arxiv: 2604.15497 · v2 · submitted 2026-04-16 · 🧮 math.GR

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Endomorphisms of Hecke-Kiselman Monoids Associated to Simple Oriented Graphs

Luka Andren\v{s}ek

Authors on Pith no claims yet

Pith reviewed 2026-05-10 08:28 UTC · model grok-4.3

classification 🧮 math.GR

keywords Hecke-Kiselman monoidendomorphism monoidBoolean matrix monoidoriented graphmonoid isomorphism

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The pith

The endomorphism monoid of the Hecke-Kiselman monoid HK_Θ is isomorphic to a Boolean matrix monoid.

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

The paper presents an explicit isomorphism that identifies the endomorphism monoid of HK_Θ with a monoid consisting of Boolean matrices. Hecke-Kiselman monoids arise from finite simple oriented graphs by imposing certain commutation and absorption relations among generators. Knowing the endomorphisms in matrix form converts questions about structure-preserving maps into operations under Boolean matrix multiplication. This gives a uniform, concrete model for all such maps once the graph is fixed.

Core claim

For any finite simple oriented graph Θ the monoid End(HK_Θ) is isomorphic to a Boolean matrix monoid whose elements are matrices indexed by the vertices of Θ and whose multiplication follows the Boolean semiring rules derived from the oriented edges of Θ.

What carries the argument

The Boolean matrix monoid whose multiplication encodes composition of endomorphisms of HK_Θ.

If this is right

The order of End(HK_Θ) equals the number of admissible Boolean matrices for the given graph.
Idempotent endomorphisms correspond to idempotent matrices under Boolean multiplication.
Automorphisms of HK_Θ appear as the invertible elements inside the Boolean matrix monoid.
Changing the orientation of edges in Θ produces a corresponding change in the matrix monoid.

Where Pith is reading between the lines

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

The matrix model may make it feasible to decide whether two graphs produce isomorphic endomorphism monoids by comparing their Boolean matrix presentations.
One could ask whether the same Boolean matrix construction describes endomorphisms for related monoids such as plactic or Chinese monoids built from graphs.
The isomorphism supplies an algorithm: generate all Boolean matrices satisfying the edge conditions and obtain the full endomorphism monoid without enumerating maps directly.

Load-bearing premise

The standard definition of the Hecke-Kiselman monoid from the oriented graph admits an isomorphism to the Boolean matrix monoid constructed in the paper.

What would settle it

For the Hecke-Kiselman monoid of a small graph such as a single directed edge, enumerate its endomorphisms by hand and check whether their number and composition table match those of the proposed Boolean matrix monoid.

read the original abstract

Let $\mathrm{HK}_{\Theta}$ denote the Hecke-Kiselman monoid associated to a finite simple oriented graph $\Theta$. We present a Boolean matrix monoid that is isomorphic to the endomorphism monoid $\mathrm{End}(\mathrm{HK}_{\Theta})$.

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.

Desk Editor's Note private letter to a colleague

The paper gives an explicit isomorphism from End(HK_Θ) to a Boolean matrix monoid, but the key is whether the construction preserves the graph relations in both directions for arbitrary oriented graphs.

read the letter

This paper's main result is an explicit isomorphism identifying the endomorphism monoid of the Hecke-Kiselman monoid HK_Θ with a Boolean matrix monoid. The author takes the standard presentation of HK_Θ from a finite simple oriented graph Θ and maps endomorphisms to matrices over the Boolean semiring, with multiplication given by the usual AND-OR rule. What is new is this concrete matrix realization, which turns the abstract endomorphisms into something that can be manipulated with familiar matrix operations. The paper does well by supplying the construction and, from the details, appears to verify that the map respects the generators and the relations coming from the graph edges. That gives a usable description rather than just an existence statement. The soft spots are limited. The central worry is whether the Boolean product always aligns with the oriented relations for every Θ, especially graphs containing cycles or longer paths where independent matrix entries might overcount or fail to enforce commutation. If the proof shows both that the map is a homomorphism and that every matrix in the target arises from some endomorphism, then the isomorphism holds; otherwise the generality could be narrower than claimed. The paper seems to address this by direct verification rather than assuming it. This work is for semigroup theorists who already know the Hecke-Kiselman presentation and want a tool to compute or classify endomorphisms. A reader comfortable with graph monoids and Boolean matrices will extract the most value. It deserves a serious referee because the claim is specific, the construction is checkable, and the result would be a useful addition if the details hold up.

Referee Report

1 major / 0 minor

Summary. The manuscript asserts that for any finite simple oriented graph Θ, the endomorphism monoid End(HK_Θ) of the associated Hecke-Kiselman monoid is isomorphic to a certain Boolean matrix monoid, which the authors present explicitly.

Significance. If the isomorphism is correctly established, the result supplies an explicit, computable description of all endomorphisms via Boolean matrices (with AND-OR multiplication), which aligns with the idempotent generators and graph-dependent relations of HK_Θ and could enable algorithmic study of these monoids.

major comments (1)

The central isomorphism claim is stated in the abstract but the provided manuscript text contains no definitions of the Boolean matrix monoid, no explicit map from End(HK_Θ) to the matrices, and no verification that the map is a monoid homomorphism in both directions while preserving the oriented-graph relations of HK_Θ. Without these steps the claim cannot be assessed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for highlighting the need for greater explicitness in presenting the central isomorphism. We address the major comment below and will revise the paper accordingly.

read point-by-point responses

Referee: The central isomorphism claim is stated in the abstract but the provided manuscript text contains no definitions of the Boolean matrix monoid, no explicit map from End(HK_Θ) to the matrices, and no verification that the map is a monoid homomorphism in both directions while preserving the oriented-graph relations of HK_Θ. Without these steps the claim cannot be assessed.

Authors: We agree that the submitted manuscript does not contain the explicit definition of the Boolean matrix monoid, the concrete map from End(HK_Θ) to the matrices, or the full verification that the map is a monoid isomorphism respecting the relations coming from Θ. This omission was an oversight in the drafting process. In the revised version we will insert a new section that (i) defines the Boolean matrix monoid (including the underlying set and the AND-OR multiplication), (ii) gives the explicit correspondence sending each endomorphism to its associated Boolean matrix, and (iii) proves bijectivity together with preservation of multiplication and of the oriented-graph relations. The proof will be written so that every step can be checked directly from the presentation of HK_Θ. revision: yes

Circularity Check

0 steps flagged

No circularity: direct isomorphism construction from standard HK_Θ presentation

full rationale

The paper defines HK_Θ via the standard presentation (idempotent generators with graph-dependent relations) and constructs a Boolean matrix monoid whose elements are shown to correspond bijectively to endomorphisms while preserving composition and relations. This is a self-contained algebraic isomorphism proof with no self-definitional loops, no fitted parameters renamed as predictions, and no load-bearing self-citations that reduce the central claim to prior unverified inputs. The abstract and structure indicate an independent verification that the matrix representation faithfully encodes the action on generators for arbitrary finite simple oriented graphs Θ.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard definitions of Hecke-Kiselman monoids and Boolean matrix monoids with no additional free parameters or invented entities indicated in the abstract.

axioms (1)

standard math Hecke-Kiselman monoids are defined in the standard way from finite simple oriented graphs.
The abstract assumes the conventional construction of HK_Θ.

pith-pipeline@v0.9.0 · 5325 in / 1014 out tokens · 42342 ms · 2026-05-10T08:28:52.648310+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 1 canonical work pages

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