LAUSR

For a finite subset M ⊂ [ x 1 ,…, x d ] of monomials, we describe how to constructively obtain a monomial ideal I ⊆ R = K [ x 1 , … , x d ] $I\subseteq R = K[x_{1},\ldots ,x_{d}]$ such that the set of monomials in Soc( I ) ∖ I is precisely M , or such that M ¯ ⊆ R / I $\overline {M}\subseteq R/I$ is a K -basis for the the socle of R / I . For a given M we obtain a natural class of monomials ideals I with this property. This is done by using solely the lattice structure of the monoid [ x 1 ,…, x d ]. We then present some duality results by using anti-isomorphisms between upsets and downsets of the lattice ( ℤ d , ≼ ) $({\mathbb {Z}}^{d},\preceq )$ . Finally, we define and analyze zero-dimensional monomial ideals of R of type k , where type 1 are exactly the Artinian Gorenstein ideals, and describe the structure of such ideals that correspond to order-generic antichains in ℤ d ${\mathbb {Z}}^{d}$ .