Integral Kannappan-Sine subtraction and addition law on semigroups
Ajebbar Omar, Elqorachi Elhoucien, Jafar Ahmed
TL;DR
This work classifies complex-valued solutions to the integral Kannappan-Sine subtraction and addition laws on semigroups $S$ equipped with a discrete measure $\mu$ and an involutive automorphism $\sigma$. The authors reduce the problem to combinations of semigroup exponentials and special sine-type equations, expressing the solutions $(f,g)$ in terms of exponentials $\chi$ (and $\chi\circ\sigma$), as well as auxiliary functions $\phi_{\chi}$ and $\Phi_{\chi}$ that solve the corresponding additive laws. They provide a complete list of solution families for both laws (subtraction: eight families; addition: several explicit families), with explicit forms and conditions on $\int_S \chi(t) \, d\mu(t)$ and similar integrals. The paper also establishes continuity results for topological semigroups, showing that the associated exponential and auxiliary functions inherit continuity when $f,g$ are continuous. Overall, the results extend known trigonometric functional equation frameworks from groups to semigroups with involution, highlighting the abelian structure of solutions via exponentials.
Abstract
Let $S$ be a semigroup, $μ$ a discrete measure on $S$ and $σ:S \longrightarrow S$ is an involutive automorphism. We determine the complex-valued solutions of the integral Kannappan-Sine subtraction law $$\int_{S}f(xσ(y)t)dμ(t)=f(x)g(y)-f(y)g(x),\; x,y \in S,$$ and the integral Kannappan-Sine addition law $$\int_{S}f(xσ(y)t)dμ(t)=f(x)g(y)+f(y)g(x),\; x,y \in S.$$ We express the solutions by means of exponentials on S, the solutions of the special sine addition law $f(xy)=f(x)χ(y)+f(y)χ(x),$ $x,y\in S$ and the solutions of of the special case of the integral Kannappan-Sine addition law $\int_{S}f(xσ(y)t)dμ(t)=[f(x)χ(y)+f(y)χ(x)]\int_{S}χ(t)dμ(t), $ $x,y\in S$, and where $χ$: $S\longrightarrow \mathbb{C}$ is an exponential. The continuous solutions on topological semigroups are also given.