D'Alembert's functional equation on monoids with both an endomorphism and an anti-endomorphism

Let S be a semigroup, ℍ be the skew field of quaternions, and ψ: S → S be an anti-endomorphism. We determine the general solution of the functional equation g ( x y ) - g ( x ψ ( y ) ) = 2 g ( x ) g ( y ) , x , y ∈ S , g\left( {xy} \right) - g\left( {x\psi \left( y \right)} \right) = 2g\left( x \right)g\left( y \right),\,\,\,\,x,y \in S, where g : S → ℂ is the unknown function. And when S = M is a monoid, we solve the functional equation g ( x y ) + g ( x ψ ( y ) ) = 2 g ( x ) g ( y ) , x , y ∈ M , g\left( {xy} \right) + g\left( {x\psi \left( y \right)} \right) = 2g\left( x \right)g\left( y \right),\,\,\,\,x,y \in M, where g : M → ℍ is the unknown function.

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D'Alembert's functional equation on monoids with both an endomorphism and an anti-endomorphism

Cited by 3 publications

References 0 publications

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