Solutions of the system of functional equations f(x o y) = f(x)g(y) + g(x)f(y) and g(x o y) = g(x)g(y) - f(x)f(y) on semigroups

Abstract

Let (S, ๐) be any semigroup, (F, +, .) be a field of characteristic different from 2. We determine all pairs f,g:S -> such that f(x๐y) = f(x)g(y)+g(x)f(y), g(x๐y) = g(x)g(y)-f(x)f(y)}(*) for all x,y ใน S. It is proved that if F contains an element i such that i2 = -1 then f,g are of the form f(x) = i/2(phi1(x)-phi2(x)), g(x) = 1/2(phi1(x)+phi2(x)),}(**) where phi1, phi2 are any homomorphisms from (S, ๐) into (F,.). In the case that F does not contain such element i, we can extend it to F that contains such element. In this case f,g are of the form f(x) = i/2(phi(x)-phi(x)), g(x) = 1/2(phi(x)+phi(x)),}(***) where phi is a homomorphism (S,๐) into (F,.) and phi is defined by phi(x) = phi(x), the conjugate of phi(x). In case (S, ๐) is a topological semigroup, (F,+,.) is a topological field of characteristic different from 2, we determine all pairs of continuous functions f,g:S -> F such that (*) holds for all x,y in S. It is proved that if F contains an element i such that i2 = -1 then f,g are of the form (**) where phi1, phi2 are any continous homomorphisms from (S, ๐) into (F,.). In the case that F does not contain such element i, we can extend it to F that contains such element. In this case f,g are of the form (***) where phi is a continuous homomorphism from (S, ๐) into (F,.) and phi is defined by phi(x) = phi(x), the conjugate of phi(x).