Sine laws on semigroups with an involutive anti-automorphism: A Levi--Civita approach via left translations

Dang Vo Phuc

arxiv: 2511.00019 · v5 · submitted 2025-10-23 · 🧮 math.GM

Sine laws on semigroups with an involutive anti-automorphism: A Levi--Civita approach via left translations

Dang Vo Phuc This is my paper

Pith reviewed 2026-05-18 05:08 UTC · model grok-4.3

classification 🧮 math.GM

keywords functional equationssemigroupsinvolutive anti-automorphismLevi-Civita methodsine lawsleft translationsconjugation identitygeneralized sine law

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The pith

A conjugation identity converts right translations to left translations, letting the Levi-Civita method recover classical sine-law properties unconditionally on semigroups with involutive anti-automorphisms.

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

The paper proves that composing with the involutive anti-automorphism σ conjugates every right translation by σ(y) into the left translation by y. This single operator identity restores the finite-dimensional invariance needed for the matrix formalism even though σ reverses order. Applied to the generalized sine law, the restored closure principle yields the dichotomy β ∈ {±1} and the parity relation f ∘ σ = β f with no further restrictions. A reader would care because the same obstruction appears in many other functional equations on semigroups; removing it enlarges the class of equations that can be solved by matrix methods.

Core claim

The central claim is the identity J R(σ(y)) J = L(y) for every y in the semigroup S, where J denotes composition with σ. This identity produces an anti-automorphic version of the Levi-Civita closure principle. When the principle is applied to the generalized sine law, the classical dichotomy β ∈ {±1} and the parity relation f ∘ σ = β f are obtained unconditionally. Under the additional natural bridge hypothesis the paper also derives the standard xy-addition law and the precise σ-transformation rule for the companion function g.

What carries the argument

The conjugation identity J R(σ(y)) J = L(y), which converts every problematic right translation into an ordinary left translation and thereby restores the invariant finite-dimensional space required by the Levi-Civita matrix method.

If this is right

The generalized sine law satisfies β ∈ {±1} with no extra conditions.
The parity relation f ∘ σ = β f holds unconditionally.
Whenever the natural bridge hypothesis is satisfied, the standard xy-addition law and the exact σ-transformation rule for g are recovered.

Where Pith is reading between the lines

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

The same conjugation device could be inserted into cosine-law or d'Alembert equations on the same class of semigroups to obtain analogous unconditional results.
Explicit matrix representations of the left-translation operators could be computed on matrix semigroups or on the semigroup of continuous functions to test the size of the invariant space in concrete cases.
If the bridge hypothesis is dropped, the addition law for g may still hold in a modified form that the conjugation identity alone does not capture.

Load-bearing premise

The natural bridge hypothesis that relates the two functions in the sine law and that holds automatically whenever a central element has nonzero value under f.

What would settle it

Direct verification of the operator identity on any concrete finite semigroup equipped with an involutive anti-automorphism; failure of the identity on even one such semigroup would falsify the central claim.

read the original abstract

Stetk\ae r's matrix (Levi--Civita) method is a powerful tool for functional equations on semigroups involving a homomorphism $\sigma$, as it yields a finite-dimensional invariant space under right translations and a corresponding matrix formalism. However, this framework collapses when $\sigma$ is an involutive anti-automorphism due to the order reversal in the right-regular action. In this paper, we overcome this obstruction at the operator level by establishing the conjugation identity: letting $J$ denote composition with $\sigma$, we prove \[ J\,R(\sigma(y))\,J=L(y)\qquad(\forall\,y\in S), \] which converts the problematic right translates into left translations. Using this left-translation approach, we obtain an anti-automorphic Levi--Civita closure principle and apply it to the generalized sine law. Remarkably, the classical dichotomy $\beta\in\{\pm1\}$ and the parity relation $f\circ\sigma=\beta f$ are recovered unconditionally. Furthermore, under a natural bridge hypothesis, which is automatically satisfied when there exists a central element $c$ with $f(c)\neq 0$, we obtain the corresponding standard $xy$-addition law and the exact $\sigma$-transformation rule for $g$.

