Monomial Ideals From a Semiring Perspective

We formulate the theory of monomial ideals of a ring of polynomials with coefficients in a field through the theory of semirings. In particular, we focus on the theory of integrally closed monomial ideals. We prove that there exist isomorphisms of lattice-ordered semirings between the semiring of antichains of Nn equipped with the product order (of its usual order), and the semiring of monomial ideals. In the same way, we prove that there is an isomorphism between the semiring of integrally closed monomial ideals and the family of vertex sets of Nn equipped with the product order. These isomorphisms are independent of the choice of a base field for the polynomial ring.