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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2511.00019 |
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| _version_ | 1866915945253961728 |
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| author | Phuc, Dang Vo |
| author_facet | Phuc, Dang Vo |
| contents | Stetkær's matrix (Levi--Civita) method is a powerful tool for functional equations on semigroups involving a homomorphism $σ$, as it yields a finite-dimensional invariant space under right translations and a corresponding matrix formalism. However, this framework collapses when $σ$ 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 $σ$, we prove \[ J\,R(σ(y))\,J=L(y)\qquad(\forall\,y\in S), \] 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 $β\in\{\pm1\}$ and the parity relation $f\circσ=βf$ 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 $σ$-transformation rule for $g$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_00019 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sine laws on semigroups with an involutive anti-automorphism: A Levi--Civita approach via left translations Phuc, Dang Vo General Mathematics 39B52, 20M15 Stetkær's matrix (Levi--Civita) method is a powerful tool for functional equations on semigroups involving a homomorphism $σ$, as it yields a finite-dimensional invariant space under right translations and a corresponding matrix formalism. However, this framework collapses when $σ$ 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 $σ$, we prove \[ J\,R(σ(y))\,J=L(y)\qquad(\forall\,y\in S), \] 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 $β\in\{\pm1\}$ and the parity relation $f\circσ=βf$ 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 $σ$-transformation rule for $g$. |
| title | Sine laws on semigroups with an involutive anti-automorphism: A Levi--Civita approach via left translations |
| topic | General Mathematics 39B52, 20M15 |
| url | https://arxiv.org/abs/2511.00019 |