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 direct operator fix for the Levi-Civita method under involutive anti-automorphisms by proving a conjugation identity that turns right translations into left ones.

read the letter

The main takeaway is that the authors establish a conjugation identity that converts right translations into left translations when dealing with an involutive anti-automorphism on a semigroup. This allows them to adapt the Levi-Civita approach successfully where it previously collapsed due to order reversal. They prove that J composed with the right translation by σ(y) and then J again equals the left translation by y, for every y. The identity follows directly from the anti-automorphism property that σ(xy) equals σ(y)σ(x) and the fact that σ squared is the identity. A direct calculation with the operators shows it reduces to the left action, and this holds without any additional hypotheses. Applying the resulting anti-automorphic Levi-Civita closure principle to the generalized sine law yields the classical dichotomy for β in plus or minus one and the parity relation f composed with σ equals β f without any assumptions. That unconditional recovery is the paper's clearest win and removes the obstruction mentioned in the abstract. The bridge hypothesis appears only when deriving the standard xy-addition law and the exact σ-transformation rule for g. The authors note it holds automatically in the presence of a suitable central element where f does not vanish, which is a practical observation that covers many standard cases. The work is technically sound on its own terms and avoids circular reasoning or fitted parameters. The focus stays on explicit calculations from the semigroup and anti-automorphism axioms. This paper is for specialists in functional equations on algebraic structures such as semigroups and groups with involutions. Anyone extending matrix methods to anti-automorphisms or involutions will find the conjugation trick relevant and potentially reusable. I would send this to peer review. The core result looks solid based on the operator-level argument, and a referee can confirm the details in the full derivations as well as check for any overlooked edge cases in the bridge hypothesis.

Referee Report

2 major / 3 minor

Summary. The manuscript develops an operator-theoretic extension of Stetkær's Levi-Civita matrix method to semigroups equipped with an involutive anti-automorphism σ. By proving the conjugation identity J R(σ(y)) J = L(y) for all y in S, where J is composition with σ, the authors convert right translations into left translations. This yields an anti-automorphic closure principle that is applied to the generalized sine law, recovering the classical dichotomy β ∈ {±1} together with the parity relation f ∘ σ = β f unconditionally. Under an additional natural bridge hypothesis (automatically satisfied when a central element c with f(c) ≠ 0 exists), the standard xy-addition law and the exact σ-transformation rule for the companion function g are also obtained.

Significance. If the derivations hold, the work is significant because it removes a structural obstruction that previously prevented direct application of the Levi-Civita formalism to anti-automorphic cases. The explicit, axiom-based conjugation identity and the resulting unconditional recovery of the dichotomy constitute a parameter-free advance. The approach is grounded in the semigroup and anti-automorphism axioms alone for the core sine-law result, which is a clear methodological strength.

major comments (2)

[Abstract] Abstract, displayed equation for the conjugation identity: the claim that J R(σ(y)) J = L(y) converts right translates into left translations is load-bearing for the entire subsequent closure argument. The manuscript must supply the explicit operator calculation (J R(σ(y)) J f)(x) = f(y x) = (L(y) f)(x) and confirm that it uses only the anti-automorphism property σ(xy) = σ(y)σ(x) together with σ² = id, without tacit associativity or other semigroup assumptions.
[Application to generalized sine law] Section on application to the generalized sine law (presumably §4 or §5): the unconditional recovery of β ∈ {±1} and f ∘ σ = β f is asserted after invoking the anti-automorphic Levi-Civita closure. The precise statement of the generalized sine law being solved must be displayed, together with the exact invocation of the closure principle, so that the reader can verify that no hidden regularity or centrality assumption enters the dichotomy step.

minor comments (3)

The natural bridge hypothesis is described only qualitatively. Provide its formal statement as a numbered assumption and give the explicit verification that it holds automatically when a central element c with f(c) ≠ 0 exists.
Operator symbols J, R(·), and L(·) should be defined at their first appearance in the main text rather than only in the abstract.
A brief remark on whether the conjugation identity extends verbatim to non-associative magmas would clarify the precise scope of the semigroup setting.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the precise comments that will improve the clarity and verifiability of the manuscript. We address each major comment below and have revised the paper accordingly.

read point-by-point responses

Referee: [Abstract] Abstract, displayed equation for the conjugation identity: the claim that J R(σ(y)) J = L(y) converts right translates into left translations is load-bearing for the entire subsequent closure argument. The manuscript must supply the explicit operator calculation (J R(σ(y)) J f)(x) = f(y x) = (L(y) f)(x) and confirm that it uses only the anti-automorphism property σ(xy) = σ(y)σ(x) together with σ² = id, without tacit associativity or other semigroup assumptions.

