Subtractive Ideals in Finite Cyclic Commutative Semirings
Dr. Francisco Alarcón
This summer, my research advisor and I are investigating a re-characterization of the subtractive ideals of a particular commutative semiring, B(n,i)[x], which is the semiring of polynomials with coefficients in B(n,i). It is well founded that, for a semiring, the subtractive ideals are the substructures that are mapped to zero by a function which preserves the properties of the semiring. However, this is not a very constructive definition. The subtractive ideals of B(n,i) have previously been fully re-characterized by Alarcón and Polkowska. Our goal is to develop a similar characterization of these subtractive ideals for B(n,i)[x]. Furthermore, we wish to investigate any potential lattice structure of these ideals and generalize our results to R[x], where R is any zero-sum-free semiring, whenever possible.
Semirings have important applications to areas of computer science and mathematics such as logic, probability, and automata theory. While the subject of our research has no direct applications currently, it is becoming apparent that semirings of this type are fundamental in semiring theory. However, as a pure mathematics research topic, the interest lies largely in the topic itself, outside of any potential applications.