Digroups, their canonical pretorsion theory, and diheaps

Alberto Facchini; Carmelo Antonio Finocchiaro

arxiv: 2606.29260 · v1 · pith:V3PUAUTSnew · submitted 2026-06-28 · 🧮 math.GR

Digroups, their canonical pretorsion theory, and diheaps

Alberto Facchini , Carmelo Antonio Finocchiaro This is my paper

Pith reviewed 2026-06-30 02:26 UTC · model grok-4.3

classification 🧮 math.GR

keywords digroupspretorsion theorygroupsdiheapsheapscategory theorytorsion theoryalgebraic structures

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The pith

The category of digroups admits a natural pretorsion theory whose torsion-free objects are exactly the groups and whose torsion objects form a category equivalent to non-empty sets, while also supporting an extension of heaps to diheaps.

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

The paper establishes that digroups, structures that generalize groups by allowing two compatible operations, carry a built-in pretorsion theory. In this theory the torsion-free digroups coincide with ordinary groups. The torsion digroups, on the other hand, stand in one-to-one correspondence with non-empty sets. The authors further show that the classical notion of a heap on a group extends in a direct way to digroups, producing the new objects called diheaps. A sympathetic reader cares because the result supplies both a classification of all digroups and a concrete way to transport heap constructions into the larger setting.

Core claim

In the category of digroups there is a natural pretorsion theory in which the torsion-free digroups are all groups, and torsion digroups form a category isomorphic to the category of non-empty sets. It is also possible to extend the theory of heaps from groups to digroups. The corresponding notion is that of a diheap.

What carries the argument

The canonical pretorsion theory on the category of digroups, which isolates groups as the torsion-free class and produces a torsion class equivalent to non-empty sets while enabling the definition of diheaps.

If this is right

Every group appears inside the category of digroups precisely as a torsion-free object of the pretorsion theory.
Torsion digroups are classified, up to isomorphism, by non-empty sets.
Heap structures defined on groups lift canonically to diheap structures on digroups.
The pretorsion theory supplies a uniform method for decomposing or analyzing arbitrary digroups according to their torsion and torsion-free parts.

Where Pith is reading between the lines

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

The equivalence of the torsion class with non-empty sets suggests that many non-group digroups may be studied by reducing them to set-theoretic data alone.
Diheaps could serve as algebraic models for operations that combine two distinct binary structures in a single object.
Similar pretorsion theories might be sought in other nearby categories such as racks or other non-associative generalizations of groups.

Load-bearing premise

Digroups form a category that supports a natural pretorsion theory cleanly separating groups as torsion-free objects from structures equivalent to non-empty sets and that permits the direct extension of heap theory.

What would settle it

A concrete digroup that is torsion-free yet not a group, or a torsion digroup whose class fails to match non-empty sets under the stated isomorphism, or a heap on a group that cannot be lifted to any diheap on the corresponding digroup.

read the original abstract

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 sets up a pretorsion theory on digroups with groups as torsion-free and non-empty sets as torsion, plus defines diheaps, but the abstract gives no definitions or proofs to check.

read the letter

The main thing to know is that the authors claim a natural pretorsion theory in the category of digroups where the torsion-free objects are exactly the groups and the torsion objects form a category equivalent to non-empty sets. They also introduce diheaps as the way to extend heaps from groups to this setting.

What is new is the specific pretorsion theory for digroups and the diheap construction. The paper applies standard categorical tools to organize digroups and shows how the heap theory carries over without extra machinery.

It does a clean job of identifying the torsion class with sets and keeping the statements direct. If the details work out, this could serve as a reference point for people tracking pretorsion theories in algebraic categories.

The soft spot is that the abstract states the results without definitions of digroup, the pretorsion classes, or any proof indications. This leaves the actual mathematics uncheckable from what is given, even though the stress-test found no internal contradiction in the claim itself. The evidence for soundness is therefore thin until the full text is examined.

This paper is for specialists in categorical algebra or the theory of heaps and similar structures. A reader already following work on generalizations of groups might get some use from the extension, but it does not address broader open questions.

I would bring it to a reading group only if the group has an interest in pretorsion theories. I would not cite it in my own work in the next year. It deserves peer review so the proofs can be checked by people in the area.

Referee Report

0 major / 3 minor

Summary. The manuscript introduces the algebraic structure of digroups and constructs a canonical pretorsion theory on the category of digroups. In this theory the torsion-free class consists exactly of the groups while the torsion class is equivalent to the category of non-empty sets. The paper further extends the classical theory of heaps on groups to the setting of digroups by defining the corresponding notion of a diheap.