Authors: We agree that an explicit verification of the identity strengthens the exposition. In the revised manuscript we insert the following direct calculation immediately after the displayed identity, both in the abstract and in the main text where the identity is first proved. Let Jf := f ∘ σ. Then [J R(σ(y)) J f](x) = [R(σ(y)) (Jf)](σ(x)) = (Jf)(σ(x) ⋅ σ(y)) = f(σ(σ(x) ⋅ σ(y))). By the anti-automorphism property, σ(a b) = σ(b) σ(a) with a = σ(x) and b = σ(y), so σ(σ(x) ⋅ σ(y)) = σ(σ(y)) σ(σ(x)) = y x, using σ² = id. Hence [J R(σ(y)) J f](x) = f(y x) = [L(y) f](x). The argument relies solely on the definitions of the translations, the anti-automorphism axiom, and σ² = id; no associativity beyond the semigroup operation or any other hidden assumptions are used. revision: yes
Referee: [Application to generalized sine law] Section on application to the generalized sine law (presumably §4 or §5): the unconditional recovery of β ∈ {±1} and f ∘ σ = β f is asserted after invoking the anti-automorphic Levi-Civita closure. The precise statement of the generalized sine law being solved must be displayed, together with the exact invocation of the closure principle, so that the reader can verify that no hidden regularity or centrality assumption enters the dichotomy step.

Authors: We accept the referee’s suggestion. In the revised version, at the opening of the section that applies the anti-automorphic Levi-Civita closure, we now display the exact functional equation under consideration (the generalized sine law on the semigroup with involutive anti-automorphism). We then spell out the precise step at which the closure principle is invoked, showing how the resulting finite-dimensional invariant subspace yields the matrix equation whose solutions directly produce the dichotomy β ∈ {±1} together with the parity relation f ∘ σ = β f. The argument uses only the semigroup axioms and the anti-automorphism; no centrality, regularity, or bridge hypothesis is required for this step. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained; no circular reductions identified

full rationale

The core conjugation identity J R(σ(y)) J = L(y) is established directly by explicit operator expansion using only the given semigroup multiplication and the anti-automorphism plus involutivity axioms of σ; the calculation maps f(x)mapsto f(yx) without any fitted parameters, self-citations, or ansatz smuggling. This identity then yields the anti-automorphic Levi-Civita closure, from which the sine-law results (β∈{±1} and f∘σ=βf) follow unconditionally. The separate bridge hypothesis is invoked only for the xy-addition law and g-rule and is not required for the primary claims. All steps remain independent of the target conclusions and rest on external axioms rather than internal redefinitions or prior author results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard semigroup axioms, the definition of an involutive anti-automorphism, and the existence of left and right translation operators; no free parameters or invented entities are introduced.

axioms (1)

domain assumption S is a semigroup equipped with an involutive anti-automorphism σ
Invoked throughout the abstract as the setting for the functional equations and the conjugation identity.

pith-pipeline@v0.9.0 · 5755 in / 1371 out tokens · 32221 ms · 2026-05-18T05:08:56.593549+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

J R(σ(y)) J = L(y) (∀ y ∈ S) ... anti-automorphic Levi–Civita closure principle
IndisputableMonolith/Foundation/GeneralizedDAlembert.lean dAlembert_cosh_solution_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

f(xσ(y)) = f(x)g(y) + β g(x)f(y) + γ f(x)f(y) ... β ∈ {±1}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

[1]

Acz´ el and J

J. Acz´ el and J. Dhombres,Functional equations in several variables: with applications to mathematics, information theory and to the natural and social sciences., Encyclopedia of Mathematics and its Applications, Vol. 31, Cambridge University Press, 1989

work page 1989
[2]

Aserrar and E

Y. Aserrar and E. Elqorachi, Cosine and sine addition and subtraction law with an automorphism,Ann. Math. Sil.38(2) (2024), 155–176. DOI:https://doi.org/10.2478/amsil-2023-0021

work page doi:10.2478/amsil-2023-0021 2024
[3]

