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arxiv: 2606.09224 · v1 · pith:DMDKJLUZnew · submitted 2026-06-08 · 🧮 math.GR

Right groups, left quasigroups, and right heaps

Andrea Albano , Alberto Facchini , Marzia Mazzotta , Paola Stefanelli This is my paper

Pith reviewed 2026-06-27 14:57 UTC · model grok-4.3

classification 🧮 math.GR
keywords right groupsleft quasigroupsright heapsYang-Baxter equationskew left trussesset-theoretic solutionssemigroups
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The pith

Right groups allow a natural definition of right heaps that produces an analogue of skew left trusses for Yang-Baxter solutions.

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

The paper develops the theory of heaps by replacing ordinary groups with right groups, which are semigroups in which every equation a·x=b has a unique solution x. This replacement yields a direct definition of right heap. The same starting point works when groups are further relaxed to left quasigroups. The explicit motivation is the study of left non-degenerate set-theoretic solutions of the Yang-Baxter equation, and the construction supplies a structure that stands as the analogue of skew left trusses.

Core claim

A right group is a semigroup (S,·) such that for every a,b in S there is a unique x in S satisfying a·x=b. Beginning with such a right group instead of a group produces a right heap; the same development is possible when the starting object is only a left quasigroup. The resulting right-heap structures supply an analogue of skew left trusses that is adapted to left non-degenerate set-theoretic solutions of the Yang-Baxter equation.

What carries the argument

The right heap obtained directly from a right group, whose unique right divisibility replaces the two-sided divisibility of ordinary heaps and thereby generates the truss analogue.

If this is right

  • A natural definition of right heap follows immediately from the definition of right group.
  • Part of the heap theory can be developed when the starting object is only a left quasigroup.
  • The right-heap construction produces a direct analogue of skew left trusses.
  • The analogue applies specifically to left non-degenerate set-theoretic solutions of the Yang-Baxter equation.

Where Pith is reading between the lines

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

  • Right heaps may import semigroup-theoretic results into the classification of Yang-Baxter solutions.
  • The left-quasigroup starting point could enlarge the pool of algebraic objects that encode such solutions.
  • Similar divisibility-based replacements might apply to other non-associative structures arising from combinatorial equations.

Load-bearing premise

The algebraic structures obtained by replacing groups with right groups or left quasigroups remain sufficiently rich and natural to advance the study of left non-degenerate set-theoretic Yang-Baxter solutions.

What would settle it

An explicit left non-degenerate set-theoretic solution of the Yang-Baxter equation whose associated right-multiplication structure cannot be equipped with a right heap or the corresponding truss analogue.

read the original abstract

A right group is a semigroup $(S,\cdot)$ in which, for every $a,b\in S$, there is a unique $x\in S$ such that $a\cdot x=b$. In this article, we develop the theory of heaps starting not from groups, but from right groups. We thus get a natural definition of right heap. It is even possible to develop part of the theory starting from a left quasigroup, which is the non-associative analogue of a right group. Our motivation for this study is the investigation of left non-degenerate set-theoretic solutions of the Yang--Baxter equation. Thus, we are led to an analogue of the skew left trusses introduced by T.~Brzezi\'nski.

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

1 major / 2 minor

Summary. The paper develops part of the theory of heaps by replacing the usual group axioms with those of right groups (semigroups in which left multiplication is bijective), thereby obtaining a natural definition of a right heap; it further shows that an analogous development is possible starting from left quasigroups and that the resulting structures yield an analogue of skew left trusses relevant to left non-degenerate set-theoretic solutions of the Yang-Baxter equation.

Significance. If the constructions are carried through as claimed, the work supplies a systematic definitional extension of heap theory that may furnish new algebraic tools for the study of set-theoretic Yang-Baxter solutions. The explicit motivation via right groups and left quasigroups, together with the production of a truss analogue, constitutes a coherent contribution to the algebraic literature on these structures.

major comments (1)
  1. [final section] The central claim that the right-heap construction produces a usable analogue of skew left trusses for YBE solutions is load-bearing; however, the manuscript only sketches the correspondence without exhibiting an explicit functor or a concrete left non-degenerate solution arising from a right heap (cf. the final section). A single worked example or a precise statement of the induced map would make the motivation fully operational.
minor comments (2)
  1. Notation for the right-group operation and the induced heap operation should be introduced once and used consistently; the current alternation between · and the heap bracket risks confusion in longer proofs.
  2. The statement that left-quasigroup axioms suffice for part of the theory would benefit from an explicit list of which heap axioms survive and which fail, perhaps in a short table.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation and the constructive suggestion regarding the final section. We address the single major comment below.

read point-by-point responses

  1. Referee: [final section] The central claim that the right-heap construction produces a usable analogue of skew left trusses for YBE solutions is load-bearing; however, the manuscript only sketches the correspondence without exhibiting an explicit functor or a concrete left non-degenerate solution arising from a right heap (cf. the final section). A single worked example or a precise statement of the induced map would make the motivation fully operational.

    Authors: We agree that the correspondence between right heaps and the skew left truss analogue is presented at a sketch level in the final section and that an explicit illustration would strengthen the link to left non-degenerate set-theoretic Yang-Baxter solutions. In the revised version we have inserted a short worked example (a concrete right heap on a four-element set) together with the explicit map sending the right heap to the associated left non-degenerate solution; we have also added a precise statement of the induced functorial correspondence. These additions make the motivation fully operational while preserving the original length and focus of the section. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper develops algebraic structures by explicit definitions: a right heap is introduced directly from the given definition of a right group (a semigroup with unique solvability for a·x=b), and an extension to left quasigroups is likewise definitional. No fitted parameters, statistical predictions, or self-citation chains are invoked to justify the central constructions. The motivation regarding Yang-Baxter solutions is stated explicitly as context rather than as a derived claim that reduces to prior fitted data. The derivation chain consists of structural definitions and properties that are independent of the target results; the paper is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities beyond the definitional move itself.

axioms (1)
  • standard math Standard axioms of semigroups and quasigroups are assumed without restatement.
    The definitions of right group and left quasigroup rely on the usual associative or non-associative multiplication axioms.
invented entities (1)
  • right heap no independent evidence
    purpose: Heap-like structure built on right groups rather than groups.
    New object introduced to support the Yang-Baxter application; no independent evidence supplied in abstract.

pith-pipeline@v0.9.1-grok · 5654 in / 1204 out tokens · 17132 ms · 2026-06-27T14:57:22.145704+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Digroups, their canonical pretorsion theory, and diheaps

    math.GR 2026-06 unverdicted novelty 6.0

    Establishes a canonical pretorsion theory for digroups with groups as torsion-free and non-empty sets as torsion objects, and defines diheaps as an extension of heaps.

Reference graph

Works this paper leans on

14 extracted references · 1 linked inside Pith · cited by 1 Pith paper

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