Constructions of Macaulay Posets and Macaulay Rings
Penelope Beall, Erenay Boyali, Nancy Chen, Ellen Chlachidze, Trong Toan Dao, Frederic Garvey, Mitchell Johnson, Yu Olivier Li, Nikola Kuzmanovski, Kelvin Ma, Treanungkur Mal, Rukshan Marasinghe, Quinlan Mayo, Nava Minsky-Primus, Alexandra Seceleanu, Sriram Veerapaneni
TL;DR
This work analyzes when the Macaulay property, originally defined for monomial posets, is preserved under topology-inspired poset operations: wedge, diamond, and fiber products. It establishes precise equivalences: for same-rank posets with unique extrema, Macaulayness is preserved through disjoint unions, wedge, and diamond under a union simplicial order, with additivity playing a crucial role in strengthening these results. The authors classify Macaulay behavior for wedge and diamond products of Clements–Lindström box posets, and they introduce heart-shaped fiber products to explore Macaulay-ness in fiber-product contexts, yielding algebraic corollaries for corresponding rings. The paper also presents conjectures, counterexamples, and small explicit constructions that illuminate when cartesian and tensor product operations preserve Macaulay properties, contributing to a deeper understanding of Macaulay rings via the combinatorics of monomial posets.
Abstract
A poset is Macaulay if its partial order and an additional total order interact well. Analogously, a ring is Macaulay if the partial order defined on its monomials by division interacts nicely with any total monomial order. We investigate methods of obtaining new structures through combining Macaulay rings and posets by means of certain operations inspired by topology. We examine whether these new structures retain the Macaulay property, identifying new classes of posets and rings for which the operations preserve the Macaulay property.