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arXiv · 2504.21057

Integral Kannappan-Sine subtraction and addition law on semigroups

Ajebbar Omar·Elqorachi Elhoucien·Jafar Ahmed
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Abstract

Let $S$ be a semigroup, $\mu$ a discrete measure on $S$ and $\sigma:S \longrightarrow S$ is an involutive automorphism. We determine the complex-valued solutions of the integral Kannappan-Sine subtraction law $$\int_{S}f(x\sigma(y)t)d\mu(t)=f(x)g(y)-f(y)g(x),\; x,y \in S,$$ and the integral Kannappan-Sine addition law $$\int_{S}f(x\sigma(y)t)d\mu(t)=f(x)g(y)+f(y)g(x),\; x,y \in S.$$ We express the solutions by means of exponentials on S, the solutions of the special sine addition law $f(xy)=f(x)\chi(y)+f(y)\chi(x),$ $x,y\in S$ and the solutions of of the special case of the integral Kannappan-Sine addition law $\int_{S}f(x\sigma(y)t)d\mu(t)=[f(x)\chi(y)+f(y)\chi(x)]\int_{S}\chi(t)d\mu(t), $ $x,y\in S$, and where $\chi$: $S\longrightarrow \mathbb{C}$ is an exponential. The continuous solutions on topological semigroups are also given.

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Ajebbar Omar, Elqorachi Elhoucien, Jafar Ahmed. 2025-04-29. Integral Kannappan-Sine subtraction and addition law on semigroups. https://arxiv.org/abs/2504.21057

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