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A Kannappan-sine subtraction law on semigroups

Ahmed Jafar, Omar Ajebbar, Elhoucien Elqorachi

Abstract

Let $S$ be a semigroup, $z_0$ a fixed element in $S$ and $σ:S \longrightarrow S$ an involutive automorphism. We determine the complex-valued solutions of Kannappan-sine subtraction law $f(xσ(y)z_0)=f(x)g(y)-f(y)g(x),\; x,y \in S$. As an application we solve the following variant of Kannappan-sine subtraction law viz. $f(xσ(y)z_0)=f(x)g(y)-f(y)g(x)+λg(xσ(y)z_0) ,\; x,y \in S,$ where $λ\in \mathbb{C}^{*}$. The continuous solutions on topological semigroups are given and an example to illustrate the main results is also given.

A Kannappan-sine subtraction law on semigroups

Abstract

Let be a semigroup, a fixed element in and an involutive automorphism. We determine the complex-valued solutions of Kannappan-sine subtraction law . As an application we solve the following variant of Kannappan-sine subtraction law viz. where . The continuous solutions on topological semigroups are given and an example to illustrate the main results is also given.
Paper Structure (7 sections, 9 theorems, 107 equations)

This paper contains 7 sections, 9 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Set up, notation and terminology
  3. Auxiliary results
  4. Preparatory works for Kannappan-sine subtraction law
  5. Main results
  6. Solution of the functional equation (\ref{['2']})
  7. An example

Key Result

Lemma 2.1

$p\in P_{\chi} \implies up,pv, upv \in P_{\chi}$ for all $u,v \in S\setminus I_{\chi}\ $.

Theorems & Definitions (19)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 3.1
  • proof
  • Lemma 4.1
  • proof
  • Remark 4.2
  • ...and 9 more