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arxiv: 2502.15166 · v2 · submitted 2025-02-21 · 🧮 math.CO · math.AC

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
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Nikola Kuzmanovski Kelvin Ma Treanungkur Mal Rukshan Marasinghe Mudiyanselage Quinlan Mayo Nava Minsky-Primus Alexandra Seceleanu Sriram Veerapaneni
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Pith reviewed 2026-05-23 02:59 UTC · model grok-4.3

classification 🧮 math.CO math.AC
keywords Macaulay posetsMacaulay ringsposet operationstopological constructionsmonomial ordersorder interactions
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The pith

Topology-inspired operations on Macaulay posets and rings produce new structures that retain the Macaulay property.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines operations inspired by topology that combine existing Macaulay posets or rings to form new ones. It checks whether the defining interaction between a partial order and a total order survives these combinations. The authors isolate specific classes of posets and rings where the Macaulay property is preserved under the operations. This supplies systematic constructions that enlarge the known families of such objects.

Core claim

Certain topology-inspired operations applied to Macaulay posets and rings yield new posets and rings that are themselves Macaulay, meaning their partial orders continue to interact appropriately with chosen total orders; the paper identifies concrete classes of input structures for which this preservation holds.

What carries the argument

Topology-inspired operations that combine posets or rings while allowing direct verification that the partial-order and total-order interaction remains intact.

If this is right

  • New Macaulay posets can be obtained directly from known ones via the operations.
  • The same operations generate new Macaulay rings from existing Macaulay rings.
  • The identified classes supply infinite families of examples without separate verification for each case.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • These constructions could be iterated to produce still larger families of Macaulay objects.
  • The approach may extend to other order-theoretic properties that depend on compatible partial and total orders.

Load-bearing premise

The operations are well-defined on the posets and rings so that the Macaulay interaction between partial and total orders can be checked directly.

What would settle it

An explicit example of one of the new combined structures in which the partial order and total order fail to interact in the required Macaulay manner.

Figures

Figures reproduced from arXiv: 2502.15166 by Alexandra Seceleanu, Ellen Chlachidze, Erenay Boyali, Frederic Garvey, Kelvin Ma, Mitchell Johnson, Nancy Chen, Nava Minsky-Primus, Nikola Kuzmanovski, Penelope Beall, Quinlan Mayo, Rukshan Marasinghe Mudiyanselage, Sriram Veerapaneni, Treanungkur Mal, Trong Toan Dao, Yu Olivier Li.

Figure 1
Figure 1. Figure 1: An illustration of Lemma 3.2 Theorem 3.4. Let P1, . . . ,Pn be ranked posets. Whenever the respective constructions are defined, the following are equivalent: (1) Fn i=1 Pi is Macaulay, (2) Wn i=1 Pi is Macaulay, (3) ♢ n i=1 Pci is Macaulay. Moreover, the list of properties below are related by implications (1′ ) ⇒ (2′ ) ⇒ (3′ ) (1’) Fn i=1 Pi is Macaulay, (2’) Wn i=1 Pi is Macaulay, (3’) ♢ n i=1 Pi is Mac… view at source ↗
Figure 2
Figure 2. Figure 2: A heart-shaped poset The poset M in Theorem 5.2 is depicted in [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: shows the top portion of the heart-shaped poset in [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The posets for part (1) of Proposition 6.1 [PITH_FULL_IMAGE:figures/full_fig_p027_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The posets for part (2) of Proposition 6.1 In [MP1] it is shown that the tensor product of a Macaulay ring with respect to the lexicographic order and a polynomial ring in a single variable is a Macaulay ring. Via the correspondence between Macaulay rings and posets, this says that the cartesian product of a Macaulay poset that is the monomial poset of a lex-Macaulay ring and an infinite path poset is Maca… view at source ↗
Figure 6
Figure 6. Figure 6: A counterexample to [Kuz, Conjecture 6.6.]. However, we found no counterexamples to the following adjusted version of Conjecture 6.2. Conjecture 6.4. If S is a Macaulay ring, x is a variable not appearing in S and n is stricly larger than the largest degree of any element of S. then S ⊗K K[x]/(x n ) is a Macaulay ring. References [Abd] A. Abedelfatah, Macaulay-Lex rings, J. Algebra 374 (2013), 122–131. [BK… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper defines topology-inspired operations on Macaulay posets and rings, supplies explicit definitions of the operations together with the relevant partial and total orders, and directly verifies that the Macaulay interaction condition is preserved on the resulting structures, thereby identifying new classes of Macaulay posets and rings.

Significance. The explicit constructions and preservation proofs enlarge the known families of Macaulay structures; the direct verification approach is a strength when the checks are complete.

minor comments (2)
  1. The introduction would benefit from a brief comparison table listing the new classes against previously known Macaulay posets/rings.
  2. Notation for the topology-inspired operations should be introduced once in a dedicated subsection rather than inline.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of the explicit constructions and direct verification approach as strengths, and recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity; derivations are direct verifications

full rationale

The paper defines topology-inspired operations on posets and rings explicitly, then performs direct checks that the Macaulay interaction condition holds on the resulting structures. No equations, fitted parameters, or predictions appear that reduce to the inputs by construction. No self-citations are invoked as load-bearing uniqueness theorems or ansatzes. The central results consist of independent preservation proofs on the constructed classes, making the work self-contained against external benchmarks with no detectable circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no free parameters, axioms, or invented entities are described.

pith-pipeline@v0.9.0 · 5682 in / 935 out tokens · 18082 ms · 2026-05-23T02:59:56.232840+00:00 · methodology

discussion (0)

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Reference graph

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