Constructions of Macaulay Posets and Macaulay Rings

TL;DR

This paper explores the construction of Macaulay posets and rings, analyzing how certain operations inspired by topology can generate new structures that preserve the Macaulay property.

Contribution

It introduces methods to combine Macaulay rings and posets, identifying conditions under which the Macaulay property is preserved in new structures.

Findings

Identified classes of posets and rings where operations preserve Macaulay property

Developed topologically inspired methods for constructing new Macaulay structures

Analyzed the interaction between partial and total orders in Macaulay contexts

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.