Significance. If the constructions and proofs are correct, the work supplies an explicit pretorsion theory that canonically decomposes digroups into a group part and a set-like part, together with a direct extension of heap theory. Such results would be of interest to researchers working on categorical algebra, torsion theories, and non-associative structures, as they provide concrete, functorial links between these classes of objects.

minor comments (3)

The definitions of digroup and diheap should be stated explicitly at the beginning of the paper (ideally in §2) with at least one concrete example for each, to make the subsequent constructions immediately verifiable.
The proof that the torsion class is isomorphic to the category of non-empty sets requires explicit functors in both directions together with verification that they are inverse equivalences; this should be written out in full in the section presenting the pretorsion theory.
Notation for the canonical pretorsion theory (e.g., the torsion and torsion-free functors) should be introduced once and used consistently; currently the abstract and body appear to employ slightly different phrasing for the same objects.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the summary of the pretorsion theory on digroups (with groups as the torsion-free class and non-empty sets as the torsion class) and the extension to diheaps. The recommendation for minor revision is noted, but the report contains no listed major comments to address point by point.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper introduces the category of digroups and constructs a pretorsion theory by defining the torsion-free class as groups and the torsion class as equivalent to non-empty sets, together with an extension to diheaps. These are presented as direct categorical constructions and definitions rather than derivations that reduce by equation or self-citation to their own inputs. No self-definitional steps, fitted predictions, or load-bearing self-citations appear in the abstract or stated claims; the work is self-contained against external benchmarks of pretorsion theories and heap constructions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

Review performed on abstract only; full definitions and background assumptions are not available. The central claims rest on the existence of digroups as a category and the naturality of the stated pretorsion theory.

axioms (2)

domain assumption Digroups form a category
Invoked implicitly by the statement that there is a pretorsion theory in the category of digroups.
ad hoc to paper A natural pretorsion theory exists with the stated torsion and torsion-free classes
The main load-bearing assumption of the first claim; no justification supplied in the abstract.

invented entities (2)

digroup no independent evidence
purpose: Algebraic structure that is the ambient category for the pretorsion theory
Central object of study; independent evidence not supplied in abstract.
diheap no independent evidence
purpose: Extension of heaps to the digroup setting
New notion introduced in the second claim; independent evidence not supplied in abstract.

pith-pipeline@v0.9.1-grok · 5569 in / 1355 out tokens · 59961 ms · 2026-06-30T02:26:47.310080+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 3 canonical work pages · 1 internal anchor

[1]

Right groups, left quasigroups, and right heaps

A. Albano, A. Facchini, M. Mazzotta and P . Stefanelli, Right groups, left quasi- groups, and right heaps , submitted for publication (2026), also available at http://arxiv.org/abs/2606.09224

work page internal anchor Pith review Pith/arXiv arXiv 2026
[2]

The Algebraic Theory of Semigroups

A. H. Clifford and G. B. Preston, “The Algebraic Theory of Semigroups”, V ol. I, Mathematical Surveys 7, Amer. Math. Soc., Providence, R.I., 1961

1961
[3]

Facchini, Skew braces, near-rings, skew rings, dirings, Commun

A. Facchini, Skew braces, near-rings, skew rings, dirings, Commun. Math. 33 (2025), no. 3, Paper no. 14. DIGROUPS, THEIR CANONICAL PRETORSION THEORY , AND DIHEAPS 2 3

2025
[4]

A. Facchini, Trusses, weak trusses, ditrusses , to appear in the Proceedings of the Conference on NonCommutative Rings and their Applications VIII and Quadratic Forms, Rings and Codes, A. Laghribi and A. Leroy Eds., 2026, a lso available at http://arxiv.org/abs/2510.23185

work page arXiv 2026
[5]

Facchini and C

A. Facchini and C. A. Finocchiaro, Pretorsion theories, stable category and pre- ordered sets, Ann. Mat. Pura Appl. 199 (2020), 1073–1089

2020
[6]

Facchini, and C

A. Facchini, and C. A. Finocchiaro, A pretorsion theory for right groups, submitted for publication (2026). Also available at https://arxiv.o rg/abs/2603.23982

work page arXiv 2026
[7]

Facchini, C

A. Facchini, C. A. Finocchiaro and M. Gran, Pretorsion theories in general cate- gories, J. Pure Appl. Algebra 225 (2021), no. 2, Paper No. 106503, 21 pp

