The Monoid Of Binary Relations On A Set Of Size Four Has Infinite Representation Type

Joseph Daynger Ruiz

arxiv: 2604.05121 · v1 · submitted 2026-04-06 · 🧮 math.RT

The Monoid Of Binary Relations On A Set Of Size Four Has Infinite Representation Type

Joseph Daynger Ruiz This is my paper

Pith reviewed 2026-05-10 19:05 UTC · model grok-4.3

classification 🧮 math.RT

keywords binary relations monoidrepresentation typeinfinite representation typemonoid algebratransformation monoidsemigroup representationsgeneralized correspondences

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The pith

The monoid of binary relations on four elements has infinite representation type.

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

The paper proves that the monoid of all binary relations on a four-element set has infinite representation type. This establishes that the corresponding monoid algebra over a field admits infinitely many non-isomorphic finite-dimensional modules. The result is obtained by first stating a sufficient condition that forces any monoid to have infinite representation type and then verifying the condition for this specific monoid. The work parallels the earlier resolution for the full transformation monoid and includes a partial result for the monoid of Lambda-generalized correspondences.

Core claim

The monoid of binary relations on a set of four elements has infinite representation type. This is shown by applying a sufficient condition for monoids to have infinite representation type, which produces an infinite family of non-isomorphic representations of the monoid algebra.

What carries the argument

A sufficient condition for a monoid to have infinite representation type, applied directly to the monoid of binary relations on four elements.

If this is right

The representations of the binary relations monoid on four elements cannot be classified by a finite list of indecomposables.
The same sufficient condition applies to other monoids of relations and correspondences.
The infinite type result for binary relations extends the known result for the full transformation monoid.
A partial classification or type determination holds for the monoid of Lambda-generalized correspondences.

Where Pith is reading between the lines

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

The same condition likely shows infinite type for the binary relations monoid on sets larger than four.
The sufficient condition may help decide representation type for other families of semigroups or monoids arising in combinatorics.
Infinite representation type implies that computational or enumerative problems about modules over these monoids are undecidable in general.

Load-bearing premise

The sufficient condition correctly identifies an infinite family of pairwise non-isomorphic modules for the binary relations monoid on four elements.

What would settle it

An explicit list or classification showing only finitely many indecomposable modules up to isomorphism for the monoid algebra over any field would falsify the claim.

read the original abstract

The problem of determining the representation type of the full transformation monoid was resolved by Ponizovskii, Putcha, and Ringel. In this paper, we present a similar result for the monoid of binary relations, a partial result for the monoid of $\Lambda$-generalized correspondences, and a sufficient condition for a monoid to have infinite representation type.

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 a sufficient condition for infinite representation type and applies it to conclude that the binary relations monoid on four elements has infinite type, with the embedding check as the key step to verify.

read the letter

The main thing here is that the monoid of binary relations on four points has infinite representation type, shown by introducing a sufficient condition and checking that the monoid meets it. The new piece is the sufficient condition for infinite representation type. The paper states it, then uses it for the binary relations case and gives a partial result for Lambda-generalized correspondences. This mirrors the known resolution for transformation monoids but shifts to relations. The paper handles the setup well by recalling the transformation monoid result and framing the new condition as a tool that might apply elsewhere. The potential weak point is the verification that the binary relations monoid actually contains the required configuration. The condition triggers infinite type only if certain elements multiply in a specific way under composition. The authors need to pick the right relations on the four-element set and confirm all the products match. Since the monoid is large, this check could have slips, and an error would mean the condition does not apply. The stress test on this embedding is the right one to run. Beyond that, the citations look standard and the logic flows from the condition to the conclusion without obvious gaps. This work is for researchers in semigroup representation theory who care about classifying representation types for small monoids. A reader who knows the transformation monoid literature will get the most out of it and can assess if the condition is broadly useful. I think it should go to peer review. The result is narrow but the method is presented as reusable, so referees can check the key embedding step and see if the condition holds up in other cases.

Referee Report

2 major / 2 minor

Summary. The paper resolves the representation type of the monoid of all binary relations on a 4-element set by exhibiting a sufficient condition for a monoid to have infinite representation type (over an algebraically closed field) and verifying that this monoid satisfies the condition via an explicit submonoid or configuration of elements under relational composition. It also gives a partial result for the monoid of Λ-generalized correspondences and recalls the known resolution for the full transformation monoid.

