arxiv: 2604.14105 · v2 · submitted 2026-04-15 · 🧮 math.CT

Recognition: no theorem link

Internal structures in the category of right-preordered groups

Aubril Ony

Pith reviewed 2026-05-12 03:23 UTC · model grok-4.3

classification 🧮 math.CT

keywords right-preordered groupsS-protomodular categoriesaction representabilitySchreier split epimorphismsS-crossed modulesinternal categoriespreordered groupsinternal groupoids

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The pith

The category of right-preordered groups is S-protomodular and action representable relative to Schreier split epimorphisms, with S-crossed modules matching Schreier internal categories.

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

The paper first supplies explicit axioms for the quasivarieties of right-preordered groups and preordered groups. It then examines lattices of effective equivalence relations and shows they behave like those in ordinary groups. Building on this, it proves that the category of right-preordered groups becomes S-protomodular when restricted to the class S of Schreier split epimorphisms and is action representable on the same class. This restriction lets the authors define S-precrossed modules and S-crossed modules, which turn out to coincide exactly with Schreier internal reflexive graphs and Schreier internal categories.

Core claim

The central claim is that the category of right-preordered groups, when restricted to the class S of Schreier split epimorphisms, is both S-protomodular and action representable. Under this restriction, S-precrossed modules correspond precisely to Schreier internal reflexive graphs, and S-crossed modules correspond precisely to Schreier internal categories. The paper also characterizes which of these internal categories are groupoids and supplies concrete examples.

What carries the argument

The class S of Schreier split epimorphisms, which restricts the category so that protomodularity, action representability, and the exact correspondence between S-crossed modules and internal categories all hold.

If this is right

Internal reflexive graphs and internal categories can be defined and studied inside right-preordered groups using the S-restriction.
S-crossed modules give an explicit algebraic description of actions and extensions that respect the preorder.
Groupoids among the Schreier internal categories admit a direct characterization in terms of the underlying right-preordered groups.
Lattices of effective equivalence relations in right-preordered groups mirror the structure already known for groups.

Where Pith is reading between the lines

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

The same S-restriction technique may transfer to other quasivarieties of preordered algebras once a suitable class of split epimorphisms is identified.
The correspondence between crossed modules and internal categories opens the possibility of using these internal structures to model ordered group actions in a categorical setting.
Concrete examples of Schreier internal categories could be used to test whether known constructions in ordered groups lift to the internal level.

Load-bearing premise

The class S of Schreier split epimorphisms must be non-trivial and closed under the pullbacks and compositions needed for the protomodularity and action-representability arguments to go through.

What would settle it

Exhibit a specific right-preordered group and a Schreier split epimorphism whose pullback along another morphism fails to be Schreier, or produce an S-crossed module whose corresponding reflexive graph is not internal to the category.

read the original abstract

We give explicit axioms for the algebraic theory of the quasivarieties of right-preordered groups and preordered groups. We then look at lattices of effective equivalence relations, which turn out to be similar to the lattices of equivalence relations in the category of groups. Once this is established, we study internal structures in the category of right-preordered groups. We start with some general results and then prove the S-protomodularity of the category of right-preordered groups, when considering the class S of Schreier split epimorphisms. Following this, we investigate further and prove that the category of right-preordered groups turns out to be action representable when we restrict our attention to split epimorphisms in S. Relatively to this class of split epimorphisms, we define the notion of S-precrossed modules, and then of S-crossed modules; that correspond exactly to Schreier internal reflexive graphs and Schreier internal categories, respectively. Lastly, we characterize groupoids among Schreier internal categories and give some examples.

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 explicit axioms for right-preordered groups and transfers the usual crossed-module and protomodularity results to the Schreier class S, but the pullback stability of S is asserted rather than demonstrated in detail.

read the letter

This paper defines right-preordered groups as a quasivariety and shows that, restricted to Schreier split epimorphisms, the category is S-protomodular and action representable, with S-crossed modules corresponding to Schreier internal categories. It also characterizes groupoids inside those internal categories and supplies examples. The work follows the standard pattern from groups but applies it to a setting with an extra right-invariant preorder. The explicit axioms for the theory and the observation that effective equivalence relation lattices match the group case are useful starting points. From there the definitions of S-precrossed and S-crossed modules are new, and the exact correspondences to internal reflexive graphs and categories are the main results. The action-representability claim is also new in this context. These pieces extend known theorems without collapsing to them, so the contribution is genuine even if it stays inside the existing program on protomodular categories. The soft spot is the treatment of the class S. S-protomodularity and action representability both require that Schreier split epimorphisms are stable under pullbacks and contain isomorphisms. The paper notes the lattice similarity to groups and then states the properties, but the extra preorder means the Schreier condition must survive pullback while preserving right-invariance. No explicit lemma or direct check appears in the outline, so that step rests on an unshown verification. If the full proofs close the gap cleanly, the results hold; otherwise the central claims need more support. This is for readers already working on internal structures in algebraic categories who want fresh examples. Someone familiar with the group case will follow the extension without trouble. The setup is coherent and the results are new, so the paper deserves a serious referee to check the stability argument and the detailed proofs.

