Abstract
For a finite subset M ⊂ [x1,…, xd] 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 Soc(I) ∖ 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 ideals I with this property. This is done by using solely the lattice structure of the monoid [x1,…, xd]. We then present some duality results by using anti-isomorphisms between upsets and downsets of the lattice \(({\mathbb {Z}}^{d},\preceq )\). 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 \({\mathbb {Z}}^{d}\).
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Sincere thanks to the anonymous referee for numerous good suggestions to improve the paper, in particular its exposition.
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Agnarsson, G., Epstein, N. On Monomial Ideals and Their Socles. Order 37, 341–369 (2020). https://doi.org/10.1007/s11083-019-09509-z
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DOI: https://doi.org/10.1007/s11083-019-09509-z