{"paper":{"title":"On monomial ideals and their socles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Geir Agnarsson, Neil Epstein","submitted_at":"2018-01-08T19:05:44Z","abstract_excerpt":"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 $(\\mathbb Z^d,\\preceq)$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.02644","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}