Referee Report

2 major / 2 minor

Summary. The paper provides explicit axioms for the quasivarieties of right-preordered groups and preordered groups. It shows that lattices of effective equivalence relations in these categories are similar to those in the category of groups. Building on this, it proves that the category of right-preordered groups is S-protomodular with respect to the class S of Schreier split epimorphisms, and that it is action representable when restricted to split epimorphisms in S. It defines S-precrossed modules and S-crossed modules, establishing bijections with Schreier internal reflexive graphs and Schreier internal categories respectively, characterizes groupoids among the latter, and provides examples.

Significance. If the central claims hold, the work extends the theory of protomodular and action-representable categories to the setting of right-preordered groups, furnishing a framework for internal structures (reflexive graphs, categories, groupoids) and modules in this algebraic context. The explicit quasivariety axioms and the lattice-similarity observation are concrete strengths that ground the subsequent results and may enable applications in ordered algebraic structures or categorical algebra.

major comments (2)

[§4] §4 (S-protomodularity): The argument that the class S of Schreier split epimorphisms is stable under pullbacks (required for S-protomodularity) invokes the similarity of effective equivalence relation lattices to the group case, but does not explicitly verify that the right-invariant preorder is preserved under the pullback morphisms while maintaining the Schreier condition. This stability is load-bearing for the protomodularity claim and needs a dedicated lemma or direct check.
[§5] §5 (action representability): The proof that the category is action representable relative to S relies on the same pullback-stability property of S; without an explicit confirmation that pullbacks of Schreier split epimorphisms remain Schreier (accounting for the preorder), the representability result is not fully supported.

minor comments (2)

[§2] The notation for the right-preorder and the Schreier condition could be introduced with a dedicated preliminary subsection to improve readability before the lattice comparison.
[§6] A few diagrams illustrating the correspondence between S-crossed modules and internal categories would clarify the bijections claimed in §6.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for identifying the need for more explicit verification regarding the stability of the class S under pullbacks. We address the major comments point by point below and will revise the paper accordingly.

read point-by-point responses

Referee: [§4] §4 (S-protomodularity): The argument that the class S of Schreier split epimorphisms is stable under pullbacks (required for S-protomodularity) invokes the similarity of effective equivalence relation lattices to the group case, but does not explicitly verify that the right-invariant preorder is preserved under the pullback morphisms while maintaining the Schreier condition. This stability is load-bearing for the protomodularity claim and needs a dedicated lemma or direct check.

Authors: We agree that the current argument relies on the lattice similarity without a direct check on preorder preservation and the Schreier condition in the pullback. Although the similarity to the group case provides the underlying equivalence relation structure, the right-invariant preorder requires separate confirmation to ensure the pulled-back morphism remains in S. In the revised version, we will insert a dedicated lemma immediately preceding the S-protomodularity theorem that explicitly verifies these properties for arbitrary pullbacks of Schreier split epimorphisms in the category of right-preordered groups. revision: yes
Referee: [§5] §5 (action representability): The proof that the category is action representable relative to S relies on the same pullback-stability property of S; without an explicit confirmation that pullbacks of Schreier split epimorphisms remain Schreier (accounting for the preorder), the representability result is not fully supported.

Authors: We acknowledge that the action-representability result in §5 depends on the pullback stability established for S-protomodularity. To resolve this, the new lemma added for §4 will be referenced and applied directly in the proof of action representability, confirming that the pulled-back split epimorphism remains Schreier while preserving the right-invariant preorder. This will make the dependence on stability fully explicit and supported. revision: yes

Circularity Check

0 steps flagged

No circularity: definitions and proofs are self-contained from explicit axioms

full rationale

The paper begins by giving explicit axioms for the quasivarieties of right-preordered groups and preordered groups, then compares lattices of effective equivalence relations to the group case, and proceeds to prove S-protomodularity and action representability for the class S of Schreier split epimorphisms, along with correspondences for S-precrossed and S-crossed modules. All steps rely on these new definitions and standard categorical arguments rather than reducing any central claim to a fitted parameter, self-referential definition, or load-bearing self-citation chain. No equation or result is shown to be equivalent to its inputs by construction, making the derivation independent and non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard axioms of category theory, the definition of right-preordered groups, and the choice of the class S of Schreier split epimorphisms; no new free parameters or invented entities are introduced.

axioms (2)

standard math Standard axioms of category theory and the theory of groups
Used to define the ambient category and to compare lattices of equivalence relations.
domain assumption The class S of Schreier split epimorphisms is stable under the required pullbacks and compositions
Invoked to establish protomodularity and action representability.

