On monomial ideals and their socles

For a finite subset $M\subset [x_1,\dots ,x_d]$ of monomials, we describe how to constructively obtain a monomial ideal $I\subseteq R=K[x_1,\dots ,x_d]$ such that the set of monomials in $\text{Soc}(I)\setminus I$ is precisely $M$, or such that $\bar{M}\subseteq R\mathrm{/}I$ is a $K$-basis for the the socle of $R\mathrm{/}I$. For a given $M$ we obtain a natural class of monomials $I$ with this property. This is done by using solely the lattice structure of the monoid $[x_1,\dots ,x_d]$. We then present some duality results by using anti-isomorphisms between upsets and downsets of $({\mathbb{Z}}^d,⪯)$. 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 ${\mathbb{Z}}^d$.