In this article we first compare the set of elements in the socle of an ideal of a polynomial algebra \(K[x_1,\ldots ,x_d]\) over a field K that are not in the ideal itself with Macaulay’s inverse systems of such polynomial algebras in a purely combinatorial way for monomial ideals, and then develop some closure operational properties for the related poset \({{\mathbb {N}}_0^d}\) . We then derive some algebraic propositions of \(\Gamma \) -graded rings (a natural generalization of the usual \({\mathbb {Z}}\) -grading where \(\Gamma \) is a monoid) that then have some combinatorial consequences. Interestingly, some of the results from this part that uniformly hold for polynomial rings are always false when the ring is local. We finally delve into some direct computations, in relation to a given term order of the monomials, for general zero-dimensional Gorenstein ideals, and we deduce a few explicit observations and results for the inverse systems from some recent results about socles.
