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In this article we first compare the set of elements in the socle of an idealof a polynomial algebra $K[x_1,\ldots,x_d]$ over a field $K$ that are not inthe ideal itself and Macaulay's inverse systems of such polynomial algebras ina purely combinatorial way for monomial ideals, and then develop some closureoperational properties for the related poset ${{\nats}_0^d}$. We then derivesome algebraic propositions of $\Gamma$-graded rings that then have somecombinatorial consequences. Interestingly, some of the results from this partthat uniformly hold for polynomial rings are always false when the ring islocal. We finally delve into some direct computations, w.r.t.~a given termorder of the monomials, for general zero-dimensional Gorenstein ideals anddeduce a few explicit observations and results for the inverse systems fromsome recent results about socles.

On posets, monomial ideals, Gorenstein ideals and their combinatorics