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