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

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 $g$.