Ebanks, Around the sine addition law and d’Alembert’s equation on semigroups,Results Math.77(1) (2022), Art

B. Ebanks, Around the sine addition law and d’Alembert’s equation on semigroups,Results Math.77(1) (2022), Art. 11. DOI:https://doi.org/10.1007/s00025-021-01548-6

work page doi:10.1007/s00025-021-01548-6 2022
[4]

Ebanks, Sine subtraction laws on semigroups,Ann

B. Ebanks, Sine subtraction laws on semigroups,Ann. Math. Sil.37(1) (2023), 49–66. DOI:https://doi. org/10.2478/amsil-2023-0002

work page doi:10.2478/amsil-2023-0002 2023
[5]

Stetkær,Functional Equations on Groups, World Scientific, 2013

H. Stetkær,Functional Equations on Groups, World Scientific, 2013. DOI:https://doi.org/10.1142/8830

work page doi:10.1142/8830 2013
[6]

Stetkær, On a sine addition/subtraction law on semigroups,Aequat

H. Stetkær, On a sine addition/subtraction law on semigroups,Aequat. Math.(2025), DOI:https://doi. org/10.1007/s00010-025-01179-0

work page doi:10.1007/s00010-025-01179-0 2025
[7]

Stetkær, On centrality of solutions of generalized addition and subtraction laws,Aequat

H. Stetkær, On centrality of solutions of generalized addition and subtraction laws,Aequat. Math.(2025), DOI:https://doi.org/10.1007/s00010-025-01226-w. Department of Mathematics, FPT University, Quy Nhon AI Campus, An Phu Thinh New Urban Area, Vietnam Email address:dangphuc150488@gmail.com, dangvophuc@qnu.edu.vn

work page doi:10.1007/s00010-025-01226-w 2025

[1] [1]

Acz´ el and J

J. Acz´ el and J. Dhombres,Functional equations in several variables: with applications to mathematics, information theory and to the natural and social sciences., Encyclopedia of Mathematics and its Applications, Vol. 31, Cambridge University Press, 1989

work page 1989

[2] [2]

Aserrar and E

Y. Aserrar and E. Elqorachi, Cosine and sine addition and subtraction law with an automorphism,Ann. Math. Sil.38(2) (2024), 155–176. DOI:https://doi.org/10.2478/amsil-2023-0021

work page doi:10.2478/amsil-2023-0021 2024

[3] [3]

Ebanks, Around the sine addition law and d’Alembert’s equation on semigroups,Results Math.77(1) (2022), Art

B. Ebanks, Around the sine addition law and d’Alembert’s equation on semigroups,Results Math.77(1) (2022), Art. 11. DOI:https://doi.org/10.1007/s00025-021-01548-6

work page doi:10.1007/s00025-021-01548-6 2022

[4] [4]

Ebanks, Sine subtraction laws on semigroups,Ann

B. Ebanks, Sine subtraction laws on semigroups,Ann. Math. Sil.37(1) (2023), 49–66. DOI:https://doi. org/10.2478/amsil-2023-0002

work page doi:10.2478/amsil-2023-0002 2023

[5] [5]

Stetkær,Functional Equations on Groups, World Scientific, 2013

H. Stetkær,Functional Equations on Groups, World Scientific, 2013. DOI:https://doi.org/10.1142/8830

work page doi:10.1142/8830 2013

[6] [6]

Stetkær, On a sine addition/subtraction law on semigroups,Aequat

H. Stetkær, On a sine addition/subtraction law on semigroups,Aequat. Math.(2025), DOI:https://doi. org/10.1007/s00010-025-01179-0

work page doi:10.1007/s00010-025-01179-0 2025

[7] [7]

Stetkær, On centrality of solutions of generalized addition and subtraction laws,Aequat

H. Stetkær, On centrality of solutions of generalized addition and subtraction laws,Aequat. Math.(2025), DOI:https://doi.org/10.1007/s00010-025-01226-w. Department of Mathematics, FPT University, Quy Nhon AI Campus, An Phu Thinh New Urban Area, Vietnam Email address:dangphuc150488@gmail.com, dangvophuc@qnu.edu.vn

work page doi:10.1007/s00010-025-01226-w 2025