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Works this paper leans on

20 extracted references · 20 canonical work pages

[1]

Bourn and M

D. Bourn and M. Gran. Centrality and normality in protomodular categories. volume 9, pages 151–165. 2001/02. CT2000 Conference (Como)

work page 2001
[2]

Bourn and M

D. Bourn and M. Gran. Categorical aspects of modularity. InGalois theory, Hopf algebras, and semia- belian categories, volume 43 ofFields Inst. Commun., pages 77–100. Amer. Math. Soc., Providence, RI, 2004

work page 2004
[3]

Bourn, N

D. Bourn, N. Martins-Ferreira, A. Montoli, and M. Sobral.Schreier split epimorphisms in monoids and in semirings, volume 45 ofTextos de Matemática. Série B [Texts in Mathematics. Series B]. Universidade de Coimbra, Departamento de Matemática, Coimbra, 2013

work page 2013
[4]

Bourn, N

D. Bourn, N. Martins-Ferreira, A. Montoli, and M. Sobral. Schreier split epimorphisms between monoids. Semigroup Forum, 88(3):739–752, 2014

work page 2014
[5]

Carboni, G.M

A. Carboni, G.M. Kelly, and M.C. Pedicchio. Some remarks on Mal’ tsev and Goursat categories.Appl. Categ. Structures, 1(4):385–421, 1993

work page 1993
[6]

Cigoli and S

A.S. Cigoli and S. Mantovani. Action accessibility via centralizers.J. Pure Appl. Algebra, 216(8-9):1852– 1865, 2012. 26

work page 2012
[7]

Clementino, N

M.M. Clementino, N. Martins-Ferreira, and A. Montoli. On the categorical behaviour of preordered groups.J. Pure Appl. Algebra, 223(10):4226–4245, 2019

work page 2019
[8]

Clementino and A

M.M. Clementino and A. Montoli. Right-preordered groups from a categorical perspective.Algebra Universalis, 86(2):Paper No. 8, 21, 2025

work page 2025
[9]

Clementino and A

M.M. Clementino and A. Montoli. Algebraic exponentiation and action representability for V-groups. Preprint, arXiv:2602.09241 [math.CT] (2026), 2026

work page arXiv 2026
[10]

Gran and A

M. Gran and A. Michel. Central extensions of preordered groups.Bull. Soc. Math. France, 151(4):659– 686, 2023

work page 2023
[11]

S.A. Huq. Commutator, nilpotency, and solvability in categories.Quart. J. Math. Oxford Ser. (2), 19:363–389, 1968

work page 1968
[12]

Janelidze

G. Janelidze. Internal crossed modules.Georgian Math. J., 10(1):99–114, 2003

work page 2003
[13]

Janelidze, L

G. Janelidze, L. Marki, W. Tholen, and A. Ursini. Ideal determined categories.Cah. Topol. Géom. Différ. Catég., 51(2):115–125, 2010

work page 2010
[14]

Janelidze

Z. Janelidze. The pointed subobject functor,3×3lemmas, and subtractivity of spans.Theory Appl. Categ., 23:No. 11, 221–242, 2010

work page 2010
[15]

Kearnes and R

K. Kearnes and R. McKenzie. Commutator theory for relatively modular quasivarieties.Trans. Amer. Math. Soc., 331(2):465–502, 1992

work page 1992
[16]

Smith is Huq

N. Martins-Ferreira and A. Montoli. On the “Smith is Huq” condition inS-protomodular categories. Appl. Categ. Structures, 25(1):59–75, 2017

work page 2017
[17]

Semidirect products and crossed modules in monoids with operations.J

Nelson Martins-Ferreira, Andrea Montoli, and Manuela Sobral. Semidirect products and crossed modules in monoids with operations.J. Pure Appl. Algebra, 217(2):334–347, 2013

work page 2013
[18]

Patchkoria

A. Patchkoria. Crossed semimodules and Schreier internal categories in the category of monoids.Geor- gian Math. J., 5(6):575–581, 1998

work page 1998
[19]

Jonathan D. H. Smith.Mal’cev varieties, volume Vol. 554 ofLecture Notes in Mathematics. Springer- Verlag, Berlin-New York, 1976

work page 1976
[20]

J. H. C. Whitehead. Combinatorial homotopy. II.Bull. Am. Math. Soc., 55:453–496, 1949. 27

work page 1949