu(I)\mu(S-I) = \mu(I+(S-I)), where \mu denotes the size of the minimal generating set and S-I is the dual of I in S. We will present recent results in the research of such pairs (perfect bricks) with \mu(I) > 2 and \mu(S-I) > 2. We will also show the existence of an infinite family of perfect bricks.

Author Bio

Brooke Fox is currently a senior mathematics major at Northern Arizona Universityin Flagstaff, Arizona. She is planning on attending graduate school at Northern Arizona University in the fall. Shehas been working on this project with Jeff Rushall for almost two years, start-ing as independent study and transitioning into original research.