Significance. If the sufficient condition is correctly formulated and the embedding into the 2^16-element monoid is verified, the result completes the classification for this natural monoid in the same spirit as the Ponizovskii–Putcha–Ringel theorem for transformation monoids. The sufficient condition itself may be of independent use for other monoids.

major comments (2)

[§3] §3 (sufficient condition): the precise statement of the criterion (e.g., the existence of a submonoid whose algebra is known to be of infinite type, or a family of modules parametrized by a curve) must be stated with all hypotheses; the current formulation appears to rely on an unstated base field or characteristic assumption that needs explicit listing.
[§4] §4 (application to binary relations on 4 points): the explicit elements or generators realizing the infinite-type configuration inside the monoid must be listed together with their composition table (or at least the relevant products) so that the reader can confirm the embedding; without this verification the application of the criterion is not checkable and the central claim remains unproven.

minor comments (2)

[Abstract] The abstract announces the result but supplies no indication of the method (sufficient condition or explicit construction), which is unhelpful for readers.
[Introduction] Notation for the monoid of binary relations (usually denoted B_4 or Rel_4) and for the representation category should be fixed at the first use and used consistently.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which will help improve the clarity and verifiability of our results. We address each major comment below and will incorporate the suggested changes in the revised version.

read point-by-point responses

Referee: [§3] §3 (sufficient condition): the precise statement of the criterion (e.g., the existence of a submonoid whose algebra is known to be of infinite type, or a family of modules parametrized by a curve) must be stated with all hypotheses; the current formulation appears to rely on an unstated base field or characteristic assumption that needs explicit listing.

Authors: We agree that the statement of the sufficient condition requires full precision. In the revised manuscript, we will explicitly formulate the criterion, stating all hypotheses including that the base field k is algebraically closed (as is standard for representation type questions over finite-dimensional algebras) and noting that there is no characteristic restriction in the argument. The criterion proceeds by exhibiting a submonoid whose monoid algebra is known to be of infinite representation type (via a family of modules parametrized by a curve or an explicit infinite family of indecomposables), and we will clarify the precise transfer of infinite type to the larger algebra. revision: yes
Referee: [§4] §4 (application to binary relations on 4 points): the explicit elements or generators realizing the infinite-type configuration inside the monoid must be listed together with their composition table (or at least the relevant products) so that the reader can confirm the embedding; without this verification the application of the criterion is not checkable and the central claim remains unproven.

Authors: We accept this point and will make the verification explicit. In the revised version of §4, we will list the specific elements (or a small set of generators) inside the monoid of binary relations on a 4-element set that realize the infinite-type configuration from §3. We will also tabulate the relevant products under relational composition that establish the required submonoid structure or relations, allowing direct confirmation that the criterion applies. While a complete 65536-by-65536 composition table is impractical, the minimal data needed to check the embedding will be provided in full. revision: yes

Circularity Check

0 steps flagged

No circularity; result rests on external citations and a new sufficient condition

full rationale

The abstract cites independent prior work (Ponizovskii–Putcha–Ringel) for the transformation monoid and states that the present paper supplies a similar result plus a sufficient condition for infinite representation type. No equations, fitted parameters, self-citations used as load-bearing premises, or renamings of known results appear in the supplied text. The skeptic concern about verifying the embedding of a configuration is a standard verification step, not a reduction of the claim to its own inputs by construction. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, background axioms, or new entities introduced in the proof.

pith-pipeline@v0.9.0 · 5343 in / 970 out tokens · 66953 ms · 2026-05-10T19:05:20.459887+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

[1]

Journal of Soviet Mathematics52, 3170–3178 (1990) https://doi.org/10

Ponizovskii, I.S.: Some examples of semigroup algebras of finite representation type. Journal of Soviet Mathematics52, 3170–3178 (1990) https://doi.org/10. 1007/BF02342937

work page 1990
[2]

highest weight cate- gories and quivers

Putcha, M.S.: Complex representations of finite monoids ii. highest weight cate- gories and quivers. Journal of Algebra205(1), 53–76 (1998) https://doi.org/10. 1006/jabr.1997.7395

work page arXiv 1998
[3]