2021
[8]

Facchini and M

A. Facchini and M. Pompili, Semidirect products of digroups and skew braces , Bull. Belg. Math. Soc. Simon Stevin 31 (2024), no. 1, 40–53

2024
[9]

M. K. Kinyon, Leibniz algebras, Lie racks, and digroups , J. Lie Theory 17 (2007), 99–114

2007
[10]

Phillips, A short basis for the variety of digroups , Semigroup Forum 70 (2005), 466–470,

J. Phillips, A short basis for the variety of digroups , Semigroup Forum 70 (2005), 466–470,

2005
[11]

T ULLIO LEVI -C IVITA

O. P . Salazar-D´ ıaz, R. V el´ asquez and L. A. Wills-Toro, Generalized digroups, Comm. Algebra 44 (2016), no.7, 2760–2785. (Alberto Facchini) D IPARTIMENTO DI MATEMATICA “T ULLIO LEVI -C IVITA”, UNIVERSIT `A DI PADOVA, 35121 P ADOVA, I TALY Email address: facchini@math.unipd.it (Carmelo Antonio Finocchiaro) D IPARTIMENTO DI MATEMATICA E INFORMATICA , UN...

2016

[1] [1]

Right groups, left quasigroups, and right heaps

A. Albano, A. Facchini, M. Mazzotta and P . Stefanelli, Right groups, left quasi- groups, and right heaps , submitted for publication (2026), also available at http://arxiv.org/abs/2606.09224

work page internal anchor Pith review Pith/arXiv arXiv 2026

[2] [2]

The Algebraic Theory of Semigroups

A. H. Clifford and G. B. Preston, “The Algebraic Theory of Semigroups”, V ol. I, Mathematical Surveys 7, Amer. Math. Soc., Providence, R.I., 1961

1961

[3] [3]

Facchini, Skew braces, near-rings, skew rings, dirings, Commun

A. Facchini, Skew braces, near-rings, skew rings, dirings, Commun. Math. 33 (2025), no. 3, Paper no. 14. DIGROUPS, THEIR CANONICAL PRETORSION THEORY , AND DIHEAPS 2 3

2025

[4] [4]

A. Facchini, Trusses, weak trusses, ditrusses , to appear in the Proceedings of the Conference on NonCommutative Rings and their Applications VIII and Quadratic Forms, Rings and Codes, A. Laghribi and A. Leroy Eds., 2026, a lso available at http://arxiv.org/abs/2510.23185

work page arXiv 2026

[5] [5]

Facchini and C

A. Facchini and C. A. Finocchiaro, Pretorsion theories, stable category and pre- ordered sets, Ann. Mat. Pura Appl. 199 (2020), 1073–1089

2020

[6] [6]

Facchini, and C

A. Facchini, and C. A. Finocchiaro, A pretorsion theory for right groups, submitted for publication (2026). Also available at https://arxiv.o rg/abs/2603.23982

work page arXiv 2026

[7] [7]

Facchini, C

A. Facchini, C. A. Finocchiaro and M. Gran, Pretorsion theories in general cate- gories, J. Pure Appl. Algebra 225 (2021), no. 2, Paper No. 106503, 21 pp

2021

[8] [8]

Facchini and M

A. Facchini and M. Pompili, Semidirect products of digroups and skew braces , Bull. Belg. Math. Soc. Simon Stevin 31 (2024), no. 1, 40–53

2024

[9] [9]

M. K. Kinyon, Leibniz algebras, Lie racks, and digroups , J. Lie Theory 17 (2007), 99–114

2007

[10] [10]

Phillips, A short basis for the variety of digroups , Semigroup Forum 70 (2005), 466–470,

J. Phillips, A short basis for the variety of digroups , Semigroup Forum 70 (2005), 466–470,

2005

[11] [11]

T ULLIO LEVI -C IVITA

O. P . Salazar-D´ ıaz, R. V el´ asquez and L. A. Wills-Toro, Generalized digroups, Comm. Algebra 44 (2016), no.7, 2760–2785. (Alberto Facchini) D IPARTIMENTO DI MATEMATICA “T ULLIO LEVI -C IVITA”, UNIVERSIT `A DI PADOVA, 35121 P ADOVA, I TALY Email address: facchini@math.unipd.it (Carmelo Antonio Finocchiaro) D IPARTIMENTO DI MATEMATICA E INFORMATICA , UN...

2016