The MacaulayPosets package for Macaulay2

TL;DR

The paper presents the MacaulayPosets package for Macaulay2, aimed at analyzing the Macaulay property for posets, especially monomial posets arising from graded rings. It extends the existing Posets framework by formalizing Macaulay posets, establishing connections to Hilbert functions, and enabling construction of complex posets and rings via four poset operations and two ring operations. A key contribution is providing monomial poset construction for non-monomial ideals, along with tools to check Macaulayness, obtain Macaulay orders, and visualize results. Overall, the package equips researchers with a systematic, computational pathway to explore the interplay between algebraic invariants like Hilbert functions and the combinatorial structure of posets and rings, with practical visualization capabilities.

Abstract

We introduce the package MacaulayPosets written for the computational algebra system Macaulay2. This package utilized the poset data type introduced in the Posets package and offers functionality for studying the Macaulay property for posets, particularly those which arise as monomial posets of commutative rings. A Macaulay poset is characterized by a ranked structure and a total order that interacts harmoniously with the partial order, enabling the establishment of bounds on the sizes of subsets of a given rank within an order ideal.

Figures (7)

• Figure 1: The monomial poset of $K[x,y]$ with $\{x^3,y^3\}$ and its upper shadow highlighted.
• Figure 2: Two sets (pink) and their upper shadows (green) in the monomial poset of $K[x,y]$.
• Figure 3: The monomial poset of $\mathbb{Q}[x,y]/(x^6,x^3y,y^4,x^2y^3-x^5)$.
• Figure 4: The TikZ picture generated by the Visualize package for the non-Macaulay disjoint union of the $2^5$-element Boolean lattice with the $7$-element chain.
• Figure 5: The two orders with respect to which the monomial poset of ${\mathbb Q}[x, y]/(x^5, x^2y^2, y^5)$ from \ref{['ex: macaulayOrders']} is Macaulay, with smaller elements appearing to the left of larger elements.

Theorems & Definitions (32)

• Definition 2.1
• Example 2.2
• Definition 2.3
• Definition 2.4
• Example 2.5
• Definition 2.6
• Definition 2.7
• Example 3.1
• Example 3.2
• Example 3.3