Semigroup Forum61, 429–434 (2000) https://doi.org/10.1007/PL00006040

Ringel, C.M.: The representation type of the full transforamtion semigroupt 4. Semigroup Forum61, 429–434 (2000) https://doi.org/10.1007/PL00006040

work page doi:10.1007/pl00006040 2000
[4]

Master’s thesis, Uppsala University, Algebra and Geometry (2020)

P´ erez Manr´ ıquez, A.: Quivers for semigroup algebras of binary relations of small rank. Master’s thesis, Uppsala University, Algebra and Geometry (2020). https: //urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-411481

work page 2020
[5]

Journal of Algebra521, 405– 451 (2019) https://doi.org/10.1016/j.jalgebra.2018.11.036

Guillaume, C.: Generalized correspondence functors. Journal of Algebra521, 405– 451 (2019) https://doi.org/10.1016/j.jalgebra.2018.11.036

work page doi:10.1016/j.jalgebra.2018.11.036 2019
[6]

Algebras and Representation Theory , ISBN =

Erdmann, K., Holm, T.: Representation type. In: Algebras and Representation Theory, 1st ed. 2018. edn., pp. 143–162. Springer, Cham (2018). https://doi.org/ 10.1007/978-3-319-91998-0 8

work page doi:10.1007/978-3-319-91998-0 2018
[7]

Lin- ear Algebra and its Applications37, 119–134 (1981) https://doi.org/10.1016/ 0024-3795(81)90172-5

de Caen, D., Gregory, D.A.: Primes in the semigroup of boolean matrices. Lin- ear Algebra and its Applications37, 119–134 (1981) https://doi.org/10.1016/ 0024-3795(81)90172-5

work page 1981
[8]

Steinberg, B.: Representation Theory of Finite Monoids, 1st ed. 2016. edn. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-43932-7 6

work page doi:10.1007/978-3-319-43932-7 2016

[1] [1]

Journal of Soviet Mathematics52, 3170–3178 (1990) https://doi.org/10

Ponizovskii, I.S.: Some examples of semigroup algebras of finite representation type. Journal of Soviet Mathematics52, 3170–3178 (1990) https://doi.org/10. 1007/BF02342937

work page 1990

[2] [2]

highest weight cate- gories and quivers

Putcha, M.S.: Complex representations of finite monoids ii. highest weight cate- gories and quivers. Journal of Algebra205(1), 53–76 (1998) https://doi.org/10. 1006/jabr.1997.7395

work page arXiv 1998

[3] [3]

Semigroup Forum61, 429–434 (2000) https://doi.org/10.1007/PL00006040

Ringel, C.M.: The representation type of the full transforamtion semigroupt 4. Semigroup Forum61, 429–434 (2000) https://doi.org/10.1007/PL00006040

work page doi:10.1007/pl00006040 2000

[4] [4]

Master’s thesis, Uppsala University, Algebra and Geometry (2020)

P´ erez Manr´ ıquez, A.: Quivers for semigroup algebras of binary relations of small rank. Master’s thesis, Uppsala University, Algebra and Geometry (2020). https: //urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-411481

work page 2020

[5] [5]

Journal of Algebra521, 405– 451 (2019) https://doi.org/10.1016/j.jalgebra.2018.11.036

Guillaume, C.: Generalized correspondence functors. Journal of Algebra521, 405– 451 (2019) https://doi.org/10.1016/j.jalgebra.2018.11.036

work page doi:10.1016/j.jalgebra.2018.11.036 2019

[6] [6]

Algebras and Representation Theory , ISBN =

Erdmann, K., Holm, T.: Representation type. In: Algebras and Representation Theory, 1st ed. 2018. edn., pp. 143–162. Springer, Cham (2018). https://doi.org/ 10.1007/978-3-319-91998-0 8

work page doi:10.1007/978-3-319-91998-0 2018

[7] [7]

Lin- ear Algebra and its Applications37, 119–134 (1981) https://doi.org/10.1016/ 0024-3795(81)90172-5

de Caen, D., Gregory, D.A.: Primes in the semigroup of boolean matrices. Lin- ear Algebra and its Applications37, 119–134 (1981) https://doi.org/10.1016/ 0024-3795(81)90172-5

work page 1981

[8] [8]

Steinberg, B.: Representation Theory of Finite Monoids, 1st ed. 2016. edn. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-43932-7 6

work page doi:10.1007/978-3-319-43932-7 2016